{ "updatedAt": "2026-10-11", "coverageNote": "Question-index snapshot: 13,197 indexed question records across 990 of the 1,143 catalogue papers. The paper catalogue covers CIE 9709 and Pearson Edexcel IAL; the question index is a growing subset, with complete sets and report-backed challenge picks identified below. The paper catalogue currently contains 840 CIE 9709 and 303 Pearson Edexcel IAL papers (1,143 total). Full question sets are indexed for four WMA12/01A China regional Paper A sessions (May/June 2025, October 2025, January 2026 and May/June 2026); four other June 2025 Paper A units (WMA11/01A, WMA13/01A, WMA14/01A and WME02/01A); all eight catalogue-listed October 2025 IAL Paper A units (WMA11/01A, WMA12/01A, WMA13/01A, WMA14/01A, WME01/01A, WME02/01A, WST01/01A and WST02/01A); and all 11 catalogue-listed January 2026 IAL Paper A units (WDM11, WMA11–WMA14, WME01–WME03 and WST01–WST03, all /01A). All numbered questions are also indexed for the 24 CIE 9709 October/November 2025 components: 199 component question records covering 1,400 component marks, or 191 unique question texts and 1,350 unique-content marks because /23 and /25 have separate printed codes but identical substantive content. The index also includes 37 report-backed challenge questions across all 13 catalogue-listed Pearson IAL May/June 2024 papers, including regional P1R/P2R; 41 report-backed challenge parts across all 12 catalogue-listed CIE May/June 2024 components; and selected challenge items from 13 other June 2025 IAL papers and 20 CIE 9709 May/June 2025 components. No verified Cambridge Principal Examiner Report was found for CIE October/November 2025, so its difficulty fields are unranked. October 2025, January 2026 and May/June 2026 IAL Paper A sets have no report-backed difficulty ranking. The May/June 2025 WMA12/01A China regional Paper A has seven question priorities backed by Pearson’s matching Summer 2025 PEF. June 2024 IAL and CIE priorities and selected June 2025 challenge priorities are report-backed. Unindexed questions are excluded from topic totals. Priority numbers rank challenge items within their own paper and are not a normalized cross-board difficulty score. Full question sets are also indexed for all six catalogue-listed Pearson Edexcel May/June 2026 Paper A units WMA11/01A, WMA12/01A, WMA13/01A, WMA14/01A, WME01/01A and WST01/01A (137 question records across the six units); no matching Paper A examiner report was verified for this session, so these sets remain unranked. All numbered question parts are indexed for the eight catalogue-listed Cambridge 9709 May/June 2026 Pure Mathematics 1/2 components 9709/11, /12, /13, /15, /21, /22, /23 and /25 (137 part records). Seven question parts are flagged from directly matched sections of the Cambridge-authored June 2026 Principal Examiner Report; the report copy was accessible as a mirror. The public /12 listing cannot be identified as the cancelled sitting or June 9 replacement, and the separate replacement form is not represented; /22, /23 and /25 retain separate catalogue entries despite matching substantive source content. All numbered question parts are also indexed for CIE Cambridge 9709 May/June 2026 Probability & Statistics 1 components /51, /52, /53 and /55 (70 records, 50 marks each); no matching June 2026 Principal Examiner Report was verified, so these remain unranked. Cambridge says zone 3/4 candidates for /52 receive assessed marks based on completed components after premature sharing; the notice does not identify a replacement /52 paper. Catalogue mirror PDFs carry PapaCambridge watermarks. All printed question parts are indexed for the four CIE 9709 May/June 2026 Statistics 2 components /61, /62, /63 and /65 (62 records; 50 marks each). These remain unranked because the visible Cambridge-authored report mirror contains no matching S2 sections. The /61 Q3(b) source is internally inconsistent: its “largest n” inequality has no finite maximum, while the mark scheme gives 607 as the smallest qualifying value; the record preserves this issue in the guide. All printed question parts are indexed for CIE Cambridge 9709 May/June 2026 Pure Mathematics 3 components /31, /32, /33 and /35 (82 records; 75 marks each). These remain unranked because no matching May/June 2026 Principal Examiner Report was directly verified. Cambridge confirms /32 was replaced for zones 3/4, but the catalogue listing does not identify whether its PDF is the original or replacement form; see the [Cambridge /32 notice](https://www.cambridgeinternational.org/exam-administration/cambridge-exams-officers-guide/update-a-level-mathematics-paper-32-june-exam-series/). All printed question parts are indexed for the four CIE 9709 May/June 2026 Mechanics components /41, /42, /43 and /45 (51 records; 50 marks each). They remain unranked because no matching May/June 2026 Principal Examiner Report was verified. The paired /45 Q4(b) question paper and mark scheme materially disagree; its answer guide follows the QP inputs and explicitly warns that the paired MS solves a different set of values. The paper catalogue link audit repaired 29 of 30 rows that had been missing a question-paper or mark-scheme link. The repairs use title/code/session-checked Pearson PDFs, a verified January 2026 WME02/01A Paper A mark scheme mirror, and Cambridge-authored combined mark schemes for selected 2003–2004 sessions. One public file remains unverified: the October/November 2001 9709/1 question paper; its link stays blank rather than pointing to a mismatched file. Full question-part sets are also indexed for all 11 Pearson Edexcel IAL January 2025 regular units (276 parts, 75 marks per paper); 47 questions carry priority flags supported by their matching Pearson examiner reports. Report-backed challenge questions are also indexed for all 18 catalogue-listed CIE 9709 May/June 2023 components (94 parts across 18 papers; 354 of 1,050 component marks), selected directly from Cambridge-authored examiner-report findings. All numbered question parts are also indexed for all 18 catalogue-listed CIE 9709 October/November 2024 components (319 parts across 18 papers; 1,050 of 1,050 marks). Question-paper and mark-scheme page spans were checked against the PDFs. No matching Cambridge-authored October/November 2024 Examiner Report was directly verified, so these questions remain unranked. Directly examiner-report-backed challenge questions are indexed for all 18 catalogue-listed CIE 9709 May/June 2022 components (68 parts across 18 components; 244 of 1,050 component marks). The /63 report section did not directly single out a challenge part, so that component has no priority-tagged record. Full question-part sets are also indexed for all eight Pearson Edexcel IAL October 2024 Mathematics units (205 parts across eight 75-mark papers); 41 parts have Pearson examiner-report-backed priority. The timetable confirms WMA11–WMA14, WME01–WME02 and WST01–WST02 as the eight units; WDM11, WME03 and WST03 had no sitting. WME01 uses a checked catalogue mirror for its question paper because Pearson’s QP endpoint was unavailable. Priority is withheld for WMA11 Q6 because the report part label does not match the QP notation. WME02 and WST02 QP filenames show dates one day after the timetable dates, while printed references match. Cambridge-authored October/November 2023 challenge parts are also indexed across all 18 catalogue-listed CIE 9709 components (70 parts across 17 papers; 260 of 1,050 component marks). The /61 cohort was too small for a meaningful report, so it has no priority-tagged questions. Report-backed challenge parts are indexed for all six catalogue-listed Pearson IAL May/June 2022 Mathematics units (28 parts; 101 selected marks). Their A Level Copilot QP/MS PDFs were SHA-256 matched to the Pearson originals; each ranked part links to its matching Pearson report, and the catalogue QP filenames are one day after the printed sitting dates. Report-backed challenge questions are indexed for all 11 timetable-listed regular Pearson IAL May/June 2023 units (48 parts across 11 papers; 172 selected marks). Five missing catalogue rows were added after Question Paper, mark scheme, and Principal Examiner Feedback checks; Further Mathematics WFM units are excluded. The October 2022 IAL index covers all eight timetable-listed Mathematics units (28 report-backed parts across eight papers; 130 selected marks); two missing catalogue rows, WME02 and WST02, were added after QP, mark scheme and examiner-report checks. The October 2023 IAL index covers all eight timetable-listed units (30 parts; 8 papers), with WME02 and WST02 added to the catalogue. The WME02 examiner feedback has a publication-code typo; its report title and paper log number match the paper, and the caveat is retained. Report-backed challenge parts are indexed for all 11 timetable-listed regular Pearson IAL Mathematics units from January 2023 (37 parts across 11 papers); five missing Mathematics catalogue rows were added after QP/MS/report checks. WFM01-WFM03 remain outside the Mathematics catalogue as Further Mathematics units. WDM11 QP filename carries a January 26 suffix although the timetable date is January 13; its cover, mark scheme and examiner report match WDM11/01. Cambridge-authored October/November 2022 Principal Examiner Report challenge parts are indexed across all 18 CIE 9709 components (68 parts across 16 papers). Cambridge-authored May/June 2021 report-backed challenge parts are indexed across all 18 CIE 9709 components (53 parts; 190 selected marks). Components 9709/61 and /63 remain unranked because their matching report sections did not directly flag a specific challenging part. The accessible report mirror is Cambridge-authored; paper report links have been added. Cambridge-authored October/November 2021 challenge parts are indexed across all 18 CIE 9709 components (69 report-backed parts across 16 ranked papers; 235 selected marks). Components 9709/23 and /41 remain unranked because the /23 report narrative does not match its paper and the /41 section did not directly single out a challenge part; the matching report links are on all 18 catalogue rows. Cambridge-authored February/March 2021 challenge parts are indexed across all six CIE 9709 components (20 directly supported parts across five ranked papers; 68 selected marks). Component 9709/22 remains unranked because the report says its cohort was too small for a meaningful evaluation. Cambridge-authored February/March 2022 challenge parts are indexed across all six CIE 9709 components (27 directly supported parts across five ranked papers; 105 selected marks). Component 9709/22 remains unranked because the report found too few candidates for a meaningful evaluation. The January 2022 Pearson IAL challenge index covers all 11 final-timetable standard Mathematics units (36 report-backed parts across 11 papers; 193 selected marks); six missing catalogue rows were added. WMA11/01 uses issued QP P70482A because the existing catalogue URL was P69458A. The report overview's WMA11 4(d) label is not present on that paper; the indexed item is the matching question-specific 8(d) discussion. WME03 and WST03 use the issued forms matched to their reports; marked UNUSED forms are excluded. WME03's report-cover title anomaly is retained as a source caveat. Cambridge-authored October/November 2020 challenge parts are indexed across all 18 CIE 9709 components (68 report-backed parts across 16 ranked papers; 244 selected marks). Components /61 and /63 remain unranked because the report gives too few candidates for a meaningful report on /61 and does not directly identify a challenge part on /63. October 2021 Pearson IAL challenge records cover all 11 final-timetable standard Mathematics units (47 report-backed parts across 11 units; 188 selected marks), with five missing catalogue rows added. The Pearson-branded timetable copy was hosted by British Council Qatar; WFM Further Mathematics and regional/Paper A variants were not listed. Pearson QP asset suffix dates are one day after sitting dates. WMA13/WMA14 archive mark-scheme mirrors match the code and page count but not Pearson's file hashes; indexed questions link to Pearson's official mark schemes. Priorities are qualitative and paper-local. January 2021 Pearson IAL challenge records add 35 examiner-report-backed parts across all 11 standard Mathematics units (158 selected marks), and five missing catalogue rows were added. The records use verified Pearson QP/PEF sources and Pearson or paper-specific mirror mark schemes where Pearson endpoints were unavailable. June 2021 was cancelled and remains an explicit gap with no issued public QP/PEF set indexed; Further Mathematics units are outside this standard Mathematics batch. Cambridge-authored February/March 2023 challenge parts are indexed across all six CIE 9709 components (31 report-backed parts across five ranked papers; 125 selected marks). Component /22 remains unranked because the report explicitly states that there were too few candidates for a meaningful report. Cambridge-authored May/June 2019 challenge parts are indexed across all 21 catalogue-listed CIE 9709 components (72 parts; 289 selected marks), including the 9709/71–/73 Probability & Statistics 2 papers. January 2024 Pearson IAL report-backed challenge parts cover all 11 timetable-listed regular Mathematics units (25 question parts; 125 selected marks). Five missing catalogue rows were added. Six existing A Level Copilot mirror QP/MS links were replaced with the matching direct Pearson QP/MS files, and direct Pearson examiner-report links were added to all 11 units. WDM11 includes a separate Answer Book after the questions; QP filename suffix dates may be one day after the timetable dates. Further Mathematics units are outside this standard Mathematics batch. Cambridge-authored February/March 2020 challenge parts are indexed across all six catalogue-listed CIE 9709 components (37 report-backed parts; 121 selected marks). Cambridge-authored October/November 2019 challenge parts are indexed across all 20 catalogue-listed CIE 9709 components (81 parts; 302 selected marks). The matching report PDF is Cambridge-authored and served from the XtremePapers mirror linked in the catalogue. Cambridge-authored February/March 2019 challenge parts are indexed across all seven catalogue-listed CIE 9709 components (26 parts; 104 selected marks). Catalogue PDF downloads returned 403 during audit; matching named PDFs were checked from the PapaCambridge archive, and the original catalogue URLs remain in the index. January 2020 Pearson IAL challenge index covers all four catalogue-listed standard Mathematics units (13 report-backed parts; 68 selected marks). Official Pearson examiner reports and mark schemes are linked throughout. WMA11 question marks total 75 despite a conflicting generic 125-mark line in its RMS. WMA13 retains the verified catalogue QP mirror because Pearson’s QP asset path did not return a PDF; its printed code, date, log P60568A, question content and marks match the official RMS/PEF and an independent copy. Three other units use verified direct Pearson QPs. Further Mathematics is excluded. CIE May/June 2018 selected report-backed challenge questions cover all 21 catalogue components. The General Comments overview labels the reviewed Mechanics 3 part Q4(ii), while the detailed report p.68 and QP label it Q4(iii); it is retained only with direct detailed commentary evidence. CIE February/March 2018 selected report-backed questions cover all seven catalogue papers. Cambridge's official paper titles identify /42 and /52 as Mechanics 1 and Mechanics 2, and /62 and /72 as Probability & Statistics 1 and 2; the affected catalogue labels were corrected to match the printed papers and report. Pearson IAL January 2019 selected report-backed challenge parts cover all five catalogue-listed standard Mathematics units (33 parts; 120 selected marks). The two legacy 125-mark C12/C34 papers state 125 on their covers and question totals; generic RMS front matter says 75, but question-specific allocations reconcile to 125. QP filename date suffixes are one day after the printed/timetable sitting dates; printed paper code, date and log ID were matched. Other January 2019 timetable units absent from this catalogue cohort are out of scope; Further Mathematics is excluded. CIE October/November 2018 selected examiner-report-backed challenge questions cover all 21 catalogue components (87 parts; 348 selected marks). The Cambridge-authored report is served from the catalogue's XtremePapers mirror. Its section headed 9709/22 Q6–Q7 (pp.15–17) instead aligns with the /21 and /23 Q6–Q7; only /22 Q2, Q4 and Q5(iii), whose commentary directly matches the /22 QP, are ranked. Applied-unit labels for /41–/73 were corrected against the printed paper titles. March 2017 CIE 9709 is indexed at report-backed question-part level across all seven catalogue papers; selected parts are partial coverage. The /62 report calls one printed Q7(b) item Q7(iii), and the matching task is documented in sourceChecks. May/June 2017 CIE 9709 is indexed at report-backed question-part level across all 21 catalogue components; selected parts are partial coverage. The /53 report alignment and parallel variants are documented in sourceChecks. October 2020 IAL Math now has report-backed question-part ranks across all six standard catalogue units. Pearson’s reuse of the cancelled summer 2020 papers and WMA11’s generic mark-scheme 125-mark line versus the verified 75-mark question allocation are documented in the handoff. October 2025 IAL Math covers eight standard papers with verified question allocations and PDF spans, but no question parts are ranked because no matching Pearson PEF is publicly accessible; all questions remain unranked in that sitting. CIE 9709 October/November 2016 examiner-report-backed selected question parts; 9709/52 is excluded from rankings because its examiner-report text mismatches the matching QP. CIE 9709 May/June 2016 examiner-report-backed selected question parts across all 21 components. CIE 9709 October/November 2017 report-backed challenge parts across 20 indexed components; 9709/22 is excluded from ranking because its Cambridge report section describes different questions. Edexcel IAL October 2019 WMA11/01 and WMA12/01 report-backed challenge parts (12 parts); WMA12 mark-scheme mirror and PEF cover-code caveats are documented in the source handoff. CIE O/N 2015: 70 report-ranked question parts across 21 catalogue papers (314 selected marks). /23, /61, /63 have no direct challenge comments; /71 and /72 remain unranked because report part boundaries do not match the QPs. /21 Q7(iii) uses the report difficulty statement and matching answer while the guide follows the printed gradient request. CIE M/J 2015: 54 examiner-report-ranked question parts across all 21 components (236 selected marks). Page spans, printed QP/MS totals, direct report links, and selected PDF pages were visually checked. Pearson Edexcel IAL June 2019: 17 examiner-report-ranked challenge parts across WMA11/01 and WMA12/01 (54 selected marks). Both 75-mark QPs hash-match official Pearson PDFs, the PEFs match the papers, and catalogue MS mirrors were checked page by page; the WMA12 PEF cover-code typo and lack of direct Pearson MS URLs are documented. Historical CIE 9709 component labels through 2019 now follow Cambridge’s former Paper 5 Mechanics 2, Paper 6 Probability & Statistics 1 and Paper 7 Probability & Statistics 2 mapping. October 2020 WMA12 now includes six examiner-report-ranked parts; the priority order follows the matching Pearson PEF. January 2018 legacy WMA01/01 C12 has six examiner-report-identified challenge parts; their priority order follows Pearson list order. Cambridge 9709 October/November 2014 challenge parts are indexed across all 21 paper components. Cambridge 9709 May/June 2014 indexed 48 report-backed parts across 21 papers; 9709/71 is left unranked because its examiner-report section duplicates 9709/72 and mismatches the /71 question paper. January 2018 legacy WMA02/01 C34 has 10 report-backed challenge parts, with priorities presented as a qualitative review sequence. Cambridge 9709 February/March 2016 indexed 41 report-backed question parts across seven papers; official archive mirror use is documented where Cambridge PDF links were not directly available. Cambridge 9709 May/June 2013 indexed 78 report-backed parts across all 21 components; QP labels take precedence where the /32 report uses equivalent older part labels. Pearson January 2017 legacy WMA01 and WMA02 add 13 PEF-backed parts; printed titles identify C12/C34 and WMA02 is distinct from newer WMA12 P2. 9709/52 November 2016 remains unranked after the examiner-report section was found to contradict its matching question paper. June 2017 legacy WMA01/01 Core Mathematics C12 and WMA02/01 Core Mathematics C34; ranked within each paper from Pearson examiner reports. WMA02/C34 is distinct from modern WMA12/P2. Cambridge 9709 May/June 2012: 76 report-backed parts across 21 components; priorities stay within each paper. /63 was left unranked for insufficient component-specific report support. The Cambridge-authored report was retrieved from a mirror; catalogue links remain as listed. Cambridge 9709 October/November 2012: 84 report-backed parts across all 21 components, ranked within each paper. QP/MS totals reconciled. Source PDFs and the Cambridge-branded report came from PapaCambridge static archive; catalogue URLs retain XtremePapers links. Edexcel IAL January 2015 WMA01/01, WMA02/01, WME01/01 and WST01/01: report-backed parts ranked within each paper. Legacy WMA01=C12 (P1 display) and WMA02=C34 (P2 display); these are distinct from modern units. The original Pearson filename suffix 20150231 is retained despite January cover dates. Pearson Edexcel IAL January 2016 report-backed challenge parts cover WMA01/01, WMA02/01, WME01/01 and WST01/01 legacy units (23 parts; 93 selected marks). Historic C12/C34 naming and WME01 report-cover label are documented in the audited handoff. CIE 9709 October/November 2011 indexed 82 report-backed question parts across all 21 components (335 selected marks). QP/MS totals reconcile. The Cambridge-branded report and QP/MS PDFs were retrieved from a PapaCambridge static archive mirror; the /73 printed title omits the numeral 2 while code and question content match. CIE 9709 May/June 2011 indexed 65 report-backed question parts across all 21 components (255 selected marks). QP/MS totals reconcile. /12 Q10 examiner-report subpart labels differ from nested QP labels and were mapped by prompt content; /22 and /23 retrieved QPs are text-identical but remain separately code-matched components and were retained separately. Pearson Edexcel IAL January 2017 legacy applied units WME01/01 and WST01/01: 11 report-backed challenge parts (39 selected marks). QP/MS totals and page spans were reconciled; selected QP and mark-scheme pages visually reviewed. CIE 9709 October/November 2010: 95 report-backed question parts across all 21 components (335 selected marks); /23 remains unranked because no direct challenge evidence was identified. QP/MS totals reconcile; historical PDFs and report were obtained from a PapaCambridge static mirror while the catalogue links remain XtremePapers. Pearson Edexcel IAL January 2018 legacy applied units WME01/01 and WST01/01: 12 report-backed challenge parts (45 selected marks). QP/MS allocations and selected pages were checked. The WME Pearson report cover says Chemistry in error, but its inner pages identify the matching January 2018 IAL Mathematics Mechanics unit; the retrieved WME question-paper URL returned HTML, so no binary match is claimed. CIE 9709 October/November 2009: 57 report-backed challenge records across all 14 catalogue components (197 selected marks). QP totals/MS maxima reconcile; selected pages were visually reviewed. Historical UCLES PDFs and report were retrieved from PapaCambridge while the catalogue retains XtremePapers URLs. CIE 9709 May/June 2010: 63 report-backed whole-question challenge records across all 21 components (486 selected marks). QP/MS totals match and selected pages were visually reviewed. Identical question text across /22–/23, /41–/42, /51–/52 and /71–/72 is disclosed; records remain separated by printed component code. CIE 9709 October/November 2008: 42 report-backed challenge parts across all seven catalogue components (148 selected marks). QP/MS totals reconcile, and all selected pages were visually reviewed. The Cambridge report PDF has no title page; its date, UCLES footer and seven component sections were checked. Catalogue labels /1–/7 map to PDF labels /01–/07; historical source PDFs/report came from PapaCambridge while catalogue links remain XtremePapers.\n\nComplete numbered-question-part coverage for CIE 9709 October/November 2020 S2 components /61 and /63. Each component has 18 question-part records covering all 50 paper marks. /63 is ranked from direct question-specific examiner-report comments for every part; /61 is fully indexed but all priorities/evidence are null because its report says too few candidates for meaningful commentary.\n\nComplete coverage of all eight standard /01 mathematics units in the October 2025 catalogue. One record is included per printed scored subpart where QP/MS allocate separate marks; each paper’s records reconcile to all printed question totals and 75 marks.\n\nThese are archived practice question parts from Cambridge 9709 May/June 2020 QP/MS files. Cambridge cancelled the series, no students sat, and Cambridge states that there is no Principal Examiner Report for Teachers for June 2020. The parts are indexed for practice only.\n\nCIE 9709 O/N 2021 /23,/41; 2022 /61,/63; 2023 /61. Every printed scored part: 76 records, 50 marks per paper.\n\nFull printed scored-part index for the six Pearson Edexcel IAL May/June 2021 standard Mathematics units (WMA11–WMA14, WME01 and WST01) plus May/June 2025 WME02/01. Every paper has a complete 75-mark set; these records are unranked because no directly matched public Pearson PEF was verified.\n\nComplete scored-part index for the eleven January 2026 Pearson Edexcel IAL standard /01 Mathematics papers (WDM11, WMA11–WMA14, WME01–WME03, WST01–WST03). Each paper reconciles to 75 marks. Paper A and May/June papers are outside this handoff.\n\nComplete printed scored-subpart coverage for Cambridge 9709 February/March 2021, 2022 and 2023 Pure Mathematics 2 variant 2 (9709/22). The 45 records reconcile to all 150 marks across three 50-mark papers. All difficulty fields are null: each corresponding Principal Examiner Report says there were too few candidates for a meaningful report.\n\nFull printed scored-part index for the six Pearson Edexcel IAL May/June 2026 standard Mathematics /01 papers WMA11, WMA12, WMA13, WMA14, WME01 and WST01. Each paper has all 75 marks represented. No difficulty rankings are assigned because no exact-matched public Pearson Principal Examiner Feedback was verified in this audit.\n\nCIE 9709 May/June 2022 /63 and May/June 2025 /15,/23,/25,/55; all scored parts indexed. Five papers, 84 scored parts and 275 marks. No pre-2019 papers included.\n\nAdds only the six Cambridge International AS & A Level Mathematics (9709) May/June 2024 Variant 1 components 11, 21, 31, 41, 51 and 61. All printed scored parts are indexed; report-based studyPriority ranks are used only where the exact component section of the official June 2024 Examiner Report comments on that part. May/June 2024 was administered. No pre-2019 papers were inspected or included.\n\nAll six CIE 9709 February/March 2024 and all six February/March 2025 standard Variant 2 papers (/12, /22, /32, /42, /52, /62) are indexed. Records are split at every printed scored question part and include one-based physical QP/MS pages and the matching source URLs. Difficulty ranks use only direct component-specific Cambridge examiner-report commentary; both /22 papers remain unranked because the reports state there were too few candidates for a meaningful report. February/March 2020 papers are unadministered practice and outside this batch. February/March 2026 remains a separately flagged omission and is not included.\n\nFully indexed all 25 scored parts for China regional WMA12/01A, May/June 2025; printed marks reconcile to 75. Difficulty priorities are from Pearson's official examiner report.\n\nComplete scored-part index for the four missing Pearson Edexcel IAL January 2019 standard units WME02/01, WME03/01, WST02/01 and WDM01/01. Each paper reconciles to 75 marks. Question and mark-scheme page spans were checked against the paired PDFs; only PEF-mentioned challenge parts receive study priorities. Priority ties preserve the report's grouping where it gave no relative ranking.\n\nComplete scored-part index for Cambridge International AS & A Level Mathematics 9709 February/March 2026 Variant 2 components /12, /22, /32, /42, /52 and /62. Each record represents one printed scored subpart (or one unlettered question), includes the matching QP and mark-scheme URLs with one-based physical PDF pages, and has a paraphrased prompt summary and answer guide. Difficulty priorities use only directly matching Cambridge-authored examiner-report commentary; the /22 report says there were too few candidates for a meaningful report, so all /22 parts remain unranked. Printed marks reconcile to the Cambridge covers: /12 75, /22 50, /32 75, /42 50, /52 50, /62 50 (350 total).\n\nComplete applied-subset scored-part index for the five Pearson Edexcel IAL core Mathematics May/June 2021 and May/June 2022 standard /01 units WDM11, WME02, WME03, WST02 and WST03. It is not the complete 11-unit core Mathematics session inventory: WMA11–WMA14, WME01 and WST01 are tracked separately. Every printed scored subpart is represented separately where marks are separately allocated; QP/MS marks reconcile to 75 per paper. Difficulty priorities are assigned only to parts directly discussed in the matching June 2022 Pearson Principal Examiner Feedback; all June 2021 parts and unmentioned June 2022 parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WMA14/01A, May/June 2025 China regional Paper A. Every printed separately scored question part is represented; the 22 records reconcile to all 75 marks, and QP/MS page spans were checked against the paired PDFs. Challenge priorities identify only question parts directly discussed as demanding or error-prone in Pearson’s matching Principal Examiner Feedback. Rank order reflects the strength of that report evidence within this paper; it is not a normalized cross-paper difficulty score. Other parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WMA11/01A, May/June 2025 China regional Paper A. Every printed separately scored question part is represented; 29 records reconcile to all 75 marks, with QP/MS page spans checked against the paired PDFs. The three flagged challenge items (Q6, Q5(b)(ii), Q4(i)) follow Pearson’s matching Principal Examiner Feedback; all other parts remain unranked.\n\nComplete scored-part index for the complementary six-unit core Maths cohort in Pearson Edexcel IAL May/June 2021 and May/June 2022: WMA11–WMA14, WME01 and WST01 (/01 standard variants). June 2021 uses the already reconciled complete scored-part records and has no matching public Pearson June 2021 Principal Examiner Feedback located; all June 2021 priorities remain null. June 2022 has complete QP/MS part coverage. Priorities are retained only for directly matched parts in the exact Pearson June 2022 Principal Examiner Feedback; other parts remain unranked. Together with the separate applied-subset handoff, these are the 11 standard core Mathematics units listed for each sitting; Further Mathematics is excluded.\n\nComplete applied-subset scored-part index for the five Pearson Edexcel IAL core Mathematics May/June 2021 and May/June 2022 standard /01 units WDM11, WME02, WME03, WST02 and WST03. It is not the complete 11-unit core Mathematics session inventory: WMA11–WMA14, WME01 and WST01 are tracked separately. Every printed scored subpart is represented separately where marks are separately allocated; QP/MS marks reconcile to 75 per paper. Difficulty priorities are assigned only to parts directly discussed in the matching June 2022 Pearson Principal Examiner Feedback; all June 2021 parts and unmentioned June 2022 parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WMA13/01A, May/June 2025 China regional Paper A. All 29 separately scored parts reconcile to all 75 marks against the paired QP and MS. Priority order is based on the matching Pearson Principal Examiner Feedback wording, not a statistical difficulty score; unmentioned parts remain unranked.\n\nComplete scored-part index for all 12 standard Pearson Edexcel IAL Mathematics units offered in January 2020: legacy 2013-spec WMA01, WMA02, WME02, WME03, WST01, WST02 and WDM01; and 2018-spec WMA11, WMA12, WMA13, WME01 and WDM11. The 12 papers total 1,000 marks (WMA01/WMA02 are 125 marks each; the other ten are 75 each). Every separately scored printed part is represented; grouped labels retain a single row where the QP/MS allocates a shared mark total. QP/MS spans are one-based physical pages and cover the full question section. Only directly discussed challenges in the matching Pearson January 2020 PEF have studyPriority/evidence; all other parts are unranked. WMA14 and WST03 were not offered in this session; WFM Further Mathematics is excluded. Source caveats for WMA13, WDM01 and WMA12 are documented in sourceChecks and the audit note.\n\nComplete scored-part index for every February/March 2019 Cambridge International 9709 Variant 2 component in the catalogue: 9709/12, /22, /32, /42, /52, /62 and /72. The archive inventory was checked against the seven matching QP/MS pairs and the Cambridge-branded paper covers; the /72 Paper 7 component is part of the 2019 pre-2020 numbering. Existing sparse records are fully replaced. Each record represents one printed scored part, includes paraphrased prompt and answer guidance, and links to the exact QP/MS physical PDF pages. Difficulty priorities are assigned only to parts with directly matching Cambridge examiner-report commentary; equal priorities are tied categories. Printed totals reconcile to 75, 50, 75, 50, 50, 50 and 50 marks respectively (400 total).\n\nComplete scored-part index for Pearson Edexcel IAL WME02/01A, May/June 2025 China regional Paper A. All 18 separately scored parts reconcile to all 75 marks against the paired QP and MS. Priority order uses direct challenge evidence from Pearson Principal Examiner Feedback; other parts remain unranked.\n\nComplete scored-part index for all 11 catalogue-listed Pearson Edexcel IAL Mathematics unit papers from January 2025: WDM11, WMA11–WMA14, WME01–WME03 and WST01–WST03. Every printed scored part was reconciled against its QP and MS; all 11 papers total 75 marks. Matching Pearson examiner reports provide difficulty evidence only where an individual part is explicitly identified as demanding or discriminating.\n\nComplete scored-part index for Pearson Edexcel IAL WMA12/01A, October 2025 China regional Paper A. All 24 separately scored parts reconcile to the 75 marks shown across the QP and MS. No matching public Pearson PEF was verified, so all parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WME02/01A, October 2025 China regional Paper A. All 18 scored parts reconcile to 75 marks against the paired QP and MS. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete source-reviewed scored-part index for Cambridge International AS & A Level Mathematics 9709 Paper 1 May/June 2020 Variants 1, papers 9709/11, /12 and /13. All 71 printed scored parts reconcile to 75 marks per paper (225 total). The May/June 2020 sitting was cancelled globally; these archived QPs/MSs are unadministered practice. Every difficulty-ranking field is null because candidates did not sit and Cambridge published no Principal Examiner Report for the series.\n\nComplete scored-part index for Pearson Edexcel IAL WME01/01A, October 2025 China regional Paper A. The 17 printed scored parts reconcile to all 75 marks against the paired QP and MS. The two support reactions in Q2(a) are kept together because the mark scheme scores those roman subparts jointly. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WMA11/01A, October 2025 China regional Paper A. All 19 printed scored parts reconcile to 75 marks against the paired QP and MS. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WMA13/01A, October 2025 China regional Paper A. All 23 printed scored parts reconcile to 75 marks against the paired QP and MS. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WMA14/01A, October 2025 China regional Paper A. All 19 printed scored parts reconcile to 75 marks against the paired QP and MS. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete independently scored-part index for the two sparse May/June 2019 Edexcel IAL core units WMA11/01 (Pure Mathematics 1) and WMA12/01 (Pure Mathematics 2). Each paper has 75 indexed marks; all 10 printed questions and their separately scored parts reconcile to the official MS. Priority ranks are retained only for exact-paper June 2019 Pearson examiner-report sections that identify the corresponding part as challenging or discriminating; unmentioned parts remain unranked. Further Mathematics is outside this cohort.\n\nCorrected and source-reviewed full scored-part indexes for the cancelled May/June 2020 Cambridge 9709 Pure Mathematics 2 variants 21, 22, and 23. These archived papers are unadministered practice; every difficulty/rank field is null. This handoff replaces the existing 45 rows for these three paper IDs only.\n\nComplete scored-part index for Pearson Edexcel IAL WST01/01A, October 2025 China regional Paper A. All 33 separately scored parts reconcile to 75 marks against the paired QP and MS. The two distribution subparts in Q6(b) are kept together because Pearson marks them as one combined part. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WST02/01A, October 2025 China regional Paper A. All 26 printed scored parts reconcile to 75 marks against the paired QP and MS. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete scored-part index for Pearson Edexcel IAL WST02/01A, October 2025 China regional Paper A. All 26 printed scored parts reconcile to 75 marks against the paired QP and MS. No public matching Pearson Principal Examiner Feedback was available to support difficulty ranks, so all parts remain unranked.\n\nComplete scored-part index for Cambridge International A Level Mathematics 9709/31, May/June 2020 Variant 1. This cancelled series paper is indexed as unadministered practice. All 22 scored parts reconcile to 75 marks against the archived QP and MS; no ranks are assigned.\n\nComplete Cambridge International 9709 May/June 2019 scored-part index for all 21 catalogue components: 361 printed scored subparts and 1,200 marks reconciled to the question papers and mark schemes. Every part has a paraphrased prompt, topic tags, answer guide, matching QP/MS source links and inclusive one-based physical page spans. A difficulty rank and Cambridge Principal Examiner Report evidence are supplied only for directly supported challenge parts; all other parts remain unranked. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete independently scored-part index for the sparse October 2019 Edexcel IAL core units WMA11/01 (Pure Mathematics 1) and WMA12/01 (Pure Mathematics 2). WMA11 contains 11 questions and WMA12 contains 10; each paper's scored parts reconcile to its 75-mark QP/MS. Difficulty ranks are retained only where the exact October 2019 Pearson examiner report discusses the matching question or subpart as challenging or discriminating; all other parts remain unranked. Further Mathematics is excluded.\n\nComplete scored-part indexes for Cambridge International A Level Mathematics 9709/32 and /33, May/June 2020 Variants 2 and 3. The cancelled series papers are indexed as unadministered practice. Each paper reconciles to the 75 marks printed on its QP; the /32 Q1 mark scheme lists 4 marks while the QP prints 3, so the printed QP allocation is retained and the discrepancy is documented in the audit. All difficulty ranks remain null.\n\nComplete scored-part indexes for Cambridge International A Level Mathematics 9709/41–43, May/June 2020 Variants 1–3. The cancelled series papers are indexed as unadministered practice. Each paper reconciles to its printed 50-mark QP; no difficulty ranks are assigned.\n\nComplete scored-part indexes for Cambridge International A Level Mathematics 9709/51–53 and 9709/61–63, May/June 2020. The cancelled series is indexed as unadministered practice with all difficulty ranks unset. Existing question rows were audited against archived QPs and paired mark schemes; corrupted or misassigned summaries were corrected.\n\nFull scored-part index for five incomplete January 2019 Pearson Edexcel IAL Mathematics units: legacy 2013-spec WMA01, WMA02, WME01, WST01 and 2018-spec WMA11. The official timetable lists nine regular Mathematics units in this session; the other four (WME02, WME03, WST02, WDM01) already have complete records in the existing January 2019 handoff. Further Mathematics WFM01 is excluded. Legacy and 2018-spec papers sharing Pure 1 naming are kept as distinct units. Difficulty ranks are copied only where the matching exact-code January 2019 Pearson examiner feedback directly identifies the part as challenging/discriminating; other records remain unranked.\n\nComplete Cambridge International 9709 October/November 2019 scored-part index for the 20 catalogue components (/11–/63, /72 and /73): 342 printed scored parts and 1150 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete February/March 2020 CIE 9709 Variant 2 index for papers /12, /22, /32, /42, /52 and /62. Every scored printed subpart is indexed and reconciled to its QP label and paired MS. The 37 selected records with individual Cambridge examiner-report evidence retain within-paper study priorities; all remaining scored parts are unranked.\n\nComplete scored-part index for the eight unresolved October 2020 Pearson Edexcel IAL Mathematics catalogue units: legacy WMA01/WMA02/WDM01, reformed WDM11, and 2018-specification WME02/WME03/WST02/WST03. Every QP mark-allocation group is represented, including nested printed labels where one bracket covers several leaves. The eight papers total 700 marks and 192 scored groups. Only directly discussed parts in the exact-code October 2020 Pearson PEFs are prioritized; three PEFs explicitly state no report will be published, and WME02 has no directly named challenge part.\n\nComplete Cambridge International 9709 October/November 2020 scored-part index for the 18 catalogue components (/11–/63): 322 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 May/June 2021 scored-part index for the 18 catalogue components (/11–/13, /21–/23, /31–/33, /41–/43, /51–/53 and /61–/63): 320 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete printed scored-part index for six already-catalogued standard Pearson Edexcel IAL Mathematics papers from October 2020: WMA11–WMA14, WME01 and WST01. All 134 scored groups reconcile to the six 75-mark QPs and paired MS; the 36 question parts directly identified by their exact-code October 2020 Pearson PEFs carry within-paper study priorities, and every other scored group is unranked. QP/MS/PEF links, one-based physical page spans, source hashes and complete-paper updates are included.\n\nComplete February/March 2021 CIE 9709 index for all six catalogue components: /12, /22, /32, /42, /52 and /62. Every printed scored part is represented with QP marks, paired MS and QP physical page spans, a concise prompt summary, topic tags and answer-guide paraphrase. Only the 20 individual parts directly supported by the Cambridge examiner report retain study ranks; /22 remains fully unranked because the report says there were too few candidates for a meaningful report.\n\nComplete Cambridge International 9709 February/March 2022 scored-part index for the six catalogue components (/12, /22, /32, /42, /52 and /62): 101 printed scored parts and 350 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 October/November 2021 scored-part index for the 18 catalogue components (/11–/63): 333 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. The 69 report-selected challenge groups expand to 70 directly supported printed scored parts, including the tied pair 9709/11 5(b)(i) and 5(b)(ii); all other parts remain unranked. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 February/March 2023 scored-part index for the six catalogue components (/12, /22, /32, /42, /52 and /62): 105 printed scored parts and 350 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete scored-part index for all 11 standard Pearson Edexcel IAL Mathematics units examined in January 2021: WDM11, WMA11-WMA14, WME01-WME03 and WST01-WST03. All 240 printed scored groups reconcile to 825 marks across eleven 75-mark papers. Further Mathematics units are excluded.\n\nComplete Cambridge International 9709 May/June 2022 scored-part index for the 18 catalogue components (/11–/13, /21–/23, /31–/33, /41–/43, /51–/53 and /61–/63): 318 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 October/November 2022 scored-part index for all 18 catalogue components: 315 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 May/June 2023 scored-part index for all 18 catalogue components (/11–/13, /21–/23, /31–/33, /41–/43, /51–/53 and /61–/63): 322 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 October/November 2023 scored-part index for all 18 catalogue components: 319 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete scored-part index for all 11 standard Pearson Edexcel IAL Mathematics unit papers examined in October 2021: WDM11, WMA11-WMA14, WME01-WME03 and WST01-WST03. Every printed scored group is represented once; all eleven papers reconcile to 75 marks each.\n\nFull scored-part index for January 2026 WMA12/01A Paper A (China regional). The QP and mark scheme were checked against the archived PaperLords PDFs; each question's scored parts and marks sum to the published 75 marks. No matching Pearson examiner report was available in the source set, so all difficulty ranks are intentionally unassigned.\n\nComplete CIE 9709 October/November 2024 scored-part index across all 18 catalogue components: 319 scored question parts and 1050 marks reconciled to the Cambridge-authored question papers and mark schemes. Each record includes a paraphrased task summary, answer guide, topic tags, exact catalogue QP/MS URLs, and inclusive one-based physical page spans. No difficulty ranking is assigned because no matching Cambridge-authored October/November 2024 examiner report was directly verified in public sources.\n\nComplete Cambridge International 9709 May/June 2024 scored-part index for the 206 records delivered. The 12 catalogue papers for Variants 2–3 are replaced in full; the six Variant 1 papers were independently reparsed and their existing scored-part counts, marks and labels match, so they are left out of the replacement. All 18 source QP totals and MS maxima reconcile to 1050 marks. The incomplete cached 19-page /33 mark scheme was replaced for auditing by a verified 21-page matching copy. QP and MS page spans are inclusive, physical, one-based pages. Difficulty ranks are assigned only where the matching official report directly anchors the same scored question part; parent-question commentary is not inherited by children.\n\nFull January 2023 standard Edexcel IAL Mathematics index: every printed scored question part in all eleven standard units, with QP/MS totals and physical page spans reconciled.\n\nComplete scored-part index for all 11 standard Pearson Edexcel IAL Mathematics units examined in January 2022: 263 printed scored groups and 825 marks across eleven 75-mark papers. Every record has a paraphrased prompt, answer guide, topic tags, paired Pearson QP/MS URLs and one-based physical page spans. Difficulty ranks are attached only to directly supported scored groups in the matching Pearson PEF; broader report comments are not transferred to child parts.\n\nPart-level question index for the May/June 2026 Pearson Edexcel IAL Paper A units WMA11/01A, WMA13/01A, WMA14/01A, WME01/01A and WST01/01A. WMA12/01A is covered in a companion handoff with 27 separately indexed scored parts. Each independently marked part has its own record; summaries and answer guides are paraphrased from the question papers and mark schemes. Page spans are one-based physical PDF pages. The paper source links are from the A Level Copilot archive mirror listed in the catalogue. No matching examiner report was verified, so all difficulty fields are null.\n\nFull scored-part index for the Pearson Edexcel IAL WMA12/01A May/June 2026 China regional Paper A. All 27 printed scored parts are indexed once for 75 marks. Summaries and solution routes are paraphrased. Source PDF page references are one-based physical pages. No matching Pearson examiner report was verified, so every part remains unranked.\n\nComplete Cambridge International 9709 October/November 2023 scored-part index for all 18 catalogue components: 319 printed scored parts and 1050 marks reconciled to the question papers and mark schemes. Records include concise paraphrased task descriptions, topic tags, method guidance and inclusive one-based physical QP/MS page spans. Difficulty ranks and evidence are carried only for exact parts discussed by the matching Cambridge examiner report; all other parts remain unranked. Whole-question report comments spanning multiple scored subparts remain unranked at part level. Existing paper catalogue rows are referenced by replacePaperIds and are not duplicated here.\n\nComplete Cambridge International 9709 May/June 2025 scored-part index for all 24 catalogue components. The handoff contains 419 scored-part records for 1400 marks; all QP printed labels, component totals and MS maxima reconcile. QP/MS sources are Cambridge-branded PDFs archived with URL, SHA-256, byte-count, physical-page and text-extraction provenance. Difficulty ranks are assigned only where the matching Principal Examiner Report discusses the exact scored part; all remaining parts stay unranked. The report section labelled 9709/25 Practical Test states that too few candidates sat it for a meaningful report, so no /25 difficulty ranks are assigned.\n\nComplete scored-part index for all eight standard Pearson Edexcel IAL Mathematics units confirmed for October 2022: 190 printed scored groups and 600 marks across eight 75-mark papers. Every record has a paraphrased prompt, answer guide, topic tags, paired Pearson QP/MS URLs and one-based physical page spans. PEF-based ranks are retained only for exact scored-group labels in the matching Pearson examiner report.\n\nIAL January 2026 Paper A audit: complete scored-part indexes for WMA11/01A and WMA13/01A (China regional variants). Every printed scored bracket has one record; multi-subpart brackets remain combined where Pearson assigns a single total (WMA11 Q4(a) and WMA13 Q8(a)). Neither paper has an examiner report in the catalogue, so difficulty rank and report-priority fields remain null; these two papers are indexed but unranked.\n\nFull scored-part index for the four October 2023 Pearson Edexcel IAL Pure Mathematics units WMA11-WMA14. Every printed scored group is indexed once, each QP question total and each 75-mark paper total reconciles, and exact-part examiner-report seeds are retained. Whole-question-only feedback is not projected onto child scored parts.\n\nFull scored-part index for the four October 2023 Pearson Edexcel IAL applied units WME01-WME02 and WST01-WST02. Every printed scored group is indexed once and question and paper totals reconcile. Existing examiner-report ranks are retained only when they match an exact scored-part label and mark value.\n\nFull scored-part index for eight exact January 2026 Edexcel IAL Paper A rows: WDM11/01A, WMA14/01A, WME01/01A, WME02/01A, WME03/01A, WST01/01A, WST02/01A, WST03/01A. Every scored part is indexed and each paper reconciles to 75 marks. WMA11/01A and WMA13/01A are handled separately. No matching Pearson examiner report was verified for these Paper A variants, so all difficulty fields remain null. WST02/01A uses its fully verified EdeMat mark-scheme copy; the catalogue's PaperLords URL returned a truncated, unreadable PDF and is covered by a separate metadata proposal.\n\nComplete Cambridge International 9709 October/November 2025 scored-part index: all 24 catalogue components, 417 scored groups and 1400/1400 printed marks reconciled against the question papers and mark schemes. QP/MS spans are one-based physical PDF pages. Question statements and answer guides are concise paraphrases. Components /23 and /25 are preserved as separate catalogue variants even though their question content duplicates; totals count both printed components.\n\nFull scored-part index for the 11 standard Pearson Edexcel IAL Mathematics units administered in January 2024: WDM11, WMA11-WMA14, WME01-WME03 and WST01-WST03. Every printed bracketed scored group is indexed once; shared nested prompts remain grouped under their enclosing scored label. WDM11 Question 5 has no separate part bracket and is indexed once from its printed 5-mark question total.\n\nComplete scored-group index for the 24 catalogue-listed Cambridge International AS & A Level Mathematics 9709 May/June 2026 components: 402 distinct printed scored groups, 1,400/1,400 marks. Question and mark-scheme page spans were checked against the corresponding PDFs; the recorded mark sequence for every component matches every printed QP allocation in order. Summaries and answer guides are concise paraphrases. Source PDFs are catalogue-linked Papafy/pastpapers.co archive mirrors carrying PapaCambridge footers, not direct Cambridge-hosted files. Version and source discrepancies are documented in versionNote and sourceChecks.\n\nFull scored-part index for all eight Pearson Edexcel IAL Mathematics components administered in October 2024: WMA11-WMA14, WME01-WME02 and WST01-WST02. Every printed scored mark group is listed once; nested prompts sharing one printed mark bracket stay together, and whole-question marks are not split into invented parts.\n\nFull scored-part index for the six standard Pearson Edexcel IAL Mathematics units administered in May/June 2026: WMA11-WMA14, WME01 and WST01. Every printed score bracket is represented once; nested prompts that share one bracket are kept together under that printed label.\n\nFull scored-group index for the eight standard Pearson Edexcel IAL Mathematics October 2025 papers. Each record corresponds to one printed scored group on the QP; subparts sharing one printed mark allocation are combined into one record. Paper A regional variants and Further Mathematics are excluded.\n\nComplete May/June 2023 Pearson Edexcel IAL Mathematics scored-part index for the 11 standard /01 units: WDM11, WMA11-WMA14, WME01-WME03 and WST01-WST03. Every standalone QP score bracket is represented once; all papers total 75 marks and every QP-printed per-question total is reconciled. Records include paraphrased prompts, answer guides, existing topic IDs, one-based physical QP/MS page spans and paired source hashes.\n\nComplete January 2026 Pearson Edexcel IAL Mathematics scored-part index for the seven standard units WDM11, WMA11–WMA14, WME01 and WME02. Each standalone question-paper score bracket appears once; shared brackets retain the parent label, and all paper and per-question totals reconcile to 75 marks.\n\nFull scored-part index for the January 2026 Pearson Edexcel IAL Mathematics components WME03/01, WST01/01, WST02/01 and WST03/01. Cards follow each printed question-paper score bracket: prompts sharing one bracket are kept together under the parent label, while separately bracketed parts remain separate.\n\nFull scored-part index for all 13 catalogue-listed Pearson Edexcel IAL Mathematics May/June 2024 papers: WDM11, WMA11, WMA11R, WMA12, WMA12R, WMA13, WMA14, WME01-WME03 and WST01-WST03. Each printed scored bracket is indexed once; nested prompts sharing one mark bracket remain under the enclosing scored label. The two regional papers are included; the catalogue has no June Paper A row.\n\nFull scored-part handoff for standard Edexcel IAL May/June 2025 Decision Mathematics 1 and Pure Mathematics 1–4 papers. Each record corresponds to one standalone printed QP score bracket.\n\nFull scored-group index for standard Edexcel IAL May/June 2025 applied papers WME01/01, WME02/01, WME03/01, WST01/01, WST02/01 and WST03/01. Each record corresponds to one printed QP score allocation; nested prompts sharing one bracket are grouped together.", "records": [ { "id": "cie-2018-s-11-q08bi", "paperId": "cie-2018-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find an expression for the sum of the given arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_11.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says identifying the first term proved difficult for all but higher-scoring candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common difference from the coefficient pattern, form the nth term, and sum the arithmetic progression in terms of p, q and n." }, { "id": "cie-2018-s-11-q10ii", "paperId": "cie-2018-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Find and classify the stationary value of m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_11.pdf", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this the most difficult part of the question and notes that few who found the x-value stated the corresponding m-value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative from part (i) to express m, find the stationary value, justify its nature, and state its corresponding m-value." }, { "id": "cie-2018-s-12-q04ii", "paperId": "cie-2018-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the parameter values for which the trigonometric equation has no solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Part (ii) proved much more challenging; many candidates used the discriminant rather than the trigonometric function's range.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the coefficients from part (i), then compare the function's maximum and minimum with the required level to identify when no intersection is possible." }, { "id": "cie-2018-s-12-q10iii", "paperId": "cie-2018-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Use the graph intersections to solve the trigonometric inequality on its stated domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "All but the most able struggled with part (iii); some gave the interval where the function was negative instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the sketches and intersections from earlier parts to identify the intervals where the required function is positive, then express the endpoints in the required units." }, { "id": "cie-2018-s-12-q11ii", "paperId": "cie-2018-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(ii)", "marks": 6, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the region between the curves about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates had trouble squaring the curve expression and handling the y=4 boundary; separate volumes followed by subtraction were more successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up each volume using the correct squared radius and bounds, account for the y=4 boundary, and subtract the inner volume from the outer volume." }, { "id": "cie-2018-s-13-q09i", "paperId": "cie-2018-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 2, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Explain why the given vector expression for OE is correct.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_13.pdf", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was done relatively badly, with a significant proportion of non-responses or unconvincing explanations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the diagram geometry to establish the relevant lengths or ratios, then show that the vector to E is the stated fraction of OB." }, { "id": "cie-2018-s-13-q11ii", "paperId": "cie-2018-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(ii)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the volume formed when the shaded region is rotated through 360° about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_13.pdf", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The key message says omitting the cross term was very common, even among high-scoring candidates, and caused the loss of three marks.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the volume integral with π times the square of the full curve expression, including the cross term, and integrate over the shaded bounds from the curve minimum to x = 1." }, { "id": "cie-2018-s-21-q07iii", "paperId": "cie-2018-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Integrate the linked trigonometric expression to find the required function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_21.pdf", "qpPages": [ 12, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct solutions were seen; candidates rarely linked the integrand to the earlier amplitude form and substituted θ=2x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand using the amplitude form from earlier parts, substitute 2x for θ, and integrate the resulting secant-squared expression." }, { "id": "cie-2018-s-22-q02ii", "paperId": "cie-2018-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Determine the nature of the stationary point and justify the conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was rarely done well and required supporting reasoning rather than an unsupported statement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a second derivative or compare the gradient on either side, showing the substitution and supporting value or sign change." }, { "id": "cie-2018-s-22-q07iii", "paperId": "cie-2018-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Integrate the linked trigonometric expression using the result from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_22.pdf", "qpPages": [ 12, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Correct answers were very rare; the link to part (i) was very rarely made and the required integrand form was necessary for progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the integrand as a secant-squared expression in the transformed variable, use the relation from part (i), and integrate with the substitution factor." }, { "id": "cie-2018-s-23-q02ii", "paperId": "cie-2018-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Determine the nature of the stationary point and justify the conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was rarely done well and required supporting reasoning rather than an unsupported statement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a second derivative or compare the gradient on either side, showing the substitution and supporting value or sign change." }, { "id": "cie-2018-s-23-q07iii", "paperId": "cie-2018-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Integrate the linked trigonometric expression using the result from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_23.pdf", "qpPages": [ 12, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Correct answers were very rare; the link to part (i) was very rarely made and the required integrand form was necessary for progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the integrand as a secant-squared expression in the transformed variable, use the relation from part (i), and integrate with the substitution factor." }, { "id": "cie-2018-s-31-q05i", "paperId": "cie-2018-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Establish the given integral result using a trigonometric substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_31.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 5(a), integration by substitution, was listed among the questions candidates found most difficult; few demonstrated the result completely.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the substitution, transform the square-root expression, and account for any sign change when reversing the limits." }, { "id": "cie-2018-s-31-q05ii", "paperId": "cie-2018-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate the squared cosine expression over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_31.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 5(b), integration of cos²θ, was listed among the most difficult; only a minority knew to use the double-angle identity.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity to rewrite cos²θ, integrate both terms, and apply the limits with the correct factors." }, { "id": "cie-2018-s-31-q07iv", "paperId": "cie-2018-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iv)", "marks": 1, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Prove that the triangle formed by the two roots and the origin is equilateral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_31.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 7(iv), the equilateral-triangle proof, was listed among the most difficult; most tried a less direct three-side calculation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the roots from the earlier part to show equal moduli and an included angle of 60 degrees." }, { "id": "cie-2018-s-31-q09i", "paperId": "cie-2018-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Decompose the top-heavy rational expression into partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_31.pdf", "qpPages": [ 14, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Question 9(i) was listed among the most difficult; only a small minority gained full marks because many ignored the top-heavy numerator.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First account for the polynomial part by division, then decompose the remaining proper fraction and determine all constants." }, { "id": "cie-2018-s-31-q10", "paperId": "cie-2018-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10", "marks": 10, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Solve the vector geometry problem involving the stated line relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_31.pdf", "qpPages": [ 16, 17 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Question 10 on vectors was listed among the most difficult; many candidates gave no response or struggled to complete the vector argument.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given vector relations to form the required equations, keep position and direction vectors distinct, and show the component calculations." }, { "id": "cie-2018-s-32-q07ii", "paperId": "cie-2018-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the angle subtended by the two complex points at the origin.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 7(ii) was listed among the parts causing most difficulty; very few candidates correctly demonstrated the required angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot the roots in the Argand diagram and use a supported geometric or trigonometric argument to find the angle between their position vectors." }, { "id": "cie-2018-s-32-q10iii", "paperId": "cie-2018-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane satisfying the stated vector geometry conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_32.pdf", "qpPages": [ 18, 19 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 10(iii) was listed among the parts causing most difficulty; only a minority adopted a correct strategy and many did not respond.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Determine a point on the plane and a normal vector from the line relationships, then write the scalar plane equation and check its constant term." }, { "id": "cie-2018-s-33-q04ii", "paperId": "cie-2018-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "functions", "proof" ], "primaryTopicId": "functions", "summary": "Use graph sketches to show how many roots the equation has.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_33.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 4(ii) was listed among those done least well; candidates had serious difficulty with both graphs and needed to show the separation beyond the intersection.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch both functions over the full positive domain, show their asymptotes and intersection, then use their behaviour to justify that there is only one root." }, { "id": "cie-2018-s-33-q07ii", "paperId": "cie-2018-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate the definite integral of 15/(cos θ + 2 sin θ)² using the amplitude form from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_33.pdf", "qpPages": [ 12, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists Question 7(ii), evaluation of the integral, among those done least well; the detailed comments say candidates often failed to use the secant-squared form or retain the exact phase angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite cos θ + 2 sin θ as R cos(θ−α) using part (i), convert the integrand to a secant-squared form, integrate to a tangent, and apply the stated limits exactly." }, { "id": "cie-2018-s-33-q08ii", "paperId": "cie-2018-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the point where the curve has a tangent parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_33.pdf", "qpPages": [ 14, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 8(ii) was listed among those done least well; most candidates set dy/dx to zero or one instead of treating a vertical tangent as infinite gradient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly and set the denominator of dy/dx to zero, then solve and check all coordinate pairs in the original curve." }, { "id": "cie-2018-s-33-q09b", "paperId": "cie-2018-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Sketch the complex-number region and determine the greatest possible argument.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_33.pdf", "qpPages": [ 16, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Question 9(b) was listed among those done least well; candidates often misplaced the circle centre or line and struggled with the extreme argument.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the circle |z|≤3 centred at the origin and the labelled line y=2, then use the limiting right triangle to find the extreme argument." }, { "id": "cie-2018-s-33-q10ii", "paperId": "cie-2018-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the angle condition to form and solve the required quadratic in the parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_33.pdf", "qpPages": [ 18, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Question 10(ii) was listed among those done least well; all candidates found it difficult and only the very able generally reached the quadratic with correct vectors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Construct the correct vectors from the stated points, apply the scalar-product angle condition, expand to the given quadratic, and reject the spurious root." }, { "id": "cie-2018-s-41-q06iii", "paperId": "cie-2018-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2018, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy equation to find work done against the varying resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_41.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as one of the more difficult questions; many omitted an energy term, used wrong signs, or assumed the resistance was constant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate work by the engine and loss of potential energy to kinetic-energy gain and work against resistance, keeping all four terms and their signs consistent." }, { "id": "cie-2018-s-42-q07i", "paperId": "cie-2018-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2018, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of A and the tension in the string while the two planes are smooth.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 7 was identified as the most challenging question; errors included resolving the weight component incorrectly and assigning forces to the wrong particle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law along each particle’s direction of motion: T = 1.6a for A and 2.4g sin 30° − T = 2.4a for B, then solve for a and T." }, { "id": "cie-2018-s-42-q07ii", "paperId": "cie-2018-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2018, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 9, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total distance travelled by A when B reaches the barrier and then A slows under friction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 7 was the most challenging; candidates often reused the earlier acceleration or tension, applied friction to the wrong particle, or used mass instead of weight.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the friction force on A, solve for the shared acceleration while B moves the 1 m to the barrier, and calculate their speed then. After B stops, use friction alone to find A’s additional stopping distance and add the first-stage distance." }, { "id": "cie-2018-s-43-q04", "paperId": "cie-2018-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2018, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy method for the connected-particle system to find the required speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_43.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 4 was challenging for many candidates, especially because it required an energy method; separate-particle solutions often omitted tension work.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use work-energy for both particles together so internal tension work cancels; include each particle’s kinetic-energy change and the correct gravitational potential-energy change to find the speed." }, { "id": "cie-2018-s-43-q06iii", "paperId": "cie-2018-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2018, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the condition for constant speed on the incline with speed-dependent forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_43.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 6 was identified as harder; candidates often formed a two-term equation instead of including all three forces or used the wrong direction for the driving force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve along the incline, include the correct speed-dependent driving force and resistance, and use zero acceleration for constant speed." }, { "id": "cie-2018-s-51-q05ii", "paperId": "cie-2018-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed using energy at the equilibrium position.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_51.pdf", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part proved very difficult for many candidates; it required finding equilibrium and setting up a five-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the equilibrium extension where acceleration is zero, then form the full energy equation including all relevant energy terms." }, { "id": "cie-2018-s-51-q06ii", "paperId": "cie-2018-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the string extension to find the required horizontal distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_51.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This part proved too difficult for many; the report explains that the extension-based radius and tension must be established before applying Newton's second law.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the circular radius and tension in terms of the extension, apply Newton's second law horizontally, then solve for the extension and requested length." }, { "id": "cie-2018-s-51-q07i", "paperId": "cie-2018-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use similar triangles to determine the conical tip height and removed volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_51.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 7 was the hardest on the paper; part (i) required a similar-triangle construction followed by volume calculations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the similar triangles to find the cone tip height, then calculate the relevant cone and cylinder volumes." }, { "id": "cie-2018-s-51-q07ii", "paperId": "cie-2018-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the required centre-of-mass distance after accounting for the removed volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_51.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Question 7 was the hardest and part (ii) rarely produced a correct answer; it required several volumes and a moment equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the volumes and their centres of mass, then take moments about the base to find the remaining body's centre of mass." }, { "id": "cie-2018-s-52-q03i", "paperId": "cie-2018-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Derive the centre-of-mass equation by taking moments about A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_52.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few attempted this part, and very few reached the correct result; the report specifies the required centre-of-mass distance and moment equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the perpendicular distance of the centre of mass of BC from the vertical through A, then equate moments about A and simplify." }, { "id": "cie-2018-s-52-q03ii", "paperId": "cie-2018-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Use tan θ = 1.25 to find the wire length in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_52.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates were able to solve the quadratic equation in this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute tan θ = 1.25 into the quadratic relation from part (i), solve for the relevant positive length, and use the diagram dimensions to express the wire length in terms of a." }, { "id": "cie-2018-s-52-q04ii", "paperId": "cie-2018-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the intersection height of the particle's trajectory and the specified line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Part (ii) proved too difficult for many; the report gives the two expressions in x that needed to be equated.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the two expressions for y in terms of x, solve the resulting equation, and substitute the relevant root into the height expression." }, { "id": "cie-2018-s-52-q05i", "paperId": "cie-2018-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use the diagram geometry to find the distance of the centre of mass from the required point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few solved this part; many made no attempt or tried to take moments about the base instead of using the required geometry.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given dimensions to find the angle with the vertical, then relate that angle to the requested distance." }, { "id": "cie-2018-s-52-q06ii", "paperId": "cie-2018-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the stated tension to find the greatest speed in the circular motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "This part was far too difficult for many; the report says the tension in AP had to be found before using Newton's second law.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the tension in AP when BP has its stated tension, then apply Newton's second law radially to determine the greatest speed." }, { "id": "cie-2018-s-53-q05ii", "paperId": "cie-2018-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed using energy at the equilibrium position.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_53.pdf", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part proved very difficult for many candidates; it required finding equilibrium and setting up a five-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the equilibrium extension where acceleration is zero, then form the full energy equation including all relevant energy terms." }, { "id": "cie-2018-s-53-q06ii", "paperId": "cie-2018-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the string extension to find the required horizontal distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_53.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This part proved too difficult for many; the report explains that the extension-based radius and tension must be established before applying Newton's second law.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the circular radius and tension in terms of the extension, apply Newton's second law horizontally, then solve for the extension and requested length." }, { "id": "cie-2018-s-53-q07i", "paperId": "cie-2018-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use similar triangles to determine the conical tip height and removed volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_53.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 7 was the hardest on the paper; part (i) required a similar-triangle construction followed by volume calculations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the similar triangles to find the cone tip height, then calculate the relevant cone and cylinder volumes." }, { "id": "cie-2018-s-53-q07ii", "paperId": "cie-2018-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the required centre-of-mass distance after accounting for the removed volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_53.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Question 7 was the hardest and part (ii) rarely produced a correct answer; it required several volumes and a moment equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the volumes and their centres of mass, then take moments about the base to find the remaining body's centre of mass." }, { "id": "cie-2018-s-61-q01", "paperId": "cie-2018-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the raw data to find the requested sum and sum of squares.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This intended straightforward opening question was one of the poorest attempted; many candidates failed to account for all 12 observations in the sum.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use all 12 observations when converting the centered totals to the required summations." }, { "id": "cie-2018-s-61-q07ii", "paperId": "cie-2018-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements satisfying the restriction on vowel positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_61.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Full marks were very rarely awarded; the subtraction method was highly complicated and did not lead to correct answers.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a systematic arrangement or gap method that accounts for repeated letters and counts each valid arrangement once." }, { "id": "cie-2018-s-62-q02i", "paperId": "cie-2018-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the chosen student is female or likes their curry hot.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 2 was the most challenging on the paper; few fully correct solutions were seen and candidates often omitted valid branches.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a tree or Venn diagram, or calculate P(F)+P(H)−P(F∩H), taking care to include the overlap exactly once." }, { "id": "cie-2018-s-62-q02ii", "paperId": "cie-2018-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "probability", "proof" ], "primaryTopicId": "probability", "summary": "Determine whether the two events are independent and justify the result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not show enough working to justify their conclusion; unsupported calculations received only partial credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate P(M)P(H) and P(M∩H), compare the values, and state the conclusion in context." }, { "id": "cie-2018-s-63-q01ii", "paperId": "cie-2018-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw the histogram using the correct unequal class widths and scale.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Histogram scaling caused problems and few candidates scored full marks on the question; many used incorrect class widths or boundaries.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate frequency densities, use class boundaries rather than limits, choose appropriate scales, and label the axes with units." }, { "id": "cie-2018-s-63-q07iv", "paperId": "cie-2018-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count selections satisfying the stated numbers of repeated letters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_63.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 7, particularly part (iv), was identified as the most challenging; many overcounted by treating repeated letters as distinguishable.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate cases with two, three, or four Es, account for the Ns in each case, and count each distinct selection without treating repeated letters as distinct." }, { "id": "cie-2018-s-71-q02i", "paperId": "cie-2018-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Calculate the confidence interval for the population proportion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 2 was one of the more demanding questions; candidates often confused a proportion interval with one for a mean and centred it on 20.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the sample proportion and critical z-value to calculate both endpoints and state a single interval." }, { "id": "cie-2018-s-71-q02ii", "paperId": "cie-2018-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Use the confidence interval to assess whether the die may be biased.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Part (ii) was poorly answered; candidates often failed to check whether 1/6 was inside the confidence interval or did not state the check clearly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Check whether the hypothesised proportion 1/6 lies inside the interval from part (i), then state the conclusion in context." }, { "id": "cie-2018-s-71-q06iii", "paperId": "cie-2018-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find P(X > 9) for the stated continuous distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_71.pdf", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Parts (iii) and (iv) were often omitted; integration attempts commonly used incorrect limits and some candidates incorrectly used a normal distribution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the density from the question and integrate from 9 to the upper endpoint of 10 with the correct limits." }, { "id": "cie-2018-s-71-q07iiib", "paperId": "cie-2018-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)(b)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the stated Poisson test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_71.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Part (iii)(b) was not well answered; many candidates incorrectly equated the probability of a Type I error with 5 percent.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the rejection region under the null Poisson model to calculate the probability of rejecting a true null hypothesis." }, { "id": "cie-2018-s-71-q07iiic", "paperId": "cie-2018-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Complete the Poisson test with the upper-tail probability and contextual conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_71.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Part (iii)(c) was not well answered; many candidates calculated P(X=3) instead of the required upper-tail probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate P(X>3) under the null model, compare it with the significance level, and state a non-definite conclusion in context." }, { "id": "cie-2018-s-72-q07i", "paperId": "cie-2018-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "statistical-testing", "binomial" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the binomial significance test and decide whether the spinner claim is justified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_72.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 7 was particularly poorly attempted; few candidates correctly formed the required binomial probability and comparison with 0.01.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State the hypotheses, use the correct binomial model and tail probability, compare with the 1% significance level, and conclude in context." }, { "id": "cie-2018-s-72-q07iii", "paperId": "cie-2018-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 6, "topicIds": [ "statistical-testing", "binomial" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of the stated Type II error using the critical region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_72.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Part (iii) was demanding; the critical region was rarely shown and few candidates calculated the required probability under the alternative model.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the acceptance boundary from the earlier part and the alternative binomial model with p=0.5 to calculate the probability required." }, { "id": "cie-2018-s-73-q04iii", "paperId": "cie-2018-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the conditional probability that no female students leave, given that the total number leaving is greater than 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_73.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The detailed report labels this as part (iii) and says very few obtained the correct conditional probability; many used only one event in the numerator or reused the combined mean.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form P(F=0 and M+F>3) in the numerator, use P(M+F>3) in the denominator, and account for the independent Poisson models; the male-only mean is required in the numerator event." }, { "id": "cie-2018-s-73-q07iii", "paperId": "cie-2018-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "probability", "integration" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly one sampled value is below 1.5 and the other two are above it.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_ms_73.pdf", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Fully correct answers were rare; candidates often used wrong integration limits or probabilities, and many forgot the factor of three for the possible arrangements.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the density from 1 to 1.5, use the complementary probability for values above 1.5, and multiply by the three possible positions of the single lower value." }, { "id": "cie-2018-m-12-q10i", "paperId": "cie-2018-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the ranges of f, g and the composite fg for their stated domains.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few answers were correct across the three range statements; some gave domains in x instead of ranges.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use each domain to determine the corresponding range; for fg, first determine the range of g and then apply f to those values." }, { "id": "cie-2018-m-12-q10iii", "paperId": "cie-2018-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 6, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "Solve the stated inequality involving the derivative and inverse of f, giving the full set of x-values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The equation-solving was described as a challenge; candidates who found critical values often forgot to impose the domain of f.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find f′ and f⁻¹, solve the resulting inequality carefully, then retain only values in the domain of f." }, { "id": "cie-2018-m-12-q10ii", "paperId": "cie-2018-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Explain why the composite gf is not defined.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says range and domain were often confused in explanations of this composition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the range of f with the domain of g and state that part of the range of f lies outside the domain required by g." }, { "id": "cie-2018-m-12-q03i", "paperId": "cie-2018-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use the constant daily-growth model to predict the plant height 60 days after its initial measurement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only candidates who used n = 61 obtained the correct term; using n = 60 was a reported common error.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the first-day increase of 1.2 cm in the arithmetic model and count the 60-day endpoint as the 61st term from the initial height." }, { "id": "cie-2018-m-12-q03ii", "paperId": "cie-2018-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use the constant daily-percentage-growth model to predict the plant height after 60 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report repeats that n = 60 was often used instead of n = 61 in the geometric model.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the first-day growth ratio 41.2/40 as the common ratio and take the 61st term, including the initial measurement as term 1." }, { "id": "cie-2018-m-22-q05ii", "paperId": "cie-2018-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use the equation from part (i) to verify that its positive root lies between 1.0 and 1.5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this type of root-bracketing request is often difficult and stresses the need to state the sign-change conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Define a residual function from the equation, evaluate it at 1.0 and 1.5, and state that the opposite signs show a root in the interval." }, { "id": "cie-2018-m-32-q07i", "paperId": "cie-2018-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "functions", "exponentials", "proof" ], "primaryTopicId": "functions", "summary": "Use sketches of the two exponential graphs to show that the equation has exactly one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was often omitted and few candidates earned marks; common sketch errors included wrong intercepts, asymptotes and restricted x-domains.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch each graph with correct intercepts and horizontal asymptotes, show the single intersection, and indicate its corresponding x-value." }, { "id": "cie-2018-m-32-q07iii", "paperId": "cie-2018-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "numerical-methods", "proof" ], "primaryTopicId": "numerical-methods", "summary": "Show that any convergent sequence generated by the given iteration converges to the root from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this part very difficult; many responses repeated the later calculation or applied logarithms incorrectly to terms in a sum.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Let the limit be L, equate the limiting consecutive iterates in the formula, then rearrange to recover the equation in part (i)." }, { "id": "cie-2018-m-32-q08ii", "paperId": "cie-2018-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the partial fractions from part (i) to evaluate the definite integral and express it as ln c for an integer c.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says integration of the partial fractions proved testing, especially the logarithmic coefficients and the term over x² + 9.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate every partial-fraction term with its correct coefficient, apply both limits, and combine the result into the required logarithmic form." }, { "id": "cie-2018-m-32-q04ii", "paperId": "cie-2018-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find n and A from the two given values of x and y, rounding each to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_32.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies widespread logarithm errors and says few candidates obtained even a correct method for both constants.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain a linear relation between ln y and ln x, use the two data points to solve for the constants, and preserve precision until rounding." }, { "id": "cie-2018-m-32-q01", "paperId": "cie-2018-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the stated definite integral with three trapezium-rule intervals to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says the question was generally not well done and lists frequent errors in interval count, width, units and ordinate selection.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use four equally spaced ordinates over the stated interval, apply the trapezium-rule weights, and round only the final estimate to three decimal places." }, { "id": "cie-2018-m-42-q04", "paperId": "cie-2018-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal force required for a particle on a rough incline to be on the point of moving up the slope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the paper’s most difficult question for many candidates and says many mistook a horizontal force for one acting up the plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the horizontal force both along and perpendicular to the plane, include both contributions to the normal reaction, then use limiting friction and equilibrium along the plane." }, { "id": "cie-2018-m-42-q03i", "paperId": "cie-2018-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed of a girl after sliding down a smooth slide through a given vertical drop from rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number used constant-acceleration formulae even though the motion was not in a straight line; the report identifies work-energy as the valid method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy: the loss of gravitational potential energy equals the gain in kinetic energy." }, { "id": "cie-2018-m-42-q03ii", "paperId": "cie-2018-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Compare two rough-slide journeys with the same work against friction to find the initial push speed in the second journey.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report again notes candidates incorrectly applying constant-acceleration equations to this non-linear slide motion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write an energy balance for each journey, equate the common work against friction, cancel that term and solve for the initial speed." }, { "id": "cie-2018-m-42-q06iii", "paperId": "cie-2018-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest constant speed of the car while it travels up an inclined hill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes many continued to use the acceleration from the previous part despite constant speed, and explains the three-force balance required.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the maximum driving force P/v equal to the resistive force plus the component of weight down the slope, then solve the resulting quadratic and reject the impossible root." }, { "id": "cie-2018-m-42-q05ii", "paperId": "cie-2018-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the rocket’s maximum height after its fuel stops burning at the 200 m point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates assumed the initial acceleration continued or treated the burnout time as the time to maximum height; some also omitted adding the original 200 m.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "After burnout use acceleration −g to find the additional rise from speed 40 m s⁻¹, then add that rise to 200 m." }, { "id": "cie-2018-m-52-q01", "paperId": "cie-2018-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the height of a square-based block just on the point of toppling on a rough plane inclined at 30°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this question very difficult and says most candidates could not visualise the block’s orientation and tipping point.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "At impending toppling the weight acts through the downhill contact corner; relate the centre-of-mass offset and half-height using the plane angle." }, { "id": "cie-2018-m-52-q07i", "paperId": "cie-2018-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the radius of the hole in a suspended square lamina from its equilibrium orientation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_52.pdf", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found this one of the hardest parts; the report identifies locating the centre of mass and taking moments about AD or BC as the required steps.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of the square with the circular hole removed, express its horizontal and vertical offsets from A, use the given tangent relation, and take moments about a side through A." }, { "id": "cie-2018-m-52-q04iii", "paperId": "cie-2018-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile’s greatest height and the time when it reaches that height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was rather difficult for many and gives the trajectory-differentiation route as the most direct method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the trajectory y(x), set dy/dx to zero to locate the maximum, substitute to find its height, then use horizontal motion to obtain the time." }, { "id": "cie-2018-m-52-q06iii", "paperId": "cie-2018-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find how far the object continues moving after the time-dependent applied force ends.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found this part difficult; the report says to integrate once for velocity and then use constant-acceleration formulae, rather than integrate again for distance.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate dv/dt from part (ii) to obtain the speed at t = 8, then use constant-acceleration motion under friction to find the stopping distance." }, { "id": "cie-2018-m-62-q07ii", "paperId": "cie-2018-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Estimate how many of 500 packets lie more than 1.5 standard deviations from the mean weight.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many misread which weights were required and that most considered only one tail.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the two-tail standard-normal probability outside ±1.5, multiply by 500, and round to a whole expected count." }, { "id": "cie-2018-m-62-q03iv", "paperId": "cie-2018-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(iv)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether hatchback type and red colour are independent events, with justification.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many did not complete the numerical comparison; the report also notes confusion between an intersection and the conditional probability from part (iii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare P(R ∩ H) with P(R)P(H), or compare P(R|H) with P(R), and use the numerical values to justify the decision." }, { "id": "cie-2018-m-62-q05ii", "paperId": "cie-2018-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the sum of the squares of the original data from the supplied shifted-data summaries and mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A large number failed to state the variance formula correctly for the unshifted data or used an inaccurate variance value.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First obtain n and Σ(x−20) from the mean and supplied sum, then expand x² = ((x−20)+20)² and substitute the given totals." }, { "id": "cie-2018-m-62-q03iii", "paperId": "cie-2018-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(iii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a car is red given that it is a hatchback.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many conditional-probability solutions incorrectly formed the numerator as a conditional probability rather than using the table counts directly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the red-hatchback count as the numerator and the total hatchback count as the denominator." }, { "id": "cie-2018-m-62-q02i", "paperId": "cie-2018-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count selections of three letters from COLLIDER containing exactly one L.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report identifies choosing any three letters without enforcing exactly one L as the most common misconception.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose two of the six distinct non-L letters; the repeated Ls are identical, so the selected L is not distinguished." }, { "id": "cie-2018-m-72-q07ii", "paperId": "cie-2018-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain whether the Central Limit theorem was needed for the confidence interval in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_72.pdf", "qpPages": [ 12, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this explanation as especially challenging and says candidates needed to specify that the population, rather than an unspecified quantity, was normal.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that no Central Limit theorem is needed because the population itself is normally distributed." }, { "id": "cie-2018-m-72-q03ii", "paperId": "cie-2018-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the mean and variance of the difference between two independently sampled total plant heights.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_72.pdf", "qpPages": [ 5, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says part (ii) was particularly demanding and records repeated errors in the variance terms and in scaling variance by the squared coefficient.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the means linearly, and add independent variances using Var(HA−2HB) = Var(HA)+4Var(HB)." }, { "id": "cie-2018-m-72-q07iv", "paperId": "cie-2018-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Assess whether a 95% confidence interval is more likely than a 99% interval to include the claimed mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_72.pdf", "qpPages": [ 12, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not appreciate that the 95% interval is narrower than the 99% interval.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare interval widths: at the same sample size the 95% interval is narrower, so it is less likely to contain a value excluded by the 99% interval." }, { "id": "cie-2018-m-72-q05i", "paperId": "cie-2018-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State null and alternative hypotheses for testing whether the orange-sweet proportion is below 17%.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_72.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates were unable to set up the correct hypotheses for the one-tailed test.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write H₀: p = 0.17 and H₁: p < 0.17, defining p as the population proportion of orange sweets." }, { "id": "cie-2018-m-72-q05iv", "paperId": "cie-2018-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iv)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the probability of a Type II error when the true orange-sweet proportion is 5%.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m18_ms_72.pdf", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was not well attempted and many candidates used the wrong tail instead of calculating the probability of at least three.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "A Type II error occurs when at least three orange sweets are observed and the null claim is not rejected; calculate this binomial upper-tail probability for p = 0.05." }, { "id": "cie-2018-w-11-q11aii", "paperId": "cie-2018-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)(ii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "State the range of f and find its inverse when a takes its greatest possible value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_11.pdf", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates recognised that the inverse range required the negative square root; many found an inverse expression but missed the range condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the specified greatest value of a, state the range with the correct open endpoint, and solve for the inverse using the negative square-root branch required by the original domain." }, { "id": "cie-2018-w-11-q10ii", "paperId": "cie-2018-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the rate of change of y at A when x is decreasing at 0.3 units per second.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_11.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the detailed chain-rule method distinguished stronger answers and notes that successful candidates recognised x was decreasing.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use dy/dt = (dy/dx)(dx/dt), evaluate the curve gradient at A, and keep the negative sign from the decreasing x-coordinate." }, { "id": "cie-2018-w-11-q07iii", "paperId": "cie-2018-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the area of the shaded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_11.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates commonly used the area under the curve and needed to subtract the triangle; the subtraction was not always included.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve between the stated limits and subtract the triangular area where the diagram shows it as part of the region." }, { "id": "cie-2018-w-12-q03ii", "paperId": "cie-2018-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the maximum possible length of PQ as t varies.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Almost 40% omitted this part, often after complex results in part (i); candidates who found t sometimes stopped without finding the maximum length.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the length expression from part (i), find the stationary value of t, and substitute it back to obtain the maximum length." }, { "id": "cie-2018-w-12-q11iii", "paperId": "cie-2018-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(iii)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the area of the shaded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates used incorrect limits, omitted the line, or used the tangent at A; this was the main difficulty in finding the required area.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct curve and boundary limits, include the straight-line contribution, and subtract or add areas according to the shaded diagram." }, { "id": "cie-2018-w-12-q04ii", "paperId": "cie-2018-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch the graph of the given function with its correct turning behaviour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says part (ii) was more difficult and many candidates did not draw a curve that clearly turned at both ends.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the function and the result from part (i) to show both turning ends of the curve and the correct overall shape." }, { "id": "cie-2018-w-12-q06i", "paperId": "cie-2018-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Use the two triangles to prove the stated relationship involving sin squared theta and cos theta.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_12.pdf", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says weaker candidates often found BD in one triangle but failed to make the second geometric connection needed to establish the result.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express BD from each triangle using the labelled angles and side lengths, equate the two expressions, and simplify to the required identity." }, { "id": "cie-2018-w-12-q08ii", "paperId": "cie-2018-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of the shaded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says weaker candidates often missed which angles and arc lengths were required, and some found area instead of perimeter.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the required sides and arc lengths from the diagram, find their angles and radii, and add the lengths rather than calculating the shaded area." }, { "id": "cie-2018-w-13-q11iv", "paperId": "cie-2018-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(iv)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the greatest p for which fg can be formed and give the composite function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_13.pdf", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only stronger candidates recognised the needed condition x + 3 <= 1 and went on to obtain full marks.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the domain restriction for f to the output of g, use the resulting inequality to determine the greatest allowed p, then substitute g(x) into f(x)." }, { "id": "cie-2018-w-13-q02", "paperId": "cie-2018-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "Decide whether the function is increasing, decreasing, or neither on its stated domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates reached an incorrect conclusion from one gradient check; the report required evidence of negative and positive gradients on the two sides of the minimum.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the critical values, test the derivative on intervals across the stationary minimum, and state that the function is neither increasing nor decreasing on the full domain." }, { "id": "cie-2018-w-13-q09i", "paperId": "cie-2018-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Show that the curve and line meet for every value of k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_13.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates stopped after forming the quadratic or introduced an irrelevant inequality; few completed the perfect-square argument needed for the conclusion.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the quadratic condition for intersection, show its discriminant expression is a perfect square or otherwise always non-negative, and state that real intersections exist for all k." }, { "id": "cie-2018-w-13-q10i", "paperId": "cie-2018-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(i)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume formed when the shaded region is rotated about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_13.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates struggled to find the squared fractional power, sometimes integrated y instead of y squared, and often used an incorrect lower limit or omitted limit working.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up pi times the integral of y squared over the shaded x-limits, square the full fractional-power expression carefully, integrate, and show both limit substitutions." }, { "id": "cie-2018-w-13-q06", "paperId": "cie-2018-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use a scalar product to find angle OBF in the solid shown.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_13.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A significant number found BF incorrectly by treating CG as a vertical edge; the report notes this gave an incompatible final form and candidates often did not check the error.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read the three-dimensional diagram carefully to form the correct vectors, then use their scalar product and magnitudes to calculate the angle." }, { "id": "cie-2018-w-13-q07ii", "paperId": "cie-2018-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2018, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the given trigonometric equation on the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_13.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many candidates did not identify that the numerator must be zero; others formed a quadratic in sin theta but could not solve it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity and equation established in part (i), set the appropriate numerator equal to zero, solve the quadratic in sin theta, and retain every solution in the interval." }, { "id": "cie-2018-w-21-q01ii", "paperId": "cie-2018-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use the result from part (i) and logarithms to solve for y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates used the intended link to part (i), so correct answers were rare; the report also notes a frequent inaccurate answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the solution from part (i) in the equivalent exponential relation, isolate the power involving y, and apply logarithms to obtain the requested value to 3 significant figures." }, { "id": "cie-2018-w-21-q03", "paperId": "cie-2018-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve secant squared theta = 3 cosecant theta for theta between 0 and 180 degrees.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says there were few correct responses because many candidates were unsure which trigonometric identities to use.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the equation using reciprocal and Pythagorean identities, solve the resulting trigonometric equation, and select solutions in the stated interval." }, { "id": "cie-2018-w-21-q06i", "paperId": "cie-2018-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "data-analysis", "integration" ], "primaryTopicId": "data-analysis", "summary": "Approximate the area of the region using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_21.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Use of the trapezium rule continued to cause problems; errors included incorrect interval counts, calculator mode and using x-values instead of y-values, so few correct solutions were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read the endpoints and midpoint from the graph, evaluate the three function heights, and apply the trapezium rule with the correct interval width." }, { "id": "cie-2018-w-21-q06ii", "paperId": "cie-2018-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact volume when the region is rotated about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_21.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found an integrand but failed to use an appropriate double-angle formula before integrating; the report says few exact answers were correct.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use pi times the integral of the squared curve, apply the double-angle identity to simplify the integrand, and evaluate exactly without a calculator." }, { "id": "cie-2018-w-21-q07ii", "paperId": "cie-2018-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 8, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinates of P and Q where the curve gradient is 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_21.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates found only the first solution and could not obtain the second; errors in alpha and degree/radian handling were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write dy/dx in the specified amplitude form, solve the cosine equation for both points on the stated domain, and give both coordinates to four significant figures." }, { "id": "cie-2018-w-22-q02", "paperId": "cie-2018-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find 3^x from the given equation, then determine x using logarithms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates made progress; the report says candidates often took logarithms of each term instead of first solving for 3^x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = 3^x to form and solve the quadratic in u, retain the valid positive value, then take logarithms to find x to four significant figures." }, { "id": "cie-2018-w-22-q04", "paperId": "cie-2018-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the implicit curve at the given point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates commonly missed the product rule or differentiated the right-hand constant incorrectly; simplification before substitution also caused errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate every term implicitly, apply the product rule to the exponential term, then substitute the point into the unsimplified derivative and solve for dy/dx." }, { "id": "cie-2018-w-22-q05iii", "paperId": "cie-2018-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the iterative formula to find p to five significant figures, showing each iteration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates used the calculator incorrectly, did too few iterations, or gave a final answer at the wrong accuracy.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Start with a value in the interval specified by the report, apply the correct iterative formula for enough steps, and record every iterate to seven significant figures before rounding p." }, { "id": "cie-2018-w-23-q01ii", "paperId": "cie-2018-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use the result from part (i) and logarithms to solve for y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates used the intended link to part (i), so correct answers were rare; the report also notes a frequent inaccurate answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the solution from part (i) in the equivalent exponential relation, isolate the power involving y, and apply logarithms to obtain the requested value to 3 significant figures." }, { "id": "cie-2018-w-23-q03", "paperId": "cie-2018-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve secant squared theta = 3 cosecant theta for theta between 0 and 180 degrees.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "There were few correct responses because many candidates were unsure which trigonometric identities to use.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the equation using reciprocal and Pythagorean identities, solve the resulting trigonometric equation, and select solutions in the stated interval." }, { "id": "cie-2018-w-23-q06i", "paperId": "cie-2018-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "data-analysis", "integration" ], "primaryTopicId": "data-analysis", "summary": "Approximate the area of the region using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_23.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Use of the trapezium rule continued to cause problems; errors included incorrect interval counts, calculator mode and using x-values instead of y-values, so few correct solutions were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read the endpoints and midpoint from the graph, evaluate the three function heights, and apply the trapezium rule with the correct interval width." }, { "id": "cie-2018-w-23-q06ii", "paperId": "cie-2018-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact volume when the region is rotated about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_23.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found an integrand but failed to use an appropriate double-angle formula before integrating; few exact answers were correct.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use pi times the integral of the squared curve, apply the double-angle identity to simplify the integrand, and evaluate exactly without a calculator." }, { "id": "cie-2018-w-23-q07ii", "paperId": "cie-2018-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2018, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 8, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinates of P and Q where the curve gradient is 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_23.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates found only the first solution and could not obtain the second; errors in alpha and degree/radian handling were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write dy/dx in the specified amplitude form, solve the cosine equation for both points on the stated domain, and give both coordinates to four significant figures." }, { "id": "cie-2018-w-31-q07ii", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use u = sin x to find the exact area of region R.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Nearly all candidates struggled despite the given substitution; very few obtained the correct integrand together with the correct limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute u = sin x, transform the remaining cosine terms correctly, change the limits to u-values, and integrate the resulting expression exactly." }, { "id": "cie-2018-w-31-q10iii", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two position vectors on line l at perpendicular distance 2 from plane n.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates who did not know the point-to-plane formula struggled; some stopped at the line-plane intersection and did not find points at distance 2.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the point-to-plane distance formula for a general point on l, solve for the parameter, and give both position vectors." }, { "id": "cie-2018-w-31-q01", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the modulus inequality for x in terms of positive a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this question challenging; many candidates omitted a, squared the factor 2 incorrectly or failed to reintroduce a after scaling.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Interpret the modulus inequality as two linear inequalities or square carefully while retaining all powers of a, then give the interval for x." }, { "id": "cie-2018-w-31-q04ii", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact point where the curve has a tangent parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates wrongly set dy/dx to zero, one or infinity without justification; those who chose the correct condition often stopped before finding the coordinates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise that a vertical tangent has an undefined or infinite dy/dx, use the parametric derivatives to find the correct parameter, then evaluate both coordinates." }, { "id": "cie-2018-w-31-q03i", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "functions", "proof" ], "primaryTopicId": "functions", "summary": "Show by sketching graphs that x^3 = 3 - x has exactly one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was challenging; candidates often truncated the sketches or stated there was one root without clearly marking the intersection.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch both functions over a range that shows their full relevant shapes, mark the single intersection, and state why it gives exactly one root." }, { "id": "cie-2018-w-31-q08a", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the given complex quotient in rei-theta form and give r and theta to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says few candidates obtained the correct argument; common errors included conjugate arithmetic and unsupported calculator arguments without checking the quadrant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide by multiplying through by the denominator conjugate, identify the real and imaginary parts, then calculate the modulus and the principal argument from the correct Argand quadrant." }, { "id": "cie-2018-w-31-q08b", "paperId": "cie-2018-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Sketch the complex locus and find the least modulus of its points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_31.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Some candidates gave only the sketch and did not attempt the least modulus; others placed the circle in the wrong quadrant or omitted its radius.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the circle with its centre and unit radius on an Argand diagram, then find the shortest distance from the origin to the circle." }, { "id": "cie-2018-w-32-q09b", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Sketch the complex region and find its greatest possible argument.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates did not attempt the greatest argument or used arctan where the circle geometry requires the tangent-radius sine relationship.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the circle and half-plane with the correct scales, locate the tangent from the origin that bounds the region, and use the radius-tangent right triangle to find the extreme argument." }, { "id": "cie-2018-w-32-q03ii", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Use the result from part (i) to show the stated definite integral result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number gave no response; candidates were expected to explain how the logarithmic expression changed from ln 2 to ln 4.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the antiderivative from part (i), show both limit substitutions, and use logarithm laws to obtain the required exact expression." }, { "id": "cie-2018-w-32-q07ii", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the value of a that makes the stated trigonometric integral equal to 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Nearly all candidates struggled with the integral despite the supplied substitution; poor substitution and limits meant very few obtained the correct integrand with the correct limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite cos squared x using a double-angle identity, integrate over the stated limits, equate the result to 1 and solve for a." }, { "id": "cie-2018-w-32-q04", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the given equation involving e^x and e^-x to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 19 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report lists recognition of the quadratic in e^x as a challenge; a significant minority made no progress by applying incorrect logarithm rules.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = e^x, rewrite the equation as a quadratic in u, retain the positive solution, and take its logarithm to find x." }, { "id": "cie-2018-w-32-q06", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Set up and solve the differential equation to find y in terms of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 18 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates often used the wrong proportionality constant, found integrating 1/y^2 difficult, or stopped before expressing y in terms of x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the proportionality into dy/dx = k y^2/x, find k using both given points, separate variables, integrate with a constant, and rearrange for y." }, { "id": "cie-2018-w-32-q10iii", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the line in plane p that intersects line l at right angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 16, 18 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Several candidates gave no attempt; successful candidates identified the vector perpendicular to both the line direction and plane normal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the direction vector perpendicular to both the direction of l and the normal to p, then write the line through the required point and check it lies in p." }, { "id": "cie-2018-w-32-q10i", "paperId": "cie-2018-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the position vector of point A from the given line and plane conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_32.pdf", "qpPages": [ 16, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Many candidates failed to use the relation involving r-a and the normal vector, or ignored the fixed vector in their equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the plane equation and the given geometric relation to determine the point coordinates, keeping the fixed position vector and direction vectors distinct." }, { "id": "cie-2018-w-33-q07ii", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use u = sin x to find the exact area of region R.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Nearly all candidates struggled despite the given substitution; very few obtained the correct integrand together with the correct limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute u = sin x, transform the remaining cosine terms correctly, change the limits to u-values, and integrate the resulting expression exactly." }, { "id": "cie-2018-w-33-q10iii", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two position vectors on line l at perpendicular distance 2 from plane n.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates who did not know the point-to-plane formula struggled; some stopped at the line-plane intersection and did not find points at distance 2.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the point-to-plane distance formula for a general point on l, solve for the parameter, and give both position vectors." }, { "id": "cie-2018-w-33-q01", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the modulus inequality for x in terms of positive a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this question challenging; many candidates omitted a, squared the factor 2 incorrectly or failed to reintroduce a after scaling.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Interpret the modulus inequality as two linear inequalities or square carefully while retaining all powers of a, then give the interval for x." }, { "id": "cie-2018-w-33-q04ii", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact point where the curve has a tangent parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates wrongly set dy/dx to zero, one or infinity without justification; those who chose the correct condition often stopped before finding the coordinates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise that a vertical tangent has an undefined or infinite dy/dx, use the parametric derivatives to find the correct parameter, then evaluate both coordinates." }, { "id": "cie-2018-w-33-q03i", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "functions", "proof" ], "primaryTopicId": "functions", "summary": "Show by sketching graphs that x^3 = 3 - x has exactly one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was challenging; candidates often truncated the sketches or stated there was one root without clearly marking the intersection.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch both functions over a range that shows their full relevant shapes, mark the single intersection, and state why it gives exactly one root." }, { "id": "cie-2018-w-33-q08a", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the given complex quotient in rei-theta form and give r and theta to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says few candidates obtained the correct argument; common errors included conjugate arithmetic and unsupported calculator arguments without checking the quadrant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide by multiplying through by the denominator conjugate, identify the real and imaginary parts, then calculate the modulus and the principal argument from the correct Argand quadrant." }, { "id": "cie-2018-w-33-q08b", "paperId": "cie-2018-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2018, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Sketch the complex locus and find the least modulus of its points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_33.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Some candidates gave only the sketch and did not attempt the least modulus; others placed the circle in the wrong quadrant or omitted its radius.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the circle with its centre and unit radius on an Argand diagram, then find the shortest distance from the origin to the circle." }, { "id": "cie-2018-w-41-q06i", "paperId": "cie-2018-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show the particle’s deceleration from the two timed sections of travel and find its initial speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_41.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates found this difficult; many incorrectly used P–Q and Q–R while treating the initial speed as unchanged.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 38, 38 ], "tagConfidence": "reviewed", "answerGuide": "Apply s = ut + ½at² to the P–Q and P–R displacements using the correct elapsed times. Solve the simultaneous equations, show a = −⅔ m s⁻², then substitute to find u." }, { "id": "cie-2018-w-41-q05i", "paperId": "cie-2018-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction of the resultant of three coplanar forces shown in the diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_41.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates identified the direction of the resultant correctly; the report recommends a diagram to describe the angle unambiguously.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 38, 38 ], "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, combine the components, then use the resultant magnitude and a labelled direction angle to state both requested values." }, { "id": "cie-2018-w-41-q03", "paperId": "cie-2018-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the work done by a van’s engine while it descends a hill, speeds up and works against resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates incorrectly assumed the engine force was constant; the report says this situation requires work–energy rather than Newton’s second law.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 37, 37 ], "tagConfidence": "reviewed", "answerGuide": "Use work–energy: engine work plus the loss of gravitational potential energy equals the gain in kinetic energy plus work against resistance." }, { "id": "cie-2018-w-42-q07ii", "paperId": "cie-2018-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find how long a rebounding particle takes to come to rest after rising from the bottom of a water tank.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this by far the hardest question for most candidates and says almost all assumed the particle stopped inside the water.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 43, 43 ], "tagConfidence": "reviewed", "answerGuide": "Find the acceleration and speed after the 1.25 m rise through water, check that the particle reaches the surface while still moving, then add the remaining free-flight time under gravity." }, { "id": "cie-2018-w-42-q05i", "paperId": "cie-2018-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time when a particle with a time-dependent acceleration reaches maximum velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found this difficult because of the fractional-power algebra needed to solve a = 0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 42, 42 ], "tagConfidence": "reviewed", "answerGuide": "Set the given acceleration equal to zero, solve the equation containing the fractional power carefully, and select the positive time at which acceleration changes from positive to negative." }, { "id": "cie-2018-w-42-q04ii", "paperId": "cie-2018-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of two connected particles when the applied force is reduced.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists recurring omissions and sign errors in the equations of motion; some system equations used only one particle’s mass.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Set the direction of motion, write Newton’s second-law equation for each particle with all weight, friction and tension terms, then eliminate the tension using the combined mass." }, { "id": "cie-2018-w-43-q04ii", "paperId": "cie-2018-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the cyclist’s acceleration from the stated two-stage velocity–time journey and distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_43.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part more challenging and says candidates often misread the time intervals or wrote incorrect suvat equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 45, 45 ], "tagConfidence": "reviewed", "answerGuide": "Use the area under the cyclist’s velocity–time graph to obtain the peak speed from the 96 m journey, then divide by the 10 s acceleration interval." }, { "id": "cie-2018-w-43-q07ii", "paperId": "cie-2018-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled by a particle before it reaches maximum velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_43.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates used constant-acceleration formulae for the variable-acceleration motion or integrated to the wrong time.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 46, 46 ], "tagConfidence": "reviewed", "answerGuide": "Find the time when a = 0, integrate acceleration to obtain velocity, then integrate velocity over the interval from rest to that time." }, { "id": "cie-2018-w-43-q03ii", "paperId": "cie-2018-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2018, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance travelled by a particle moving down an inclined line against resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_43.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies repeated omissions, duplicated terms and sign errors in the required work–energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 45, 45 ], "tagConfidence": "reviewed", "answerGuide": "Equate the gain in kinetic energy to the loss in gravitational potential energy minus the 25 J work against resistance, and solve for the distance along the incline." }, { "id": "cie-2018-w-51-q06ii", "paperId": "cie-2018-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the weight of a prism that is on the point of toppling under an applied force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_51.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates scored well; the report says setting up the moment equation about D proved far too difficult for many.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 48, 48 ], "tagConfidence": "reviewed", "answerGuide": "Take moments about the tipping edge D, use the perpendicular distances of the force and the weight from D, and equate clockwise and anticlockwise moments." }, { "id": "cie-2018-w-51-q07ii", "paperId": "cie-2018-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the greatest vertical height of a projectile above an inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_51.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was too difficult for many candidates and gives the required height-above-plane expression and differentiation method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 48, 48 ], "tagConfidence": "reviewed", "answerGuide": "Write the projectile’s vertical height and the plane height at horizontal displacement x, subtract to get height above the plane, then differentiate with respect to t and set the rate to zero." }, { "id": "cie-2018-w-51-q02ii", "paperId": "cie-2018-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cone’s height when a composite cone and hemisphere is on the point of toppling about A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a few candidates solved this successfully; the report identifies the required moment point and the resulting equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 47, 47 ], "tagConfidence": "reviewed", "answerGuide": "Use the centres of mass of the solid cone and hemisphere, write their weights in terms of their volumes, and balance moments about A." }, { "id": "cie-2018-w-51-q07i", "paperId": "cie-2018-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Express the projectile’s horizontal and vertical displacements in time and find when it strikes the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_51.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates recognised that impact on the 45° plane requires x = y.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 48, 48 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the launch velocity into horizontal and vertical components, write x(t) and y(t), then use the plane equation y = x for the 45° incline." }, { "id": "cie-2018-w-51-q02i", "paperId": "cie-2018-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the composite centre of mass position for a cone and hemisphere on the point of toppling about B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates found this part difficult; the report notes the geometry needed to relate the axis angle and the requested distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 47, 47 ], "tagConfidence": "reviewed", "answerGuide": "Use the component centres of mass and volume-proportional weights, then apply the limiting toppling geometry and resolve the centre-of-mass distance using the stated radii." }, { "id": "cie-2018-w-52-q06i", "paperId": "cie-2018-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a rectangle dimension in a composite lamina whose centre of mass is at O.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates solved this; the report says the centre-of-mass setup and resulting equation were extremely difficult to manipulate.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 50, 50 ], "tagConfidence": "reviewed", "answerGuide": "Use signed areas for the quarter-circle, rectangle and triangular pieces, take moments about O, and solve the resulting equation for x in terms of r." }, { "id": "cie-2018-w-52-q07ii", "paperId": "cie-2018-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction for a ring in limiting equilibrium on a rotating rough rod.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was very difficult for many; candidates needed both force resolution and Newton’s second law before using F = μR.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 50, 50 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the forces on the ring vertically and radially, use the required centripetal force equation, then apply limiting friction F = μR." }, { "id": "cie-2018-w-52-q02ii", "paperId": "cie-2018-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest applied-force multiple for which a cone-and-cylinder object will not topple.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_52.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a few candidates solved this; the report says a moment equation about a point on the base circumference was required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 49, 49 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the applied force into horizontal and vertical components, take moments about the relevant point on the circumference of the base, and impose the limiting toppling balance." }, { "id": "cie-2018-w-52-q06ii", "paperId": "cie-2018-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Decide whether the centre of mass lies within the square or quarter-circle in the alternative lamina arrangement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only a few candidates gave a satisfactory answer to this follow-up part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 50, 50 ], "tagConfidence": "reviewed", "answerGuide": "Use the centre-of-mass location from the composite-area calculation and compare its position with the boundaries of the square and quarter-circle." }, { "id": "cie-2018-w-52-q02i", "paperId": "cie-2018-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Calculate the centre-of-mass distance from the cone’s vertex for a composite cone-and-cylinder solid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_52.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report describes this part as rather difficult because the moment equation produced a complex expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 49, 49 ], "tagConfidence": "reviewed", "answerGuide": "Find the centres of mass of the cone and cylinder, weight them by their volumes, and take moments about the cone vertex to obtain the combined centre-of-mass distance." }, { "id": "cie-2018-w-53-q06ii", "paperId": "cie-2018-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the weight of a prism that is on the point of toppling under an applied force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_53.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates scored well; the report says setting up the moment equation about D proved far too difficult for many.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 52, 52 ], "tagConfidence": "reviewed", "answerGuide": "Take moments about the tipping edge D, use the perpendicular distances of the force and the weight from D, and equate clockwise and anticlockwise moments." }, { "id": "cie-2018-w-53-q07ii", "paperId": "cie-2018-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the greatest vertical height of a projectile above an inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_53.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was too difficult for many candidates and gives the required height-above-plane expression and differentiation method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 52, 52 ], "tagConfidence": "reviewed", "answerGuide": "Write the projectile’s vertical height and the plane height at horizontal displacement x, subtract to get height above the plane, then differentiate with respect to t and set the rate to zero." }, { "id": "cie-2018-w-53-q02ii", "paperId": "cie-2018-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cone’s height when a composite cone and hemisphere is on the point of toppling about A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_53.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a few candidates solved this successfully; the report identifies the required moment point and the resulting equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 51, 51 ], "tagConfidence": "reviewed", "answerGuide": "Use the centres of mass of the solid cone and hemisphere, write their weights in terms of their volumes, and balance moments about A." }, { "id": "cie-2018-w-53-q07i", "paperId": "cie-2018-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Express the projectile’s horizontal and vertical displacements in time and find when it strikes the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_53.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates recognised that impact on the 45° plane requires x = y.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 52, 52 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the launch velocity into horizontal and vertical components, write x(t) and y(t), then use the plane equation y = x for the 45° incline." }, { "id": "cie-2018-w-53-q02i", "paperId": "cie-2018-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2018, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the composite centre of mass position for a cone and hemisphere on the point of toppling about B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_53.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates found this part difficult; the report notes the geometry needed to relate the axis angle and the requested distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 51, 51 ], "tagConfidence": "reviewed", "answerGuide": "Use the component centres of mass and volume-proportional weights, then apply the limiting toppling geometry and resolve the centre-of-mass distance using the stated radii." }, { "id": "cie-2018-w-61-q06iii", "paperId": "cie-2018-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 6, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean and standard deviation from a cumulative-frequency table of rainfall data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few fully correct answers were seen; midpoint errors and stopping after the mean commonly prevented completion of the standard deviation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 55, 55 ], "tagConfidence": "reviewed", "answerGuide": "Convert cumulative frequencies into class frequencies, use class midpoints to estimate Σfx and Σfx², then calculate the grouped mean and standard deviation." }, { "id": "cie-2018-w-61-q07iv", "paperId": "cie-2018-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iv)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that three students include exactly one Music student and exactly two boys.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_61.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few solutions were fully successful; the report says candidates missed valid scenarios or treated the selections as if replacement occurred.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 55, 55 ], "tagConfidence": "reviewed", "answerGuide": "Separate the valid cases by whether the Music student is a boy or a girl, count each combination without replacement, add the cases, then divide by the number of three-student groups." }, { "id": "cie-2018-w-61-q06ii", "paperId": "cie-2018-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the cumulative-frequency graph to estimate the rainfall threshold exceeded on 100 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates interpreted the requested number of days correctly; almost all read the graph at cumulative frequency 100 instead of 150.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 55, 55 ], "tagConfidence": "reviewed", "answerGuide": "Convert “100 days or more” to cumulative frequency 250 − 100 = 150, then read the rainfall value at cumulative frequency 150 from the graph." }, { "id": "cie-2018-w-61-q03i", "paperId": "cie-2018-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count selections of six musicians meeting lower and upper limits for three instrument groups.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_61.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many solutions considered only boundary cases and omitted a valid group containing four violinists and two cellists.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 54, 54 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate all valid combinations of violinists, cellists and double-bass players that meet the “at least” and “no more than” conditions, then sum the combination counts." }, { "id": "cie-2018-w-61-q05ii", "paperId": "cie-2018-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use an appropriate approximation to find the probability that more than 166 of 200 students pass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_61.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates did not progress from part (i); common failures were omitting the continuity correction or finding the complement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 54, 55 ], "tagConfidence": "reviewed", "answerGuide": "Approximate Bin(200, 0.8) by a normal distribution with the matching mean and variance, apply the continuity correction for X ≥ 167, then standardise and find the upper-tail probability." }, { "id": "cie-2018-w-62-q07b", "paperId": "cie-2018-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find P(Y > 0) for a normal variable when its mean μ and standard deviation σ satisfy 4σ = 3μ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_62.pdf", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates found this part challenging and a significant proportion made no attempt.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 59, 59 ], "tagConfidence": "reviewed", "answerGuide": "Since σ is positive, 4σ = 3μ gives μ/σ = 4/3. Standardise 0 to obtain P(Y > 0) = Φ(4/3)." }, { "id": "cie-2018-w-62-q06iii", "paperId": "cie-2018-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for the difference of two spinner scores, given that the difference is non-zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was challenging and omitted by a significant number; many gave P(X = 1) without conditioning.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 58, 59 ], "tagConfidence": "reviewed", "answerGuide": "Use the probability distribution of X from part (i), divide P(X = 1) by P(X ≠ 0), and retain the exact conditional event." }, { "id": "cie-2018-w-63-q04ii", "paperId": "cie-2018-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a randomly chosen group of seven includes at least two girls.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_63.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was more challenging than part (i); candidates often assumed constant probability or gave the number of groups instead of a probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 62, 62 ], "tagConfidence": "reviewed", "answerGuide": "Count the favourable groups with two, three or four girls using combinations, or subtract the zero-girl and one-girl cases from all 7-person groups, then divide by the total." }, { "id": "cie-2018-w-63-q06iii", "paperId": "cie-2018-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2018, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Justify the normal approximation used for the binomial count of bulbs exceeding a lifetime threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_63.pdf", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Surprisingly few candidates understood what the question asked; only a few evaluated np and n(1 − p) and checked that both exceeded 5.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 63, 63 ], "tagConfidence": "reviewed", "answerGuide": "State the binomial parameters and evaluate np and n(1 − p); explain that both are sufficiently large for the normal approximation." }, { "id": "cie-2018-w-71-q07ii", "paperId": "cie-2018-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the second one-sided test of the mean sack mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_71.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists frequent reversals of the two means, missing sample-size scaling, wrong-tail signs and omission of the first critical-value stage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 67, 68 ], "tagConfidence": "reviewed", "answerGuide": "Use the null model to find the rejection boundary at the stated significance level, then standardise that boundary under the alternative mean of 49 kg using the known population standard deviation and sample size." }, { "id": "cie-2018-w-72-q05ii", "paperId": "cie-2018-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the South Mowland mean number of courts is below the North Mowland mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_72.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report explicitly calls this part challenging and notes missing hypotheses, incorrect test statistics, and unclear comparisons.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 70, 71 ], "tagConfidence": "reviewed", "answerGuide": "State H₀ and H₁, calculate the test statistic using the sample variance and √70, compare it with the critical value, then give a non-definite conclusion in context." }, { "id": "cie-2018-w-72-q07ii", "paperId": "cie-2018-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find P(X = 2) conditional on X + Y = 3 for independent Poisson variables.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_72.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not recognise the need for a conditional probability and few attempted the required calculation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 71, 71 ], "tagConfidence": "reviewed", "answerGuide": "Calculate P(X = 2, Y = 1) and divide by P(X + Y = 3), or use the conditional distribution of X given the total." }, { "id": "cie-2018-w-72-q06i", "paperId": "cie-2018-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the timetable change reduced the proportion of late journeys.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_72.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often omitted or misstated hypotheses and failed to justify their conclusion with a clear comparison.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 71, 71 ], "tagConfidence": "reviewed", "answerGuide": "State the hypotheses for p, use the binomial model or the stated normal approximation, compare the one-sided result with the 10% critical value, and conclude in context." }, { "id": "cie-2018-w-72-q02ii", "paperId": "cie-2018-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain why the sampling distribution can be treated as approximately normal for the stated confidence interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report explicitly says this part proved challenging because responses did not clearly identify that the population was not stated to be normal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 69, 70 ], "tagConfidence": "reviewed", "answerGuide": "Identify that normality is not assumed for the population and explain that the large sample size allows the central limit theorem to be used for the sample mean." }, { "id": "cie-2018-w-73-q07ii", "paperId": "cie-2018-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2018, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the second one-sided test of the mean sack mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_ms_73.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists frequent reversals of the two means, missing sample-size scaling, wrong-tail signs and omission of the first critical-value stage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w18_er.pdf", "difficultyReportPages": [ 74, 75 ], "tagConfidence": "reviewed", "answerGuide": "Use the null model to find the rejection boundary at the stated significance level, then standardise that boundary under the alternative mean of 49 kg using the known population standard deviation and sample size." }, { "id": "cie-2017-m-12-q10iii", "paperId": "cie-2017-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the x-coordinates where the curve crosses the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_12.pdf", "qpPages": [ 18, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says part (iii) was considerably less well done; candidates often did not multiply through by x² and treat the resulting quartic as a quadratic in x², and arithmetic errors were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the function found in part (ii), set it equal to zero, multiply through by x² to obtain a quadratic in x², then solve and retain the positive x-values because the curve is defined for x > 0." }, { "id": "cie-2017-m-12-q05ii", "paperId": "cie-2017-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the coordinates of the second intersection B of y = tan x and y = cos x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_12.pdf", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says part (ii) was less successful than part (i); candidates struggled with accuracy, radians and identifying a strategy for the coordinates of B.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the equation tan x = cos x to find the second intersection in the interval shown, keep x in radians, and substitute into either curve to obtain the y-coordinate." }, { "id": "cie-2017-m-12-q08ii", "paperId": "cie-2017-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of fg and determine its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the inverse domain was often not attempted and a common wrong answer was x ≥ 3.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form fg(x) using the unsimplified composite from part (i), rearrange y = fg(x) for x, and use the minimum value of fg on its original domain to state the inverse domain." }, { "id": "cie-2017-m-12-q06i", "paperId": "cie-2017-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the angle OAB between OA and AB using the given position vectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says a substantial number found angle AOB instead; among the rest, at least half used the wrong scalar-product vectors or reported an acute angle instead of the obtuse angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find vector BA or AB as required to construct both sides of angle OAB, then use the scalar product and select the obtuse angle at A." }, { "id": "cie-2017-m-12-q04ii", "paperId": "cie-2017-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of the shaded region formed by the two circular arcs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_12.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that some candidates found the shaded area instead of the requested perimeter and that premature rounding of BC often caused an inaccurate final answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the exact radian angle from part (i) to obtain the relevant arc lengths, find the straight or curved boundary lengths from the diagram, and add the perimeter components rather than finding the shaded area." }, { "id": "cie-2017-m-22-q02ii", "paperId": "cie-2017-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the result from part (i) to solve the trigonometric equation for 0° < θ < 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says correct solutions were few and candidates often failed to use the result from part (i) or omitted the second solution in the interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use tan²θ = 3/4, account for both positive and negative tangent values in the interval, and give both angles to one decimal place." }, { "id": "cie-2017-m-22-q03ii", "paperId": "cie-2017-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the largest integer y satisfying the linked inequality in ln y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often did not connect this part to the upper bound obtained in part (i), which was needed to find y.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the interval for x = ln y found in part (i), exponentiate its upper bound, and take the greatest integer allowed by the strict inequality." }, { "id": "cie-2017-m-22-q07ii", "paperId": "cie-2017-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of the shaded region under the trigonometric curve and above the axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates needed to find the curve's x-intercept to obtain the limits for the exact-area integral.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the x-intercept to set the integration limits, rewrite the product with the compound-angle result from part (i), and evaluate the definite integral exactly." }, { "id": "cie-2017-m-22-q04", "paperId": "cie-2017-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the implicit curve at the specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most candidates knew implicit differentiation was required, but simplification and substitution errors often prevented the correct gradient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x² sin y + cos 3y = 4 implicitly with respect to x, collect the dy/dx terms, then substitute the point and simplify carefully." }, { "id": "cie-2017-m-22-q05ii", "paperId": "cie-2017-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use an iterative formula to find a to 4 significant figures, showing each iteration to 6 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says some candidates did not attempt this linked part despite being told to use part (i), and some lost the final accuracy mark; iterations otherwise were usually at the required precision.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the rearrangement from part (i) into the stated iteration, continue until the value is stable to 4 significant figures, and show the requested 6-significant-figure iterates." }, { "id": "cie-2017-m-32-q05", "paperId": "cie-2017-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "differentiation", "trigonometry", "proof" ], "primaryTopicId": "differentiation", "summary": "Find a so the curve has exactly one horizontal tangent in the stated interval, and give its x-coordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 4, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates could not progress far; very few recognised that exactly one point requires a repeated root and therefore a zero discriminant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate e^(−ax) tan x, set the derivative to zero, rewrite the condition as a quadratic in tan x, and use the repeated-root condition to determine a and x." }, { "id": "cie-2017-m-32-q08ii", "paperId": "cie-2017-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 7, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Find the other three roots of the quartic with real coefficients, given one complex root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was difficult; many candidates failed to use the conjugate root, and the polynomial division and quadratic solution needed to be shown in detail without calculator use.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the conjugate of −1 + i as another root, form the real quadratic factor from the conjugate pair, divide the quartic by it with working shown, then solve the remaining quadratic without a calculator." }, { "id": "cie-2017-m-32-q06ii", "paperId": "cie-2017-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a plane parallel to a given line, perpendicular to another plane, and through a specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often assumed the required normal was perpendicular to the line direction and the point's position vector, instead of to the line direction and the original plane's normal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find a normal vector perpendicular to both the line's direction vector and the given plane's normal, then use the specified point to determine the constant in ax + by + cz = d." }, { "id": "cie-2017-m-32-q07ii", "paperId": "cie-2017-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Evaluate the time integral using the substitution u = 5 − √h.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says limits were often reversed or left unchanged after substitution, and candidates commonly omitted the constant 10 in the antiderivative.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Change both the integrand and the limits consistently under u = 5 − √h, integrate the resulting terms including the constant contribution, and substitute the transformed limits in the correct order." }, { "id": "cie-2017-m-42-q06ii", "paperId": "cie-2017-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 8, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled by the 0.8 kg particle in the first second, including motion after the heavier particle reaches the floor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was difficult; many candidates incorrectly assumed the lighter particle continued upward for the entire remaining 0.2 seconds instead of splitting the motion when it comes to rest.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First find the time and speed when the 1.2 kg particle hits the floor. Then split the remaining motion into the upward deceleration to rest and the final fall under gravity, and add all three distances." }, { "id": "cie-2017-m-42-q05iii", "paperId": "cie-2017-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled from the piecewise velocity function over 35 seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates who treated all three parts of the velocity-time graph as straight lines incorrectly used triangles and trapezia instead of integrating the two quadratic sections.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity on each of the three time intervals, using the constants from part (i); the middle interval can also be evaluated as a trapezium, but the two quadratic sections are curved areas." }, { "id": "cie-2017-m-42-q04iii", "paperId": "cie-2017-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the car's deceleration on CD and the resistance R as it comes to rest at D.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates tried to use the full distance AD directly; they first needed to find CD before calculating the deceleration and resistance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find AB and BC from earlier parts, use AD to determine CD, apply a constant-acceleration equation to the speed at C and stopping distance, then use Newton's second law to find R." }, { "id": "cie-2017-m-42-q03", "paperId": "cie-2017-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the least possible force P and find the greatest possible force that keeps the particle in equilibrium on the rough inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_42.pdf", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says some candidates considered only one of the two limiting cases; the greatest and least possible values require opposite directions for limiting friction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel to and perpendicular to the plane, use R = 0.6g cos 21°, and consider both limiting cases: friction acts up the plane for the least P and down the plane for the greatest P." }, { "id": "cie-2017-m-52-q06ii", "paperId": "cie-2017-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Find the possible initial speeds U given that the particle stops between x = 2.0 m and x = 2.1 m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was very difficult; candidates often integrated v instead of v²/2 and struggled to integrate 2/x before applying both stopping-position bounds.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate v dv/dx = −4 − 2/x from x = 1 to the stopping point, use v = U initially and v = 0 at rest, then evaluate the bounds at both ends of the stated interval." }, { "id": "cie-2017-m-52-q07ii", "paperId": "cie-2017-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find how far the particle moves from equilibrium before it first comes to rest after being projected downwards.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part required a four-term energy equation, while candidates commonly included only three terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply conservation of energy between the equilibrium position and the first turning point, including initial kinetic energy, change in gravitational potential energy, and elastic potential energy at the final extension." }, { "id": "cie-2017-m-52-q07iii", "paperId": "cie-2017-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed after a second particle attaches at the first turning point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates needed the new equilibrium extension and another four-term energy equation; many included only three terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the new equilibrium extension for the combined mass, then use an energy balance from the attachment position to the new equilibrium to determine the maximum speed." }, { "id": "cie-2017-m-52-q04i", "paperId": "cie-2017-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of the uniform lamina measured from AB and BC.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates often struggled to find the centre of mass from both AB and BC, commonly using incorrect distances when taking moments.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the lamina into simple uniform shapes, calculate each area and centroid location from the specified sides, and use area-weighted moments for both coordinates." }, { "id": "cie-2017-m-62-q05ii", "paperId": "cie-2017-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements of seven distinct cupcakes and four distinct brownies with no two brownies adjacent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the most successful solutions ordered the cupcakes first and inserted brownies in the gaps; the alternative subtractive approach was much more complex and often mishandled arrangements that failed the condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Arrange the seven cupcakes first to create eight gaps, choose four different gaps for the brownies, and arrange the four distinct brownies in those gaps." }, { "id": "cie-2017-m-62-q02", "paperId": "cie-2017-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly three of seven balloons selected without replacement are green.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says some candidates considered only three green balloons rather than a total of seven, while others assumed replacement or omitted the combinations needed for different selections.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a hypergeometric count: choose three of the 12 green balloons and four of the 28 non-green balloons, then divide by the number of ways to choose any seven from all 40 balloons." }, { "id": "cie-2017-m-62-q01", "paperId": "cie-2017-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the mean and standard deviation of x − 1760, then use them to find the mean and standard deviation of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates calculated the original mean and standard deviation directly or used circular reasoning instead of identifying the coded mean; some also incorrectly shifted the standard deviation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the coded values x − 1760 first; add 1760 to their mean to recover the mean of x, while the standard deviation remains unchanged by the shift." }, { "id": "cie-2017-m-62-q04ii", "paperId": "cie-2017-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Name a suitable distribution model for the cereal packet weights and justify the choice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates fully justified a normal model; the explanation needed to mention that the data was approximately symmetrical and peaked around the median.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State the normal distribution and justify it from the approximately symmetric shape of the displayed data, with values concentrated around the median." }, { "id": "cie-2017-m-62-q07b", "paperId": "cie-2017-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a probability using a normal variable whose mean equals its standard deviation, with the cutoff at 1.5 times the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates needed to state the final probability after standardising the cutoff 1.5μ; some stopped after finding the z-score.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Let both the mean and standard deviation be μ, standardise X < 1.5μ to obtain the corresponding z-value, then use the standard normal distribution." }, { "id": "cie-2017-m-72-q02ii", "paperId": "cie-2017-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use the test statistic to state the conclusion of the two-tailed 1% test for the insect lifespan mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says conclusions were sometimes given without a valid comparison and required a contextual statement rather than simply accepting the null hypothesis.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare |z| = 2.43 with the two-tailed 1% critical value, then state in context whether there is sufficient evidence that the population mean differs from 6.4 weeks." }, { "id": "cie-2017-m-72-q05aii", "paperId": "cie-2017-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Explain why each of two displayed functions cannot be a probability density function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_72.pdf", "qpPages": [ 6, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates clearly explained why f2 and f3 were invalid; some showed little understanding of the defining properties of a probability density function.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Check both defining conditions: a density cannot be negative, and its total area must equal one. Identify which condition each graph violates." }, { "id": "cie-2017-m-72-q06i", "paperId": "cie-2017-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the percentage of standard crane loads that exceed the safe limit, stating a necessary assumption.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_72.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few candidates stated a necessary assumption, and variance calculations often incorrectly squared the products of carton counts and standard deviations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Assume individual carton masses are independent so the sum is normal; add the means and variances of eight sugar and three flour cartons, then find the upper-tail probability above 900 kg." }, { "id": "cie-2017-m-72-q03ii", "paperId": "cie-2017-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Explain why the Central Limit Theorem was needed to model the sample mean of snake lengths.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_72.pdf", "qpPages": [ 4, 4 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says answers showed a lack of understanding; vague statements that the population was not normally distributed or that it needed to be normal were insufficient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the population distribution of individual snake lengths is not specified as normal, so the large sample size is needed for the sample mean to be approximately normal." }, { "id": "cie-2017-m-72-q07i", "paperId": "cie-2017-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "State two assumptions required for a Poisson model of daytime plane arrivals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_72.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says the two assumptions for the Poisson model were often not stated in the context; textbook definitions without the airport setting were not accepted.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that planes arrive at a constant mean rate during the period and that arrivals occur randomly and independently, expressed in the airport context." }, { "id": "cie-2017-m-72-q02iii", "paperId": "cie-2017-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(iii)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Explain when a one-tailed test is appropriate instead of a two-tailed test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says candidates were required to explain when a Type I error one-tail test would be used, and many did not give a full answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a one-tailed test when the research question specifies a preplanned increase or decrease in the population mean, rather than any change in either direction." }, { "id": "cie-2017-s-11-q06i", "paperId": "cie-2017-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Express the prism height in terms of x and establish the stated total surface-area formula.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_11.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this unexpectedly difficult and says some candidates found h but omitted faces; completing the algebra for all five faces required strong reasoning.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the triangular prism, calculate the base area and height, account for all five faces, then simplify to the required expression." }, { "id": "cie-2017-s-11-q10ii", "paperId": "cie-2017-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the shaded region about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_11.pdf", "qpPages": [ 18, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says integration produced fewer correct solutions and that an attempt to integrate y² had to be shown for credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the disc method with π times the square of the curve, integrate over the shaded interval, and show the lower-limit substitution as well as the upper limit." }, { "id": "cie-2017-s-12-q03i", "paperId": "cie-2017-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the trigonometric identity by simplifying its left-hand side.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_12.pdf", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a real challenge for many; poor expansion and difficulty factorising the numerator into the required factors were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a common denominator, expand the numerator correctly, apply a suitable trigonometric identity, and factor the result into the required form." }, { "id": "cie-2017-s-12-q06", "paperId": "cie-2017-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume generated by rotating the region between the line and curve about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 4, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number found the volume-between-two-curves method difficult; common errors included subtracting before squaring or adding the volumes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the two squared radii over the intersection interval and subtract the inner volume from the outer volume; do not square the difference of the curves." }, { "id": "cie-2017-s-12-q08ii", "paperId": "cie-2017-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find p and q so that AB is parallel to OC.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says part (ii) proved much more difficult; candidates often mishandled signs, equated lengths, or failed to use all component relationships implied by parallel vectors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form AB as OB−OA, set it equal to a scalar multiple of OC, then equate all three components and solve consistently for p and q." }, { "id": "cie-2017-s-12-q10iii", "paperId": "cie-2017-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch the function and its inverse on the same axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only the very best candidates obtained full marks; many extended the tangent graph beyond the restricted domain or failed to draw its correct shape.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the original function only on its one-to-one domain, then reflect that branch in y=x and preserve the matching endpoints and asymptotic behaviour." }, { "id": "cie-2017-s-13-q02ii", "paperId": "cie-2017-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the set of possible values of S from the stated restriction on r.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was not done well because candidates often failed to use the restriction on r when finding S=2−r.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the relation S=2−r and translate the allowed values of r into the corresponding interval for S, preserving strict endpoints." }, { "id": "cie-2017-s-13-q09i", "paperId": "cie-2017-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Express the quadratic in the required completed-square form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_13.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes that many did not present the expression in the required form and could lose up to two marks for this.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Factor the leading coefficient into a squared bracket, match the x and constant terms, and state the resulting constants clearly." }, { "id": "cie-2017-s-13-q11iii", "paperId": "cie-2017-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(iii)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Recover f(x) from its derivative and find the minimum value of f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_13.pdf", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often failed to carry constants of integration through the earlier parts; the report also notes errors using f(0) and x=2 to find the minimum value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate f′(x), use the value of f(0) found in part (ii) to determine the constant, then evaluate f(2) for the stated minimum." }, { "id": "cie-2017-s-21-q04ii", "paperId": "cie-2017-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Determine the exact value of α from the fixed-point relation for the iteration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates knew how to obtain the exact value; many continued iterating instead of solving the fixed-point relation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set α equal to the iteration formula with each xₙ replaced by α, clear the denominator, solve the resulting equation and select the root consistent with the given convergence." }, { "id": "cie-2017-s-21-q06ii", "paperId": "cie-2017-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact volume formed by rotating the area under y=tan 2x about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_21.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates could not make meaningful progress with the integration; only a few produced the correct exact volume.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the disc method, π∫tan²(2x) dx, over the interval shown, and evaluate the integral and limits exactly." }, { "id": "cie-2017-s-21-q07ii", "paperId": "cie-2017-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates where the specified line is the curve’s normal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_21.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Most candidates identified the gradient relationship but could not proceed to determine the point.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the normal’s gradient to obtain the tangent gradient, differentiate the curve, solve for the point parameter, and verify it lies on both the curve and line." }, { "id": "cie-2017-s-21-q08ii", "paperId": "cie-2017-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of the minimum point M on the logarithmic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_21.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates found the exact coordinates of M; errors in the logarithmic differentiation and solving for the stationary point were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the product of x² and the logarithm, set the derivative to zero to find the positive x-coordinate, and substitute it into the curve for y." }, { "id": "cie-2017-s-22-q05", "paperId": "cie-2017-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Find K and a in y=K/a^(2x) from the straight-line graph of ln y against x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 4, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as probably the most demanding question; many read the gradient as ln a rather than 2 ln a and then found K incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain ln y=ln K−2x ln a, use the gradient to find a, then use a point to determine K and round both values as instructed." }, { "id": "cie-2017-s-22-q08i", "paperId": "cie-2017-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric curve to establish the stated dy/dx expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates found dy/dt correctly, and many struggled with the powers in the trigonometric expressions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y separately with respect to t, form (dy/dt)/(dx/dt), and simplify the trigonometric powers carefully." }, { "id": "cie-2017-s-22-q08iii", "paperId": "cie-2017-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradient of the normal at the point where x=2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not obtain the exact answer and some gave the tangent gradient instead of the normal gradient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the parameter value corresponding to x=2, evaluate dy/dx there using the earlier result, and take the negative reciprocal for the normal gradient." }, { "id": "cie-2017-s-23-q05", "paperId": "cie-2017-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Find K and a in y=K/a^(2x) from the straight-line graph of ln y against x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 4, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as probably the most demanding question; many read the gradient as ln a rather than 2 ln a and then found K incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain ln y=ln K−2x ln a, use the gradient to find a, then use a point to determine K and round both values as instructed." }, { "id": "cie-2017-s-23-q08i", "paperId": "cie-2017-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric curve to establish the stated dy/dx expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_23.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates found dy/dt correctly, and many struggled with the powers in the trigonometric expressions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y separately with respect to t, form (dy/dt)/(dx/dt), and simplify the trigonometric powers carefully." }, { "id": "cie-2017-s-23-q08iii", "paperId": "cie-2017-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradient of the normal at the point where x=2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_23.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not obtain the exact answer and some gave the tangent gradient instead of the normal gradient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the parameter value corresponding to x=2, evaluate dy/dx there using the earlier result, and take the negative reciprocal for the normal gradient." }, { "id": "cie-2017-s-31-q06ii", "paperId": "cie-2017-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane parallel to the given plane and one unit closer to the origin.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_31.pdf", "qpPages": [ 10, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this one of the paper’s most difficult questions; many candidates did not use the normal-vector and plane-distance approach needed to progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Keep the given plane’s normal vector, use the parallel-plane distance formula to determine the new constant, and choose the plane on the side nearer the origin." }, { "id": "cie-2017-s-31-q07ii", "paperId": "cie-2017-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Prove the stated angle between OA and OB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_31.pdf", "qpPages": [ 12, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates recognised that the angle could be obtained as the argument of the ratio of the complex numbers.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the relevant complex ratio in modulus-argument form and use its argument to identify the directed angle, accounting for the correct quadrant." }, { "id": "cie-2017-s-31-q10i", "paperId": "cie-2017-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use u=cos x to find the exact area bounded by the curve and x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_31.pdf", "qpPages": [ 18, 19 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says fully correct solutions were unusual; candidates struggled with the trigonometric substitution and exact limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the specified substitution, transform the integrand and limits consistently, then evaluate the resulting exact integral over the shaded interval." }, { "id": "cie-2017-s-32-q05ii", "paperId": "cie-2017-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find the limiting mass of B and state the long-term behaviour of the mass of A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a few candidates gave both required limiting statements; vague claims about A tending to zero were not enough without a valid explanation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the linked differential-equation model to determine the steady-state values, and explain what happens to A as time increases without bound." }, { "id": "cie-2017-s-32-q06i", "paperId": "cie-2017-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the real constants a and b using the stated complex root of a cubic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many found the complex roots, but the simultaneous-equation route needed to determine a and b was rarely successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the given complex number into the cubic, expand its powers, and equate real and imaginary parts to solve for a and b." }, { "id": "cie-2017-s-32-q09ii", "paperId": "cie-2017-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane through the origin containing line l.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority identified the two directions needed to define the plane, and very few recognised that a plane through the origin has constant term zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the line’s direction and a second independent vector in the plane to obtain a normal vector, then use the origin to set the plane constant to zero." }, { "id": "cie-2017-s-33-q05ii", "paperId": "cie-2017-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate where the curve has gradient −1, to 3 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_33.pdf", "qpPages": [ 6, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few candidates formed a suitable equation, and few of those who did solved the resulting three-term quadratic correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set the derivative equal to −1, rearrange to a solvable quadratic, and select the solution in the stated domain." }, { "id": "cie-2017-s-33-q06ii", "paperId": "cie-2017-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show that any convergent sequence generated by the iteration has the stated limit.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_33.pdf", "qpPages": [ 8, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates had difficulty resolving the addition or removal of π in the inverse-tangent relation, and few completed the argument adequately.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the limit equal to the right-hand side of the iteration, use the inverse-tangent principal range to identify the correct fixed point, and establish convergence on the given interval." }, { "id": "cie-2017-s-33-q07iii", "paperId": "cie-2017-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 1, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Explain whether the four-interval trapezium estimate is greater or less than E, the two-interval estimate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_33.pdf", "qpPages": [ 10, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates could explain which estimate would be greater; many omitted the part or guessed without using the curve’s shape.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the convex shape of y=eˣ/x on the interval: both trapezium estimates lie above the curve, and the narrower four-interval trapezia give a smaller estimate than E." }, { "id": "cie-2017-s-33-q11a", "paperId": "cie-2017-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the simultaneous equations for z and w and give each in x+iy form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_33.pdf", "qpPages": [ 18, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most candidates did not perform well and only a minority completed this complex simultaneous-equation part for full marks.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Eliminate one unknown, solve the resulting complex equation, substitute back for the other value, and separate real and imaginary parts." }, { "id": "cie-2017-s-41-q01", "paperId": "cie-2017-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the work done against air resistance during the particle’s fall.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many assumed air resistance was a constant force and used Newton’s second law; full marks required the work-energy principle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply conservation of energy: compare the loss in gravitational potential energy with the gain in kinetic energy; their difference is work against resistance." }, { "id": "cie-2017-s-42-q03iii", "paperId": "cie-2017-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(iii)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the minimum distance between the two particles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_42.pdf", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates successfully completed this minimum-distance part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express their separation as a function of time, differentiate to find when the separation is stationary, and check that this gives the minimum distance." }, { "id": "cie-2017-s-42-q05", "paperId": "cie-2017-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 8, "topicIds": [ "mechanics", "proof" ], "primaryTopicId": "mechanics", "summary": "Find the range of force magnitudes P that can maintain equilibrium on the rough plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates considered only one limiting case; common errors included using the wrong reaction force and failing to state the complete range.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the plane, use the limiting-friction cases for impending motion in each direction, and combine them into an interval for P." }, { "id": "cie-2017-s-43-q05i", "paperId": "cie-2017-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the time required in the two-particle projection scenario.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_43.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was more challenging because of the two-second delay between the projections.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write each particle’s displacement using its own elapsed time, account for the two-second delay between projections, and solve the stated meeting condition." }, { "id": "cie-2017-s-43-q07ii", "paperId": "cie-2017-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 8, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "For m=3, find the total distance A travels before it comes momentarily to rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_43.pdf", "qpPages": [ 10, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies the final question as challenging, with two stages of motion and force equations needed to complete it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the first stage while the string is taut, identify when the other particle reaches the ground, then model A’s second stage separately and add the distances." }, { "id": "cie-2017-s-51-q02ii", "paperId": "cie-2017-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find m from the particle’s motion after the horizontal force is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this energy-equation part very difficult.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the change in elastic potential energy to the change in gravitational potential energy between release and the next instant of rest, using the initial and final spring extensions." }, { "id": "cie-2017-s-51-q03ii", "paperId": "cie-2017-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Calculate the weight of H from the composite centre-of-mass setup.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_51.pdf", "qpPages": [ 6, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates obtained the correct weight; the report describes the centre-of-mass calculation as difficult.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments or use the centre-of-mass formula for the complete system, including the stated components, then solve for H." }, { "id": "cie-2017-s-51-q05ii", "paperId": "cie-2017-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle with the horizontal of the force exerted by the pivot at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_51.pdf", "qpPages": [ 10, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates found the correct angle; the report highlights difficulty resolving the reaction force at the peg or pivot.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the equilibrium equations for the lamina, resolve the pivot reaction into horizontal and vertical components, and obtain its direction from their ratio." }, { "id": "cie-2017-s-52-q03ii", "paperId": "cie-2017-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the lid angle when the box system’s centre of mass reaches the stated height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a very difficult question and says very few candidates found the distance of the lid’s centre of mass correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the combined centre-of-mass height in terms of the lid angle using the lid and box geometry, then solve the given height condition." }, { "id": "cie-2017-s-52-q04ii", "paperId": "cie-2017-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "integration", "kinematics" ], "primaryTopicId": "integration", "summary": "Express v in terms of x using the given differential equation for dv/dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Too many candidates did not recognise that the separated integral produced a logarithmic form.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables in dv/dx=10−0.5v², use partial fractions to integrate the v-expression, then apply the initial condition to obtain v as a function of x." }, { "id": "cie-2017-s-53-q02ii", "paperId": "cie-2017-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find m from the particle’s motion after the horizontal force is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_53.pdf", "qpPages": [ 4, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates found this energy-equation part very difficult.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the change in elastic potential energy to the change in gravitational potential energy between release and the next instant of rest, using the initial and final spring extensions." }, { "id": "cie-2017-s-53-q03ii", "paperId": "cie-2017-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Calculate the weight of H from the composite centre-of-mass setup.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_53.pdf", "qpPages": [ 6, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates obtained the correct weight; the report describes the centre-of-mass calculation as difficult.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments or use the centre-of-mass formula for the complete system, including the stated components, then solve for H." }, { "id": "cie-2017-s-53-q05ii", "paperId": "cie-2017-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle with the horizontal of the force exerted by the pivot at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_53.pdf", "qpPages": [ 10, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates found the correct angle; the report highlights difficulty resolving the reaction force at the peg or pivot.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the equilibrium equations for the lamina, resolve the pivot reaction into horizontal and vertical components, and obtain its direction from their ratio." }, { "id": "cie-2017-s-61-q01", "paperId": "cie-2017-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the coded totals and known mean to find k, then find the standard deviation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes this as one of the worst-attempted questions; candidates struggled to move between coded and uncoded mean and variance expressions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the coded first-moment sum into the ordinary total to solve for k, then expand the coded squared-deviation total carefully and use the variance formula before taking the square root." }, { "id": "cie-2017-s-62-q04", "paperId": "cie-2017-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Determine p, q and r from the score probabilities for two identical spinners.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 5, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this unstructured probability question was challenging; stronger solutions organised the cases with a tree diagram and algebra.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the score-6 and score-5 events to form probability equations, combine them with p+q+r=1, and keep only valid non-negative probabilities." }, { "id": "cie-2017-s-62-q06", "paperId": "cie-2017-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements of the repeated books for the stated end-book condition and for the conditions that all A books are together while no B books are adjacent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates were confused by the identical-book context and showed weak understanding of how to count the allowed arrangements.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat identical copies as indistinguishable; count the allowed end cases separately in part (i), then treat the A block as one unit and place the B books with gaps in part (ii). Avoid overcounting repeated books." }, { "id": "cie-2017-s-63-q05i", "paperId": "cie-2017-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "State two conditions required for a binomial distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_63.pdf", "qpPages": [ 6, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates stated both required conditions; most misunderstood the independence requirement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that there is a fixed number of independent trials and that each trial has two outcomes with the same probability of success." }, { "id": "cie-2017-s-63-q06bii", "paperId": "cie-2017-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that fewer than three chocolate biscuits are selected from the six chosen biscuits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls the permutations-and-combinations question the paper’s most challenging; many overcomplicated the count and omitted the final probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the favourable selections with zero, one or two chocolate biscuits using combinations, add those cases, and divide by the total number of six-biscuit selections." }, { "id": "cie-2017-s-71-q03b", "paperId": "cie-2017-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Find the probability that all three independent confidence intervals lie entirely above the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_71.pdf", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates correctly answered this confidence-interval probability part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For one two-sided 92% confidence interval, identify the upper-tail probability of lying wholly above the true mean, then raise it to the third power for three independent samples." }, { "id": "cie-2017-s-71-q06i", "paperId": "cie-2017-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region and the probability of a Type I error for the test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_71.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 6 was not well attempted; the report says candidates needed the cumulative probabilities for X≤1 and X≤2 and had to justify the critical region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the null distribution’s cumulative probabilities to choose the most extreme outcomes that keep the significance level within bounds, then state the actual Type I error probability." }, { "id": "cie-2017-s-72-q02ii", "paperId": "cie-2017-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Explain why the results in part (i) agree with a Poisson approximation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was not well attempted; candidates often ignored the link to part (i) and failed to explain why the mean and variance were approximately equal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Connect the mean and variance found in part (i) to the Poisson property that these quantities are equal, allowing for approximation and rounding." }, { "id": "cie-2017-s-72-q05i", "paperId": "cie-2017-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find P(X<2), the parameter a and the density function f(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_72.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The density-function question proved challenging for many; candidates made repeated false starts and often obtained invalid probabilities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the total-area condition for a probability density to find a, integrate the relevant density interval for P(X<2), and state the piecewise density accurately." }, { "id": "cie-2017-s-73-q07iii", "paperId": "cie-2017-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for a new sample under the changed mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_ms_73.pdf", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the paper’s most challenging part; candidates needed to use the earlier critical region and calculate the Type II error under the new mean.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s17_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Carry forward the critical region from the earlier part, identify the acceptance outcomes, and evaluate their probability under the new distribution and mean." }, { "id": "cie-2016-w-11-q10ii", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the derivative and use a point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a few candidates who completed part (i) successfully could finish this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate f′(x), include the constant of integration, use the given point to find it, and substitute into the required curve relation." }, { "id": "cie-2016-w-11-q10i", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the derived equation and reject any solution introduced by squaring.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the extra solution x = 1 introduced by squaring was rarely discarded.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Square only as required, solve the resulting quadratic, and check each candidate in the original equation before accepting it." }, { "id": "cie-2016-w-11-q08ii", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the inverse function and state its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Correct expressions for the domain were rare; only a few candidates realised that x = 0 could not be included.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the inverse algebraically, then use the range of the original function to state the inverse domain with the correct endpoints." }, { "id": "cie-2016-w-11-q07ii", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the area under the curve y² = 1 − 2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says integrating (1 − 2x)^(1/2) proved challenging, especially the division by 3/2.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stated limits and integrate the appropriate positive square-root branch, handling the factor from the substitution correctly." }, { "id": "cie-2016-w-11-q03ii", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the sector areas to find the required angle in radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Some candidates treated the shaded region as a sector; converting to degrees rarely succeeded, and errors in squaring 2α prevented correct answers.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the difference between the sector areas equal to the given value, keep the angle in radians, and square the full angle expression carefully." }, { "id": "cie-2016-w-11-q05", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use two geometric-series sums to determine the required parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says the formula for the sum of n terms was often seen but very rarely successful; poor algebra also spoiled otherwise promising work.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express each stated sum using α and r, use separate terms to eliminate α, then solve the resulting relation accurately." }, { "id": "cie-2016-w-11-q11ii", "paperId": "cie-2016-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(ii)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Classify the stationary point at x = 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 2, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says second-derivative or sign-table methods identified the turning point; using the first derivative at x = 1 did not establish its nature.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the second derivative or a sign table for the first derivative on either side of x = 2 to determine its nature." }, { "id": "cie-2016-w-12-q10iv", "paperId": "cie-2016-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iv)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Restrict the domain so the function has an inverse and state the inverse relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This was the worst-done part of the paper; only the best candidates recognised that the function had to be restricted to be one-to-one.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose an interval on which the function is one-to-one, sketch or describe that branch, and express the inverse consistently with that restriction." }, { "id": "cie-2016-w-12-q07iv", "paperId": "cie-2016-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Use the chain rule while accounting for the decreasing x-coordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says most candidates who used the chain rule failed to account for the fact that the x-coordinate was decreasing.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the function with respect to x, multiply by dx/dt, and use the negative value of dx/dt because x is decreasing." }, { "id": "cie-2016-w-12-q04", "paperId": "cie-2016-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Determine the unknown binomial parameters from the specified coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates failed to see that both the x and x² coefficients came from two terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify every expansion term contributing to each stated coefficient, form both coefficient equations, and solve for the integer parameters." }, { "id": "cie-2016-w-12-q02ii", "paperId": "cie-2016-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find both trigonometric solutions in the required angular range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates correctly identified the fourth-quadrant solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the inverse tangent result and check the signs and quadrants so both the first- and fourth-quadrant solutions are included." }, { "id": "cie-2016-w-12-q05", "paperId": "cie-2016-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the line parameters using all three geometric conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates found this question difficult and often ignored the midpoint condition, attempting only to solve the two given line equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use both line equations and the condition on the midpoint of AB to form the required simultaneous equations." }, { "id": "cie-2016-w-12-q06iii", "paperId": "cie-2016-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the complete shaded area made from the sector, triangle and semicircle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 8 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this part was challenging because many candidates did not include the whole required area.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the sector, triangle and semicircle areas and combine all three using the correct angle for the major sector." }, { "id": "cie-2016-w-13-q06ii", "paperId": "cie-2016-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the line parameters from the perpendicular-gradient and point conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes this as a challenging question; long methods often failed, while the direct simultaneous-equation method was most effective.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the gradient condition to form one equation in the parameters and substitute the point coordinates into the second line condition to form another." }, { "id": "cie-2016-w-13-q04", "paperId": "cie-2016-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the least value of n for which the derivative satisfies the stated condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found the critical values but then gave the wrong least n, often selecting −1 or answering 4.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find and interpret the critical x-values, then use the condition on the least value of n rather than selecting the wrong critical value." }, { "id": "cie-2016-w-13-q01", "paperId": "cie-2016-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the discriminant inequality for k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates found the discriminant but gave an incorrect region for k, including the interval between the critical values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Eliminate y, form the quadratic in x, set the discriminant condition, and solve the resulting inequality in both regions." }, { "id": "cie-2016-w-13-q11i", "paperId": "cie-2016-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(i)", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the stationary values and retain both solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says a disappointing number of candidates forgot the ± sign after reaching the squared equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate both terms carefully, solve the resulting squared equation and take both signs before determining the stationary values." }, { "id": "cie-2016-w-13-q05ii", "paperId": "cie-2016-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the area using both the minor and major sectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Some candidates were uncomfortable using 2π − 1.8 and consequently used indirect methods that often did not reach the correct area.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the angle 1.8 for the minor sector and 2π − 1.8 for the major sector, then subtract or add the sector areas as required." }, { "id": "cie-2016-w-21-q01i", "paperId": "cie-2016-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(i)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the equation by treating it as a quadratic in 3ˣ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates recognised that the equation was quadratic in 3ˣ or found a valid value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = 3ˣ, solve the resulting quadratic for its valid positive value, and then solve for x." }, { "id": "cie-2016-w-21-q07iib", "paperId": "cie-2016-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the polynomial factors to solve the related secant equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this the third part; the printed QP labels it 7(ii)(b). Very few candidates attempted it because they could not relate sec x to the linear factor from the earlier parts.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate sec x to the linear factor found earlier, use the factorisation from part (ii), and solve for all angles in the stated range." }, { "id": "cie-2016-w-21-q03", "paperId": "cie-2016-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Differentiate and solve the trigonometric equation in radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Completely correct differentiation was rare; few simplified to tan 2x and some answered in degrees despite the radian interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate with respect to x, simplify fully to an equation in tan 2x, and use the radian interval given in the question." }, { "id": "cie-2016-w-21-q05ii", "paperId": "cie-2016-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Use the identity proved earlier to evaluate the integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates who used the wrong doubled angle in the integrand commonly failed to reach the required answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the exact result from part (i) and integrate the correct expression in cos 2x over the stated limits." }, { "id": "cie-2016-w-21-q02", "paperId": "cie-2016-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Determine the exponential-model parameters from the transformed straight-line information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates showed limited understanding of the logarithmic transformation and could not eliminate A correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain the linear relation, use the gradient to find p, and substitute a point to determine A." }, { "id": "cie-2016-w-21-q04i", "paperId": "cie-2016-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Integrate the given expression and rearrange it into the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many candidates substituted into the integrand without integrating, and others could not rearrange the correct integral into the required form.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate each term, substitute both limits, and rearrange the result carefully to match the stated form." }, { "id": "cie-2016-w-22-q04iii", "paperId": "cie-2016-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor the polynomial fully after making the required substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was rarely done well; candidates often stopped after the first factorisation or tried to factor a sixth-degree expression directly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Factor the original polynomial first, substitute the stated power of x, and factor the resulting expression without attempting a direct sixth-degree factorisation." }, { "id": "cie-2016-w-22-q06iii", "paperId": "cie-2016-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate the integral using the result established earlier.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Correct solutions were rare; many candidates did not use the earlier identity and others integrated the wrong secant expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the result from part (i), integrate the correct expression and apply the stated limits exactly." }, { "id": "cie-2016-w-22-q02i", "paperId": "cie-2016-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation using a quadratic substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The question was rarely done correctly because candidates did not recognise the quadratic substitution or discarded the validity check.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite 4ʸ as (2ʸ)², solve the resulting quadratic in 2ʸ and retain only the valid positive value." }, { "id": "cie-2016-w-22-q05i", "paperId": "cie-2016-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate and rearrange the expression into the required logarithmic form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Rearrangement was less successful than differentiation and exposed weaknesses with logarithms and algebra.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the quotient rule, simplify the derivative, and use logarithm laws and careful algebra to reach the stated form." }, { "id": "cie-2016-w-22-q07i", "paperId": "cie-2016-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric equations and determine k exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Compound-angle formula errors and algebraic slips meant few candidates obtained the correct value of k.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find dx/dθ and dy/dθ, form dy/dx, use the compound-angle identity and solve for k without losing exactness." }, { "id": "cie-2016-w-22-q07ii", "paperId": "cie-2016-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact equation of the line using the result for k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Candidates often lost the final mark by giving m or c in decimal form instead of the required exact values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the exact parameter from part (i), use the given point and tangent condition, and retain exact coefficients." }, { "id": "cie-2016-w-23-q01ii", "paperId": "cie-2016-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "State the exact value associated with the numerical iteration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates noticed the exact-answer requirement and many gave a decimal instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the exact relationship established by the iteration and express the result in the requested exact form, not as a decimal." }, { "id": "cie-2016-w-23-q04ii", "paperId": "cie-2016-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Show that the remaining quadratic has no real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates stopped after writing the quadratic and did not show that it had no real roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "After factorisation, use the discriminant of the remaining quadratic and state explicitly why it has no real roots." }, { "id": "cie-2016-w-23-q05iii", "paperId": "cie-2016-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the integral exactly using the stated limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Some candidates used the wrong upper limit and many used a calculator despite the requirement for an exact result.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the exponential expression and substitute the limits 0 and 6, keeping the answer exact throughout." }, { "id": "cie-2016-w-23-q07iii", "paperId": "cie-2016-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find both solutions in the required interval to the trigonometric equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates obtained both solutions and errors in numerical accuracy were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the amplitude form from the earlier parts, solve for all angles in the stated range, and give the required accuracy." }, { "id": "cie-2016-w-23-q06i", "paperId": "cie-2016-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric expression for y with respect to t.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates missed that y was a product of two functions of t and did not apply the product rule.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the product rule to the two t-dependent factors and then use the chain rule for the logarithmic term." }, { "id": "cie-2016-w-23-q02", "paperId": "cie-2016-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the parameters of the exponential model from its logarithmic linear form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many candidates misused the logarithms of the given values after rewriting the exponential equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms, use the straight-line gradient to find p, and substitute one point to determine K." }, { "id": "cie-2016-w-31-q04", "paperId": "cie-2016-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the implicit curve and solve for the required stationary condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this question challenging; product-rule errors and differentiating a constant with respect to x prevented progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate both product terms implicitly, differentiate the constant multiple of a with respect to x as zero, and collect the dy/dx terms carefully." }, { "id": "cie-2016-w-31-q10i", "paperId": "cie-2016-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 9, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Separate variables and integrate the differential equation using partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not recognise that the rational integral required partial fractions and used an incorrect logarithmic antiderivative.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the variables, decompose the rational term into partial fractions, integrate both sides, and apply the initial condition." }, { "id": "cie-2016-w-31-q05ii", "paperId": "cie-2016-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use the identity from part (i) to evaluate the integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not see the link between the parts or recognise the standard integral despite the form of the given answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given identity to rewrite the integrand as a standard logarithmic integral and show the limit substitution." }, { "id": "cie-2016-w-31-q06i", "paperId": "cie-2016-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Show that the equation has one root in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates mentioned the intersection of the graphs when showing that a root existed in the interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the graph intersection and evaluate the relevant function at the interval endpoints to establish the change in sign." }, { "id": "cie-2016-w-32-q04", "paperId": "cie-2016-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the implicit curve and solve for the required stationary condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says product-rule errors and differentiating a constant with respect to x were common and made progress difficult.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate both product terms implicitly and collect the dy/dx terms carefully, treating constants as constants." }, { "id": "cie-2016-w-32-q10i", "paperId": "cie-2016-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 9, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Separate variables and integrate the differential equation using partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not recognise the partial-fractions integral and used an incorrect antiderivative.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the variables, decompose the rational term into partial fractions, integrate both sides, and apply the initial condition." }, { "id": "cie-2016-w-32-q05ii", "paperId": "cie-2016-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use the identity from part (i) to evaluate the integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not see the link between the parts or recognise the standard integral despite the form of the given answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given identity to rewrite the integrand as a standard logarithmic integral and apply the limits clearly." }, { "id": "cie-2016-w-32-q06i", "paperId": "cie-2016-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Show that the equation has one root in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates mentioned the intersection of the graphs when showing that a root existed in the interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the graph intersection and evaluate the relevant function at the interval endpoints to establish the change in sign." }, { "id": "cie-2016-w-33-q09i", "paperId": "cie-2016-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the condition for the two curves to touch at their point of intersection.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates equated the y-values but did not recognise that the gradients also had to be equal; few differentiated the reciprocal curve successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the y-values and equate the derivatives at the touching point, differentiating both functions correctly." }, { "id": "cie-2016-w-33-q04ii", "paperId": "cie-2016-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the discriminant to show that the quadratic factor has no real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates stopped after obtaining the quadratic and did not show that it had no real roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "After factorising the polynomial, evaluate the discriminant of the quadratic and conclude that it is negative." }, { "id": "cie-2016-w-33-q06i", "paperId": "cie-2016-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the stated substitution to transform the definite integral and its limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says a given-answer integral required every substitution step to be shown, including the changed limits and cancellation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Change both the variable and limits, show the differential substitution and cancel the common factor before integrating." }, { "id": "cie-2016-w-33-q05i", "paperId": "cie-2016-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 1, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx and use it in the differential-equation model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not include dy/dx in their expression and could not progress from the stated relation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated relation implicitly, isolate dy/dx, and then substitute the given expression for the related rate." }, { "id": "cie-2016-w-41-q07iii", "paperId": "cie-2016-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find when the velocity becomes zero in the second stage of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates completed this part correctly; integrating A/t², including the integration constant and matching velocity were common barriers.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the second acceleration expression, include a constant, match velocity at t = 5 and solve the resulting velocity equation." }, { "id": "cie-2016-w-41-q05iii", "paperId": "cie-2016-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the vertical distance descended using the motion on the slope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found this part more challenging and often omitted kinetic-energy or force terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use Newton’s second law along the slope or a complete work-energy equation, including the change in kinetic energy and work against friction." }, { "id": "cie-2016-w-41-q06iv", "paperId": "cie-2016-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iv)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the time taken for the block to stop while travelling up the slope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Errors were often seen in this part, especially omitted energy terms or forces in the Newton’s-law method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a complete work-energy equation or resolve all forces on the slope, then use the constant-acceleration relation to find time." }, { "id": "cie-2016-w-42-q06iii", "paperId": "cie-2016-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the new tension when the driving force varies at the stated instant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates struggled; the varying driving force meant constant-acceleration and usual work-energy methods did not apply.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law separately to the van and trailer at the stated speed, include the acceleration and weight component, and solve simultaneously." }, { "id": "cie-2016-w-42-q07ii", "paperId": "cie-2016-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the motorcycle’s acceleration from the linked travel times and distances.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found this part quite difficult because it required several linked time and distance deductions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the car’s total travel time, infer the motorcycle’s moving intervals, use equal total distances to find its speed and then calculate the acceleration." }, { "id": "cie-2016-w-42-q05", "paperId": "cie-2016-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the mass and friction coefficient from two limiting-equilibrium cases.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates made errors by treating the two cases as one or omitting friction; even correct equations were often difficult to solve simultaneously.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write a separate three-force equilibrium equation for each applied force, use opposing friction directions, and solve the simultaneous equations." }, { "id": "cie-2016-w-42-q07iii", "paperId": "cie-2016-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the displacement-time graph for the three stages of the car’s motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates drew a velocity-time graph instead of the required displacement-time graph.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Show the initial concave-up quadratic section, the middle straight line and the final concave-down quadratic section, with the given boundary points." }, { "id": "cie-2016-w-43-q07ii", "paperId": "cie-2016-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use an energy method over both stages of the box’s motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part challenged some of the strongest candidates and fully correct solutions were infrequent; many considered only the first 5 m or double-counted gravitational work.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write one work-energy equation for the full 10 m journey, including potential-energy loss and work against resistance without double-counting either contribution." }, { "id": "cie-2016-w-43-q04i", "paperId": "cie-2016-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time at which two particles are at the same height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Some candidates overlooked the one-second time difference and others confused equal height with equal distance travelled.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use each particle’s own time interval in the constant-acceleration equation and set the two heights equal." }, { "id": "cie-2016-w-51-q07i", "paperId": "cie-2016-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Determine the launch parameter by using the two stated points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Solving the two equations for the launch parameters proved too difficult for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute both points into the given projectile equation, form two equations and solve them simultaneously." }, { "id": "cie-2016-w-51-q07iii", "paperId": "cie-2016-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the speed and direction from the horizontal and vertical velocity components.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates struggled because both velocity components were needed before finding the speed and angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find both velocity components at the stated time, then use Pythagoras and trigonometry to obtain speed and direction." }, { "id": "cie-2016-w-51-q06ii", "paperId": "cie-2016-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use the energy equation when the elastic string becomes slack.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes the energy question as historically difficult and specifies that a second four-term energy equation is required for the slack-string stage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the arc length to determine the angle at which the string becomes slack, then write a second complete energy equation for that position." }, { "id": "cie-2016-w-53-q07i", "paperId": "cie-2016-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Determine the launch parameter by using the two stated points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Solving the two equations for the launch parameters proved too difficult for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute both points into the given projectile equation, form two equations and solve them simultaneously." }, { "id": "cie-2016-w-53-q07iii", "paperId": "cie-2016-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the speed and direction from the horizontal and vertical velocity components.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates struggled because both velocity components were needed before finding the speed and angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find both velocity components at the stated time, then use Pythagoras and trigonometry to obtain speed and direction." }, { "id": "cie-2016-w-53-q06ii", "paperId": "cie-2016-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use the energy equation when the elastic string becomes slack.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes the energy question as historically difficult and specifies that a second four-term energy equation is required for the slack-string stage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the arc length to determine the angle at which the string becomes slack, then write a second complete energy equation for that position." }, { "id": "cie-2016-w-61-q05a", "paperId": "cie-2016-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the arrangements with repeated letters and constrained vowel positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates struggled to identify the six possible positions for the vowels and used 4! instead of 6P4.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Arrange the consonants with repeated-letter corrections, then count the valid insertion positions for the vowels and apply the repeated-letter divisor." }, { "id": "cie-2016-w-61-q06i", "paperId": "cie-2016-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability after transferring a pen between the two pockets.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates who attempted this part failed to update the denominator from 4 to 5 after the transfer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a tree diagram and update the number of pens in the destination pocket after the first transfer before multiplying the branch probabilities." }, { "id": "cie-2016-w-61-q02", "paperId": "cie-2016-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the expected difference using the joint outcomes and their probabilities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates built the outcome table but failed to assign probabilities to each outcome or complete the expected-value calculation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "List the possible score differences with their probabilities, then multiply each difference by its probability and add the results." }, { "id": "cie-2016-w-62-q05i", "paperId": "cie-2016-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Construct the histogram using class boundaries and frequency density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates did not proceed beyond the frequency distribution and often used incorrect class widths or axes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the class boundaries and widths, calculate the frequency distribution or density required, choose a usable scale and label the axes with units." }, { "id": "cie-2016-w-62-q05iii", "paperId": "cie-2016-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the classes containing the median and lower and upper quartiles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Weaker candidates often did not attempt this part, and the lower-quartile class was less often identified correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the cumulative frequencies to locate the required observations and state the corresponding classes in the table’s notation." }, { "id": "cie-2016-w-62-q06iii", "paperId": "cie-2016-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the valid selections using combinations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates missed that “selected” required combinations and often failed to list all valid cases.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use combinations for each allowed case, state the options being counted, and add the separate case totals." }, { "id": "cie-2016-w-63-q07ii", "paperId": "cie-2016-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability of the specified meal pattern over five days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this one of the most challenging questions; only the strongest candidates included the required multinomial arrangement factor.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Multiply the three meal probabilities with their stated powers and multiply by 5!/(2!2!) to count the possible orderings." }, { "id": "cie-2016-w-63-q01", "paperId": "cie-2016-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the selections without separating the people into unnecessary groups.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes this as a challenging first question and says enumerating separate male/female cases rarely led to the correct answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the valid selections directly using the stated restriction, then divide by the total number of equally likely selections." }, { "id": "cie-2016-w-71-q05bi", "paperId": "cie-2016-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the percentage α from the stated probability for the new sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates who found the standardised value were able to convert it into the correct α percentage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the standard deviation for the new sample, standardise the stated value, and use the normal distribution to determine α." }, { "id": "cie-2016-w-71-q07ii", "paperId": "cie-2016-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the stated alternative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates recognised that they first needed the critical value before calculating the Type II error probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the critical value for the test under the null hypothesis, then use it to calculate the probability under the alternative distribution." }, { "id": "cie-2016-w-71-q02iii", "paperId": "cie-2016-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(iii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the smallest sample size that meets the stated test condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often obtained 16.4 but incorrectly chose 16 rather than the smallest valid integer 17.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the inequality for n, reverse the inequality correctly when dividing by the negative logarithm, and round up to the next integer." }, { "id": "cie-2016-w-72-q05bi", "paperId": "cie-2016-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the percentage α from the stated probability for the new sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates who found the standardised value were able to convert it into the correct α percentage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the standard deviation for the new sample, standardise the stated value, and use the normal distribution to determine α." }, { "id": "cie-2016-w-72-q07ii", "paperId": "cie-2016-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the stated alternative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates recognised that they first needed the critical value before calculating the Type II error probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the critical value for the test under the null hypothesis, then use it to calculate the probability under the alternative distribution." }, { "id": "cie-2016-w-72-q02iii", "paperId": "cie-2016-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(iii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the smallest sample size that meets the stated test condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often obtained 16.4 but incorrectly chose 16 rather than the smallest valid integer 17.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the inequality for n, reverse the inequality correctly when dividing by the negative logarithm, and round up to the next integer." }, { "id": "cie-2016-w-73-q05ii", "paperId": "cie-2016-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Determine the Type I error probability from the binomial critical region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Candidates often omitted the comparison with 0.05 or calculated only one of the two required binomial sums.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate both adjacent lower-tail binomial sums, compare each with the 5% significance level, identify the critical region, and state its probability." }, { "id": "cie-2016-w-73-q08iia", "paperId": "cie-2016-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)(a)", "marks": 3, "topicIds": [ "data-analysis", "integration" ], "primaryTopicId": "data-analysis", "summary": "Show the variance and standard deviation of the stated continuous distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_73.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says marks were lost when candidates omitted the zero mean, skipped integration steps, or failed to state the standard deviation after finding the variance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use symmetry to establish the mean, integrate x²f(x) for the variance, and state the square root as the standard deviation." }, { "id": "cie-2016-w-73-q04", "paperId": "cie-2016-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Find the possible sample sizes from the two stated significance levels.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Some candidates gave only an inequality instead of listing all the integer sample sizes required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use both critical z-values to form bounds for n and list every integer in the resulting interval explicitly." }, { "id": "cie-2016-s-11-q04i", "paperId": "cie-2016-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use the chain rule to find the rate of change of y as the point crosses the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the chain-rule route was not widely recognised; many candidates could not connect the given rate of change with the derivative of the curve.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use dy/dt = (dy/dx)(dx/dt), substitute x = 0 into the given curve gradient, and multiply by the stated rate dx/dt = 0.3." }, { "id": "cie-2016-s-12-q09", "paperId": "cie-2016-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9", "marks": 9, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model the amount of water lost from a tank first by an arithmetic sequence and then by a geometric sequence.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this by far the worst-answered question; candidates often confused nth terms with sums, and only the strongest candidates usually completed the final geometric-series part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the nth-term formula for the amount lost on the specified day, distinguish that from the cumulative sum, and use the appropriate arithmetic or geometric sum to determine when the tank empties and the final percentage." }, { "id": "cie-2016-s-12-q11ii", "paperId": "cie-2016-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(ii)", "marks": 3, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Show that the tangent line to the quadratic has the stated relation between its gradient and intercept.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as the most challenging part of the question; candidates needed to equate and rearrange expressions before using the discriminant, while others differentiated and equated gradients.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the derivative of the quadratic equal to the tangent gradient to locate the point of contact, substitute into the curve to find the intercept, then eliminate the contact point to obtain the required relation." }, { "id": "cie-2016-s-13-q07", "paperId": "cie-2016-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Relate the rates of change of x and y to find the possible values of x on the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question challenged even some high-ability candidates; many became confused by the chain-rule notation, and some then squared a relation involving a square root incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the constant rate of change into a relation between dx/dt and dy/dt, use the chain rule to obtain dy/dx, then solve the resulting equation carefully for x." }, { "id": "cie-2016-s-13-q10ii", "paperId": "cie-2016-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Determine the maximum value of q from the range condition on the quadratic function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was quite challenging for most candidates; relatively few chose the negative value for the maximum of q.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the quadratic's vertex and the stated range condition, then choose the negative root that gives the maximum admissible value." }, { "id": "cie-2016-s-21-q04iii", "paperId": "cie-2016-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 1, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Solve the modulus equation that follows from the factorisation in the earlier parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct solutions were seen; many candidates reused the three roots from the factorisation and ignored the modulus sign.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the modulus to each factor, solve the resulting absolute-value equation, and retain all values satisfying the original condition." }, { "id": "cie-2016-s-21-q07a", "paperId": "cie-2016-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the first trigonometric integral exactly using the identities needed to simplify its integrand.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates recognised the required sequence of trigonometric rewrites, and very few correct solutions were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand using the double-angle identity and secant identity before integrating; keep the final expression exact." }, { "id": "cie-2016-s-21-q07b", "paperId": "cie-2016-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the logarithmic definite integral and express the exact result in the requested form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often found the logarithmic antiderivative but few converted the final constant to the required exact form.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the indicated substitution or antiderivative, evaluate both limits, apply logarithm laws, and rewrite the result in the exact exponential form requested." }, { "id": "cie-2016-s-22-q05ii", "paperId": "cie-2016-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Show that the root of the given equation lies in the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this type of interval-root question repeatedly causes difficulty; many stopped after substituting the endpoints without stating the sign-change conclusion.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the expression from part (i) to evaluate the relevant function at both endpoints, show the change in sign, and state that a root lies between them." }, { "id": "cie-2016-s-22-q06a", "paperId": "cie-2016-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the given expression involving exponential terms after simplifying it.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few correct solutions were seen; candidates often did not rewrite the integrand in the required form before integrating.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide each numerator term by the exponential denominator to rewrite the integrand as a sum of exponential terms, then integrate each term with its correct coefficient and include the constant of integration." }, { "id": "cie-2016-s-22-q07iii", "paperId": "cie-2016-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find the gradients of the parametric curve at P and Q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates continued to this part; those who did often lost accuracy through premature approximation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the parameter values for P and Q from the earlier parts, substitute each into dy/dx, and retain enough precision before rounding." }, { "id": "cie-2016-s-23-q05ii", "paperId": "cie-2016-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Show that the root of the given equation lies in the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this type of interval-root question repeatedly causes difficulty; many stopped after substituting the endpoints without stating the sign-change conclusion.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the expression from part (i) to evaluate the relevant function at both endpoints, show the change in sign, and state that a root lies between them." }, { "id": "cie-2016-s-23-q06a", "paperId": "cie-2016-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the given expression involving exponential terms after simplifying it.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few correct solutions were seen; candidates often did not rewrite the integrand in the required form before integrating.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide each numerator term by the exponential denominator to rewrite the integrand as a sum of exponential terms, then integrate each term with its correct coefficient and include the constant of integration." }, { "id": "cie-2016-s-23-q07iii", "paperId": "cie-2016-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find the gradients of the parametric curve at P and Q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates continued to this part; those who did often lost accuracy through premature approximation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the parameter values for P and Q from the earlier parts, substitute each into dy/dx, and retain enough precision before rounding." }, { "id": "cie-2016-s-31-q04", "paperId": "cie-2016-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the differential equation with the stated boundary conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists the variable-separable equation among the least well-done questions; candidates often mishandled the integral of the rational term.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the variables, split the rational expression before integrating, apply the limits or boundary data, and solve for the constant without carrying an incorrect additive constant into the exponential form." }, { "id": "cie-2016-s-31-q08ii", "paperId": "cie-2016-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the function in ascending powers of x through the term in x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a small number handled both negative-power terms correctly and reached the required expansion coefficient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand each partial-fraction term with its correct negative-binomial form, track the powers and coefficients separately, and combine the terms through x²." }, { "id": "cie-2016-s-31-q09i", "paperId": "cie-2016-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(i)", "marks": 6, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane containing the three stated points in the required Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists vector geometry among the least well-done questions and warns that D was sometimes included even though it was not stated to lie in the plane; the plane must be determined from A, B and C only.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use two independent vectors in the plane to find a normal vector, then substitute one of the given points to determine the constant in the plane equation." }, { "id": "cie-2016-s-31-q10bii", "paperId": "cie-2016-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the least possible value of |w − z| for points on the stated complex loci.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a small number of candidates earned the final two marks by correctly identifying the point where the complex-number difference was least.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the Argand diagram and the two locus conditions to identify the closest possible pair, then calculate their separation and state the least distance." }, { "id": "cie-2016-s-32-q01", "paperId": "cie-2016-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation to the specified accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists the indicial-equation question among those done least well and describes common bracket and logarithm errors that affected the final accuracy.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides while preserving the brackets and powers, isolate x exactly, then delay approximation until the final rounding." }, { "id": "cie-2016-s-32-q07i", "paperId": "cie-2016-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Express the rational function as partial fractions with the correct numerator forms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was not well answered; many used incomplete or invalid partial-fraction forms, while only a substantial minority chose the appropriate decomposition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose a constant numerator over each linear factor, include the repeated-factor term, clear denominators, and equate coefficients to determine the constants." }, { "id": "cie-2016-s-32-q09i", "paperId": "cie-2016-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the fourth vertex of the stated parallelogram and establish that it is a rhombus.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists vector geometry among the least well-done questions and describes common diagram-order and vector-equality errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use vector addition or equal diagonal midpoints to find the missing vertex, then prove the rhombus condition using adjacent side lengths or perpendicular diagonals." }, { "id": "cie-2016-s-32-q10a", "paperId": "cie-2016-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the quadratic equation in the complex variable and give both roots in the requested form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report lists complex numbers among the least well-done questions and notes frequent errors in the discriminant and in converting the roots to the required form.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Simplify the discriminant exactly, solve for both roots, and convert any complex denominator into the requested standard form while showing the algebra." }, { "id": "cie-2016-s-33-q02", "paperId": "cie-2016-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the logarithmic transformation to find the exact gradient and solve for the requested variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies logarithms as one of the least well-done questions; many replaced exact coefficients with decimals and could not complete the exact solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms and retain exact logarithmic coefficients, compare with the straight-line form, then combine logarithms rather than introducing decimals before the final answer." }, { "id": "cie-2016-s-33-q05", "paperId": "cie-2016-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 8, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the variable-separable differential equation and find y at the stated value of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this question was usually done very well or very badly; many failed to reach an e^(2y) term or recognise the Pythagorean identity needed for the tan²x integral, and accuracy errors were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate e^(2y) dy and use tan²x = sec²x − 1 for the x integral, apply y = 0 when x = 0, and keep enough accuracy for the final value." }, { "id": "cie-2016-s-33-q07ii", "paperId": "cie-2016-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral after using the substitution given in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists this exact integral among the least well-done parts; candidates commonly used an incorrect power and many replaced the exact result with a decimal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the transformed integrand into three terms, integrate each with the correct power, substitute the new limits, and retain the exact logarithmic result instead of a decimal approximation." }, { "id": "cie-2016-s-33-q08ii", "paperId": "cie-2016-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact perpendicular distance from the point to the line in three dimensions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies this as among the least well-done parts; candidates often used the wrong vector and introduced decimals where an exact answer was required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Construct the vector from a point on the line to the given point, project it onto the line direction, and use the exact Pythagorean remainder to obtain the perpendicular distance." }, { "id": "cie-2016-s-33-q10ii", "paperId": "cie-2016-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand f(x) in ascending powers of x through the term in x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report describes repeated copying and sign errors in rewriting the denominators before the binomial expansions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite each partial-fraction denominator in the form (1+u)^n, preserve signs and coefficients, and combine each series through x²." }, { "id": "cie-2016-s-41-q05", "paperId": "cie-2016-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and tension for two connected particles moving under gravity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this question difficult; errors often came from inconsistent assumed directions for the two particles.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw consistent force diagrams for both particles, resolve along the direction of motion, and solve the simultaneous equations for acceleration and tension." }, { "id": "cie-2016-s-41-q07iib", "paperId": "cie-2016-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration when the driving force is angled and friction changes with the normal reaction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was difficult for many candidates because the angled driving force changed the normal reaction and therefore the friction force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the angled driving force both parallel and perpendicular to the plane, find the changed normal reaction and friction, then apply Newton's second law along the plane." }, { "id": "cie-2016-s-42-q05", "paperId": "cie-2016-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension needed to keep a block in equilibrium on a rough inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found this question quite challenging; a frequent error was omitting the component of tension from the normal reaction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the tension and weight parallel and perpendicular to the plane, include the tension's contribution to the normal reaction, and apply limiting friction with zero acceleration." }, { "id": "cie-2016-s-42-q07ii", "paperId": "cie-2016-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total distance travelled after the connected motion changes when one particle reaches the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was quite difficult; many reused the first-stage acceleration or time instead of finding the new motion after the particle reached the plane.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat the motion in two stages, find the speed at the transition, then apply the correct frictional force and constant-acceleration equations to the second stage." }, { "id": "cie-2016-s-43-q02ii", "paperId": "cie-2016-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance OP in the runners problem after accounting for their different starting times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this part as one of the most challenging; many used the wrong time difference because they treated the two speed changes as five rather than ten seconds apart.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write each runner’s distance using the correct elapsed-time origin, account for the ten-second difference between the changes in speed, solve for the meeting time, and find OP." }, { "id": "cie-2016-s-43-q05iii", "paperId": "cie-2016-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the car's acceleration down the hill after its engine power is reduced.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists this among the most challenging parts; many omitted the weight component or resistance or used the wrong sign for the downhill motion.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the reduced power to find the driving force at the stated speed, then include both the downhill weight component and resistance with a consistent sign convention." }, { "id": "cie-2016-s-43-q07ii", "paperId": "cie-2016-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use the acceleration function and initial conditions to find the displacement function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies this as one of the most challenging parts; candidates often could not determine both integration constants and confused the two integrations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration once to obtain velocity using the stated initial condition, integrate again using the displacement condition, and keep separate constants for the two integrations." }, { "id": "cie-2016-s-51-q02i", "paperId": "cie-2016-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the required centre-of-mass distance for a wire bent into the stated shape.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Question 2 caused many problems; some candidates used the centre-of-mass formula for a lamina instead of a wire, and a clear diagram was needed to identify the distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the centre of mass of the wire rather than a semicircular lamina and resolve its position into the required direction using the given geometry." }, { "id": "cie-2016-s-51-q04ii", "paperId": "cie-2016-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the greatest height of a cylinder before the composite solid topples on the inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part proved rather difficult and outlines the centre-of-mass and moment calculation required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of the composite solid in terms of the unknown height, then take moments about the centre of the cylinder's base at the limiting position." }, { "id": "cie-2016-s-51-q07ii", "paperId": "cie-2016-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use energy conservation to find the initial kinetic energy for motion of the elastic string system.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls the elastic-string question the hardest on the paper and says a four-term energy equation was required here.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the four-term energy equation with kinetic energy, gravitational potential energy and both elastic-energy contributions, using the correct extension at each position." }, { "id": "cie-2016-s-51-q07iii", "paperId": "cie-2016-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed after the elastic string system moves from its new equilibrium position.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this is still part of the paper's hardest question and requires a second four-term energy equation after finding the new equilibrium.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the new equilibrium using Hooke's law, then write the four-term energy equation with the new extension and solve for the greatest speed." }, { "id": "cie-2016-s-52-q02ii", "paperId": "cie-2016-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed of projection using the elastic-energy, kinetic-energy and potential-energy changes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this part rather difficult; it required a three-term energy equation involving potential, kinetic and elastic energy.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the three-term energy equation, including the elastic energy stored before projection, the gain in kinetic energy and the change in gravitational potential energy." }, { "id": "cie-2016-s-52-q06ii", "paperId": "cie-2016-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension when a particle moves in a vertical circle at the stated condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates found the new radius, so few obtained correct solutions for this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the radial equation at the stated position, account for the change in speed and radius, and solve for the new tension rather than reusing the previous circular-motion values." }, { "id": "cie-2016-s-53-q02i", "paperId": "cie-2016-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the required centre-of-mass distance for a wire bent into the stated shape.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Question 2 caused many problems; some candidates used the centre-of-mass formula for a lamina instead of a wire, and a clear diagram was needed to identify the distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the centre of mass of the wire rather than a semicircular lamina and resolve its position into the required direction using the given geometry." }, { "id": "cie-2016-s-53-q04ii", "paperId": "cie-2016-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the greatest height of a cylinder before the composite solid topples on the inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part proved rather difficult and outlines the centre-of-mass and moment calculation required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of the composite solid in terms of the unknown height, then take moments about the centre of the cylinder's base at the limiting position." }, { "id": "cie-2016-s-53-q07ii", "paperId": "cie-2016-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use energy conservation to find the initial kinetic energy for motion of the elastic string system.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls the elastic-string question the hardest on the paper and says a four-term energy equation was required here.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the four-term energy equation with kinetic energy, gravitational potential energy and both elastic-energy contributions, using the correct extension at each position." }, { "id": "cie-2016-s-53-q07iii", "paperId": "cie-2016-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed after the elastic string system moves from its new equilibrium position.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this is still part of the paper's hardest question and requires a second four-term energy equation after finding the new equilibrium.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the new equilibrium using Hooke's law, then write the four-term energy equation with the new extension and solve for the greatest speed." }, { "id": "cie-2016-s-61-q04", "paperId": "cie-2016-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the mean and variance for the number of green sweets in a sample drawn without replacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question was difficult for many; candidates often failed to recognise sampling without replacement and used a binomial model instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise the sampling as without replacement, use the appropriate hypergeometric probabilities for the possible counts, and calculate the mean and variance from that distribution." }, { "id": "cie-2016-s-61-q05ii", "paperId": "cie-2016-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the normal approximation to a binomial distribution to find the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says approximately half of those who answered part (i) successfully were unable to use the normal approximation in this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the binomial mean and variance, apply the appropriate continuity correction, and standardise the required probability before using the normal table." }, { "id": "cie-2016-s-62-q04ii", "paperId": "cie-2016-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Solve the inequality for the number of trials required to meet the stated binomial probability condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates gave a single value, often by trial and improvement, but misinterpreted the inequality and gained little credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the probability requirement into the stated inequality, take logarithms with the correct inequality direction, and interpret the resulting integer threshold." }, { "id": "cie-2016-s-62-q07aii", "paperId": "cie-2016-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the arrangements of repeated letters that satisfy the stated positional conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates found this question challenging and often failed to account for the different positions available to the repeated W letters.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the cases according to the positions of the repeated W letters, account for the remaining repeated letters, and explain the counting method at each stage." }, { "id": "cie-2016-s-63-q04ii", "paperId": "cie-2016-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Find the variance of the nine apartment rental prices from the given offset sums.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says part (ii) caused the most difficulty; uncoded calculations often expanded the squared deviations incorrectly, and some candidates gave the standard deviation instead of the variance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the mean to find c. Since x − 2205 = (x − c) − 205, expand the squared deviations from the mean using both supplied sums, then divide by 9; do not take a square root." }, { "id": "cie-2016-s-71-q02iii", "paperId": "cie-2016-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(iii)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "State the population for the sampling investigation in the context of the workers' journeys.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the contextual definition of population was not well answered; many candidates confused the employees with the population of distances travelled.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Describe the complete set of distances travelled by all employees at the firm, rather than naming the employees themselves." }, { "id": "cie-2016-s-71-q03ii", "paperId": "cie-2016-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the confidence level corresponding to the value of the standardised test statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates often found the z-value, but very few continued to calculate the correct confidence level.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the standard normal distribution to convert the given z-value into the two-sided confidence level, keeping the tail areas consistent with the question." }, { "id": "cie-2016-s-71-q07i", "paperId": "cie-2016-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that at least one spoon and at least one fork are defective.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates did not recognise the separate Poisson probabilities needed and often failed to multiply them.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the Poisson mean for each type of cutlery, calculate the probability of at least one defect in each, and multiply the independent probabilities." }, { "id": "cie-2016-s-72-q02i", "paperId": "cie-2016-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 1, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Explain the Type I error in the context of the customer-satisfaction test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates often gave only a textbook definition of a Type I error and failed to explain both the test conclusion and the reality in the given context.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State the conclusion made by the test and the opposite truth about the population proportion, both in the context of customer satisfaction." }, { "id": "cie-2016-s-72-q06ii", "paperId": "cie-2016-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probabilities that stocks of four and five hair dryers will each be enough to meet weekly demand.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates failed to distinguish the two cumulative probabilities required here from probabilities of exactly four and five items.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the Poisson distribution for weekly demand and calculate the required cumulative probabilities for stocks of four and five, rather than the probabilities of exactly those counts." }, { "id": "cie-2016-s-72-q06iii", "paperId": "cie-2016-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the smallest number of hair dryers for which the cumulative Poisson probability reaches the stated level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates used the point probability P(n) instead of the cumulative probability P(X≤n).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Evaluate the cumulative probability P(X≤n) for successive integer values and stop at the first n that reaches the required threshold." }, { "id": "cie-2016-s-73-q04i", "paperId": "cie-2016-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the discrete significance test and state the evidence supported by the critical region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this topic remains difficult; many candidates used a normal distribution, failed to test after finding a critical region, or compared a point probability with the significance level.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the critical region from the relevant discrete probabilities, compare the observed count with that region, and state the conclusion in context." }, { "id": "cie-2016-s-73-q05iii", "paperId": "cie-2016-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that T lies between its expected value and its median.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was much more challenging; candidates often had to solve a cubic for the median and some lost accuracy through insufficient precision.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the median from the cumulative distribution, evaluate the cumulative probability at E(T), and subtract one half while retaining enough accuracy." }, { "id": "cie-2016-s-73-q06iv", "paperId": "cie-2016-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iv)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Explain why the Central Limit Theorem is needed for the sampling distribution in this question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this the most challenging part of the paper; many incorrectly claimed the theorem was unnecessary or that the original distribution was unknown.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the original variable is Poisson rather than normally distributed, so the Central Limit Theorem justifies the approximate normal distribution of the sample mean." }, { "id": "cie-2016-s-73-q07ii", "paperId": "cie-2016-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the masses of two full boxes differ by less than the stated amount.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a minority obtained all the marks; many calculated a one-sided difference and did not interpret the required absolute difference.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the distribution of the difference between the two box masses, use its variance correctly, and calculate the two-sided probability for the absolute difference bound." }, { "id": "cie-2017-w-11-q07bii", "paperId": "cie-2017-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation for every angle in the stated degree interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_11.pdf", "qpPages": [ 13, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few solutions were fully correct because candidates often omitted the negative solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the result from the previous part, solve for all permitted cosine values, and include the negative-angle solution as well as the positive one." }, { "id": "cie-2017-w-11-q02", "paperId": "cie-2017-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the largest domain endpoint that keeps the cubic function increasing.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a minority connected the derivative range with the given domain and selected its negative limit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate, find where the derivative changes sign, and choose the negative boundary that is compatible with the given domain x < a." }, { "id": "cie-2017-w-11-q07a", "paperId": "cie-2017-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Read the vertical shift and amplitude from the displayed sine graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_11.pdf", "qpPages": [ 12, 12 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For most candidates, finding the amplitude or using the indicated point to determine b proved elusive.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the graph’s midline for a and its distance from the midline to a maximum or minimum for |b|, then determine the sign of b from the curve." }, { "id": "cie-2017-w-12-q06b", "paperId": "cie-2017-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the constants in a sine function from its full range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_12.pdf", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as perhaps the most challenging part of a question and says success depended on using the maximum and minimum of sine.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 4, 6 ], "tagConfidence": "reviewed", "answerGuide": "Use the midpoint of the range for the vertical shift and half its width for the amplitude, allowing either sign for the sine coefficient." }, { "id": "cie-2017-w-12-q03a", "paperId": "cie-2017-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Calculate the value of a stamp after ten annual increases of 5%.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this scenario challenged candidates; common errors included using an arithmetic model, the wrong common ratio, or the wrong number of growth steps.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 4, 6 ], "tagConfidence": "reviewed", "answerGuide": "Model the annual growth as a geometric progression with ratio 1.05, use ten growth steps, and round the result to the nearest hundred dollars." }, { "id": "cie-2017-w-12-q03b", "paperId": "cie-2017-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Recover the first term and common difference from the formula for an arithmetic-series sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often chose the arithmetic-series formula but then struggled to rearrange the resulting expression or made invalid assumptions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 4, 6 ], "tagConfidence": "reviewed", "answerGuide": "Compare the expanded given sum with S_n = n/2[2a + (n−1)d] and equate the coefficients of n² and n." }, { "id": "cie-2017-w-13-q06i", "paperId": "cie-2017-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the restricted rational function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_13.pdf", "qpPages": [ 8, 8 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates omitted the ± when solving and failed to use the original domain to select the negative branch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 7, 10 ], "tagConfidence": "reviewed", "answerGuide": "Set y equal to the function, solve the resulting quadratic for x, select the branch consistent with x < −1, and express the inverse with its implied domain." }, { "id": "cie-2017-w-13-q01", "paperId": "cie-2017-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the least integer n for which the arithmetic-series sum exceeds 3000.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates rounded the positive boundary down, even though that integer did not make the sum exceed 3000.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 7, 10 ], "tagConfidence": "reviewed", "answerGuide": "Form the quadratic inequality from the sum formula, use the positive root as a boundary, and test the next integer above it." }, { "id": "cie-2017-w-13-q09ii", "paperId": "cie-2017-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2017, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)", "marks": 4, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two position vectors of a point on the line through B and C using the earlier vector ratio.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_13.pdf", "qpPages": [ 15, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Most candidates did not carry the ratio found in part (i) into this part and instead tried unrelated algebraic approaches.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 7, 10 ], "tagConfidence": "reviewed", "answerGuide": "Apply the established length ratio along BC in both possible directions, then express each location as a position vector from the origin." }, { "id": "cie-2017-w-21-q07ii", "paperId": "cie-2017-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show that the root of the trigonometric equation lies between −4.5 and −4.0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_21.pdf", "qpPages": [ 11, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was rarely attempted and even less often justified correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 11, 13 ], "tagConfidence": "reviewed", "answerGuide": "Form the function from the equation found earlier, evaluate it at both endpoints in radians, and show that the values have opposite signs." }, { "id": "cie-2017-w-21-q06ii", "paperId": "cie-2017-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the normal where the parametric curve crosses the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_21.pdf", "qpPages": [ 9, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only very well-prepared candidates made progress on the required parameter, derivative, and perpendicular-line steps.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 11, 13 ], "tagConfidence": "reviewed", "answerGuide": "Find the parameter value where y = 0, evaluate dy/dx there, and take the negative reciprocal for the normal gradient." }, { "id": "cie-2017-w-21-q01", "paperId": "cie-2017-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation and express the answer in terms of e.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates mishandled logarithm laws, split the expression into separate terms, or failed to recognise that 1 could be written as ln e.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 11, 13 ], "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms into a single quotient, write 1 as ln e, equate the arguments after exponentiating, and solve while respecting the log domains." }, { "id": "cie-2017-w-21-q04a", "paperId": "cie-2017-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Integrate the trigonometric fraction in the first part of Question 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates who did not simplify the denominator with the relevant identity made no progress, and many responses were incorrect.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 11, 13 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity for 1−sin²θ to simplify the denominator, rewrite the fraction using standard trigonometric forms, then integrate and include a constant." }, { "id": "cie-2017-w-23-q07ii", "paperId": "cie-2017-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show that the root of the trigonometric equation lies between −4.5 and −4.0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_23.pdf", "qpPages": [ 11, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was rarely attempted and even less often justified correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 17, 19 ], "tagConfidence": "reviewed", "answerGuide": "Form the function from the equation found earlier, evaluate it at both endpoints in radians, and show that the values have opposite signs." }, { "id": "cie-2017-w-23-q06ii", "paperId": "cie-2017-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the normal where the parametric curve crosses the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_23.pdf", "qpPages": [ 9, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only very well-prepared candidates made progress on the required parameter, derivative, and perpendicular-line steps.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 17, 19 ], "tagConfidence": "reviewed", "answerGuide": "Find the parameter value where y = 0, evaluate dy/dx there, and take the negative reciprocal for the normal gradient." }, { "id": "cie-2017-w-23-q01", "paperId": "cie-2017-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation and express the answer in terms of e.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates mishandled logarithm laws, split the expression into separate terms, or failed to recognise that 1 could be written as ln e.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 17, 19 ], "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms into a single quotient, write 1 as ln e, equate the arguments after exponentiating, and solve while respecting the log domains." }, { "id": "cie-2017-w-23-q04a", "paperId": "cie-2017-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2017, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Integrate the trigonometric fraction in the first part of Question 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates who did not simplify the denominator with the relevant identity made no progress, and many responses were incorrect.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 17, 19 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity for 1−sin²θ to simplify the denominator, rewrite the fraction using standard trigonometric forms, then integrate and include a constant." }, { "id": "cie-2017-w-31-q07b", "paperId": "cie-2017-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Shade the Argand-plane region satisfying both the modulus and argument conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_31.pdf", "qpPages": [ 13, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes repeated errors in the circle position, the line boundary, and the final shaded region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 20, 23 ], "tagConfidence": "reviewed", "answerGuide": "Draw the circle with the specified centre and radius, add the boundary ray or line for the argument condition, and shade only the intersection satisfying both inequalities." }, { "id": "cie-2017-w-31-q09ii", "paperId": "cie-2017-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Verify the given exact value of the definite integral for the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_31.pdf", "qpPages": [ 17, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes sign errors across the repeated integration-by-parts steps and says the final result was not often obtained.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 20, 23 ], "tagConfidence": "reviewed", "answerGuide": "Integrate by parts twice, keep the exponential and polynomial terms organised, then substitute both limits and simplify exactly." }, { "id": "cie-2017-w-31-q01", "paperId": "cie-2017-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide x⁴ by the given quadratic and state the quotient and remainder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the long-division question was generally poorly done.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 20, 23 ], "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division, subtract each leading term in sequence, and check that divisor × quotient plus remainder reproduces x⁴." }, { "id": "cie-2017-w-31-q02", "paperId": "cie-2017-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Use plotted data from an exponential model to find its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies the logarithmic transformation and extracting the constants from the graph as a difficult question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 20, 23 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms to make the model linear in x, read the gradient and intercept from the straight-line plot, then recover the model parameters." }, { "id": "cie-2017-w-32-q06ii", "paperId": "cie-2017-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find both points where the curve has a tangent parallel to the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_32.pdf", "qpPages": [ 11, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates missed the zero-valued case and therefore failed to find both points.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 24, 28 ], "tagConfidence": "reviewed", "answerGuide": "Set the numerator of dy/dx to zero, retain every factor that can vanish, and substitute each resulting relation into the curve to obtain both points." }, { "id": "cie-2017-w-32-q05", "paperId": "cie-2017-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Solve the given first-order differential equation using the stated point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often failed to separate variables correctly and many did not complete the integration and apply the condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 24, 28 ], "tagConfidence": "reviewed", "answerGuide": "Separate the x and y terms, integrate each side, use the initial condition to determine the constant, and rearrange for y." }, { "id": "cie-2017-w-32-q01i", "paperId": "cie-2017-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the definite integral using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the trapezium rule was unfamiliar to many candidates and was not generally done well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 24, 28 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the integrand at both endpoints and the midpoint, use h/2 times the endpoint ordinates plus twice the midpoint ordinate, and round as requested." }, { "id": "cie-2017-w-32-q09ii", "paperId": "cie-2017-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Show by calculation that the required root lies between 2 and 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_32.pdf", "qpPages": [ 17, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes frequent errors when candidates used calculator values to establish the sign change.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 24, 28 ], "tagConfidence": "reviewed", "answerGuide": "Substitute x = 2 and x = 4 into the defining function and demonstrate that the values have opposite signs." }, { "id": "cie-2017-w-33-q07b", "paperId": "cie-2017-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Shade the Argand-plane region satisfying both the modulus and argument conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_QP_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_33.pdf", "qpPages": [ 13, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes repeated errors in the circle position, the line boundary, and the final shaded region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 29, 32 ], "tagConfidence": "reviewed", "answerGuide": "Draw the circle with the specified centre and radius, add the boundary ray or line for the argument condition, and shade only the intersection satisfying both inequalities." }, { "id": "cie-2017-w-33-q09ii", "paperId": "cie-2017-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Verify the given exact value of the definite integral for the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_QP_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_33.pdf", "qpPages": [ 17, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes sign errors across the repeated integration-by-parts steps and says the final result was not often obtained.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 29, 32 ], "tagConfidence": "reviewed", "answerGuide": "Integrate by parts twice, keep the exponential and polynomial terms organised, then substitute both limits and simplify exactly." }, { "id": "cie-2017-w-33-q01", "paperId": "cie-2017-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide x⁴ by the given quadratic and state the quotient and remainder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_QP_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the long-division question was generally poorly done.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 29, 32 ], "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division, subtract each leading term in sequence, and check that divisor × quotient plus remainder reproduces x⁴." }, { "id": "cie-2017-w-33-q02", "paperId": "cie-2017-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2017, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Use plotted data from an exponential model to find its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_QP_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies the logarithmic transformation and extracting the constants from the graph as a difficult question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 29, 32 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms to make the model linear in x, read the gradient and intercept from the straight-line plot, then recover the model parameters." }, { "id": "cie-2017-w-41-q07ii", "paperId": "cie-2017-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the incline angle that puts the connected particles in limiting equilibrium with friction on B’s plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_41.pdf", "qpPages": [ 11, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Question 7 among the paper’s most challenging and describes difficulty handling friction in the connected-particle equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 33, 36 ], "tagConfidence": "reviewed", "answerGuide": "Find the normal reaction and limiting friction on B, write the equilibrium equations for both particles, and eliminate the tension to solve for the angle." }, { "id": "cie-2017-w-41-q07i", "paperId": "cie-2017-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of particle B when the plane is smooth and the angle is 15°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_41.pdf", "qpPages": [ 10, 10 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 7 was one of the most challenging; candidates often omitted forces or used inconsistent directions in their equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 33, 36 ], "tagConfidence": "reviewed", "answerGuide": "Draw force diagrams for both particles, resolve along the slope and string, and solve Newton’s second-law equations with the smooth-plane assumption." }, { "id": "cie-2017-w-42-q04ii", "paperId": "cie-2017-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the times when the two particles are moving in the same direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this part as the most difficult on the paper.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 37, 40 ], "tagConfidence": "reviewed", "answerGuide": "Use the given velocity functions, identify where each velocity changes sign, and find the intervals where both signs agree." }, { "id": "cie-2017-w-42-q06ii", "paperId": "cie-2017-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the mass m when an extra force is applied to the connected particles on the inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_42.pdf", "qpPages": [ 9, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates reached the correct answer; common errors were missing force terms, wrong signs, or reusing the tension from part (i).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 37, 40 ], "tagConfidence": "reviewed", "answerGuide": "Reuse the friction force from part (i), form Newton’s second-law equations for both particles or the whole system, use the new tension, and solve for m." }, { "id": "cie-2017-w-42-q05ii", "paperId": "cie-2017-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the cyclist’s speed ten seconds after reaching the top of the hill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_42.pdf", "qpPages": [ 7, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates struggled because they treated the work-energy situation as if the force were constant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 37, 40 ], "tagConfidence": "reviewed", "answerGuide": "Use the work-energy principle over the stated interval, including the cyclist’s work, resistive work, and the changes in kinetic and gravitational potential energy." }, { "id": "cie-2017-w-43-q07ii", "paperId": "cie-2017-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy method to find the speed of P at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_43.pdf", "qpPages": [ 11, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies both parts of Question 7 among the paper’s most challenging questions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 41, 43 ], "tagConfidence": "reviewed", "answerGuide": "Relate the initial and final kinetic energies to the changes in gravitational potential energy and the work done by the stated forces." }, { "id": "cie-2017-w-43-q07i", "paperId": "cie-2017-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2017, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the least value of T that keeps the particle at rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_43.pdf", "qpPages": [ 10, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies both parts of Question 7 among the paper’s most challenging questions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 41, 43 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the forces for equilibrium and apply the limiting condition that gives the smallest admissible value of T." }, { "id": "cie-2017-w-51-q06iii", "paperId": "cie-2017-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the unknown quantity P in the final part of Question 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_51.pdf", "qpPages": [ 11, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates completed this part successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Take moments about the point where the hemisphere contacts the rough surface, use the perpendicular moment arms shown, and solve for P." }, { "id": "cie-2017-w-51-q05i", "paperId": "cie-2017-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed at the position where its acceleration is zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_51.pdf", "qpPages": [ 8, 8 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists Question 5 among the paper’s hardest questions and notes that candidates often did not identify the zero-acceleration position.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Use Newton’s second law to find the position where the resultant force is zero, then apply the energy equation between the initial and that position." }, { "id": "cie-2017-w-51-q05ii", "paperId": "cie-2017-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle moves 1.2 m while travelling upwards with constant deceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_51.pdf", "qpPages": [ 9, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part proved far too difficult for many candidates and required a three-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Start at the equilibrium position with the speed from part (i), set the speed to zero at the highest point, and use the three-term energy equation to find the rise." }, { "id": "cie-2017-w-52-q02", "paperId": "cie-2017-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of the composite cone-and-cylinder solid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question was too difficult for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Use the component volumes and their individual centre positions, then take the volume-weighted mean along the common axis." }, { "id": "cie-2017-w-52-q05ii", "paperId": "cie-2017-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the weight of the lamina from its geometry and centre-of-mass information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_52.pdf", "qpPages": [ 9, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a few candidates obtained a complete correct answer to this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the relevant lengths and areas, take moments about the stated reference point, and solve the balance equation for the weight." }, { "id": "cie-2017-w-52-q07iii", "paperId": "cie-2017-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the total time until B strikes the plane for the second time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_52.pdf", "qpPages": [ 13, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the second-flight timing as difficult for candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Find the time for each flight section from the vertical-motion equations, account for the rebound or change of direction, and add the intervals requested." }, { "id": "cie-2017-w-53-q06iii", "paperId": "cie-2017-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the unknown quantity P in the final part of Question 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_53.pdf", "qpPages": [ 11, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates completed this part successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Take moments about the point where the hemisphere contacts the rough surface, use the perpendicular moment arms shown, and solve for P." }, { "id": "cie-2017-w-53-q05i", "paperId": "cie-2017-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed at the position where its acceleration is zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_53.pdf", "qpPages": [ 8, 8 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists Question 5 among the paper’s hardest questions and notes that candidates often did not identify the zero-acceleration position.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Use Newton’s second law to find the position where the resultant force is zero, then apply the energy equation between the initial and that position." }, { "id": "cie-2017-w-53-q05ii", "paperId": "cie-2017-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2017, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle moves 1.2 m while travelling upwards with constant deceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_53.pdf", "qpPages": [ 9, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part proved far too difficult for many candidates and required a three-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Start at the equilibrium position with the speed from part (i), set the speed to zero at the highest point, and use the three-term energy equation to find the rise." }, { "id": "cie-2017-w-61-q06aii", "paperId": "cie-2017-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count seatings with Mary and Ahmad together in the front row and the other three together in a different row.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_61.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few fully correct solutions were seen; the common error was adding the two arrangement counts instead of multiplying by the remaining-row arrangements.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Count adjacent placements for Mary and Ahmad in the front row, choose one of the other seven rows for the remaining three, then count their adjacent placements and internal orders." }, { "id": "cie-2017-w-61-q06b", "paperId": "cie-2017-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count four-person teams that do not contain both of the specified people.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_61.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes that candidates often mishandled the restriction on the two named people and did not count the allowed teams correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Subtract from all four-person teams the selections that include both named people, choosing the remaining two from the other eight." }, { "id": "cie-2017-w-61-q05iii", "paperId": "cie-2017-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of economy class given that Julian did not sleep well.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_61.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates successfully completed this conditional-probability part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Use the conditional-probability ratio: the joint probability of economy class and poor sleep divided by the probability of poor sleep." }, { "id": "cie-2017-w-61-q04ii", "paperId": "cie-2017-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the pooled sample standard deviation for the two art classes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes the pooled standard-deviation calculation as involved and challenging.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Combine the supplied sample sizes, sums, and sums of squares, calculate the pooled mean and corrected sum of squares, then use the sample-variance denominator." }, { "id": "cie-2017-w-62-q07ii", "paperId": "cie-2017-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find an upper weight value from a normal-distribution percentile.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the most challenging part and notes that candidates struggled to link the probability to the normal-curve symmetry.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 53, 56 ], "tagConfidence": "reviewed", "answerGuide": "Translate the stated 25% area into the appropriate upper-tail z-score, then use k = μ + zσ with the correct positive sign." }, { "id": "cie-2017-w-62-q06aii", "paperId": "cie-2017-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count even numbers formed from the available digits when repetition is allowed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the digit-selection context caused difficulty and this part was often omitted or misinterpreted.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 53, 56 ], "tagConfidence": "reviewed", "answerGuide": "Choose an even final digit first, then count the valid choices for the other positions, accounting for the stated repetition rule." }, { "id": "cie-2017-w-62-q04iii", "paperId": "cie-2017-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a die result equals the number of heads from the coin tosses.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes Question 4’s context as challenging and notes difficulty translating its wording into probability statements.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 53, 56 ], "tagConfidence": "reviewed", "answerGuide": "Condition on the possible number of heads, use the matching die probability for each case, and sum the mutually exclusive joint probabilities." }, { "id": "cie-2017-w-63-q06ii", "paperId": "cie-2017-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the five selected cars are not all adjacent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most challenging part and says many candidates counted the all-adjacent case instead of its complement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 57, 59 ], "tagConfidence": "reviewed", "answerGuide": "Count all placements of the five chosen cars, subtract the placements where they form one consecutive block, and divide by the total number of placements." }, { "id": "cie-2017-w-63-q06iii", "paperId": "cie-2017-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count selections containing cars of at least two different colours.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_63.pdf", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates often gave a probability rather than the requested number, showing that the demand of this part was misunderstood.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 57, 59 ], "tagConfidence": "reviewed", "answerGuide": "Count all allowed selections and subtract selections consisting of only one colour, respecting the number of cars available in each colour." }, { "id": "cie-2017-w-63-q07iv", "paperId": "cie-2017-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that Josie misses the bus.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_63.pdf", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates found the wording of the later bus parts difficult and that this part was frequently left blank or misunderstood.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 57, 59 ], "tagConfidence": "reviewed", "answerGuide": "Translate the wording into the complementary event, identify the relevant arrival-time or delay probabilities, and combine the independent events as stated." }, { "id": "cie-2017-w-63-q05i", "paperId": "cie-2017-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2017, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Represent the data in a stem-and-leaf diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a minority produced a fully correct diagram, often because the data were labelled or ordered incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 57, 59 ], "tagConfidence": "reviewed", "answerGuide": "Order the leaves within each stem, include every observation once, and give a key that explains the place values." }, { "id": "cie-2017-w-71-q07i", "paperId": "cie-2017-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the Poisson hypothesis test given the observed total over five days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_71.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates frequently selected the wrong tail for the Poisson test.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 60, 63 ], "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, identify the correct tail, calculate the Poisson probability for the critical region, and conclude in context." }, { "id": "cie-2017-w-71-q08ii", "paperId": "cie-2017-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error under the stated alternative mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_71.pdf", "qpPages": [ 11, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates completed only the second-stage probability and did not first establish the critical value for the test.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 60, 63 ], "tagConfidence": "reviewed", "answerGuide": "Use the test’s acceptance region and calculate its probability under the alternative distribution, keeping the critical boundary from the earlier part." }, { "id": "cie-2017-w-71-q04i", "paperId": "cie-2017-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate a 99% confidence interval for the population parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_71.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes common errors in the confidence interval, including using an unsuitable critical value or interval width.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 60, 63 ], "tagConfidence": "reviewed", "answerGuide": "Use the relevant standard error and the 99% normal critical value, then form and report both interval endpoints to the requested precision." }, { "id": "cie-2017-w-72-q06i", "paperId": "cie-2017-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the Poisson test using the observed total from five days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_72.pdf", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the later test parts were more demanding and candidates often did not establish the critical region correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 64, 66 ], "tagConfidence": "reviewed", "answerGuide": "State the hypotheses, calculate the relevant lower- and upper-tail probabilities, choose the largest valid critical region, and conclude at the stated level." }, { "id": "cie-2017-w-72-q06ii", "paperId": "cie-2017-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the repeated test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_72.pdf", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this later part was demanding and that candidates often did not connect its probability to the null rejection region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 64, 66 ], "tagConfidence": "reviewed", "answerGuide": "Use the rejection region from the stated test and calculate its probability under the null Poisson model." }, { "id": "cie-2017-w-72-q03ii", "paperId": "cie-2017-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Explain whether the Central Limit Theorem was needed in the preceding result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_72.pdf", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this explanation was poorly attempted and many candidates did not address the reason the theorem was or was not required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 64, 66 ], "tagConfidence": "reviewed", "answerGuide": "Identify the distributional assumption given for the population and state whether the sample-mean result follows exactly or only by a large-sample approximation." }, { "id": "cie-2017-w-72-q06iii", "paperId": "cie-2017-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "statistical-testing", "proof" ], "primaryTopicId": "statistical-testing", "summary": "Explain the meaning of a Type I error in the context of the test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_72.pdf", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies this explanation as part of the more demanding test question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 64, 66 ], "tagConfidence": "reviewed", "answerGuide": "Describe rejecting the null hypothesis when it is true, using the defective-items context in the question." }, { "id": "cie-2017-w-73-q07i", "paperId": "cie-2017-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the Poisson hypothesis test given the observed total over five days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_73.pdf", "qpPages": [ 8, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates frequently selected the wrong tail for the Poisson test.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 67, 70 ], "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, identify the correct tail, calculate the Poisson probability for the critical region, and conclude in context." }, { "id": "cie-2017-w-73-q08ii", "paperId": "cie-2017-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error under the stated alternative mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_73.pdf", "qpPages": [ 11, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates completed only the second-stage probability and did not first establish the critical value for the test.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 67, 70 ], "tagConfidence": "reviewed", "answerGuide": "Use the test’s acceptance region and calculate its probability under the alternative distribution, keeping the critical boundary from the earlier part." }, { "id": "cie-2017-w-73-q04i", "paperId": "cie-2017-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2017, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate a 99% confidence interval for the population parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_ms_73.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes common errors in the confidence interval, including using an unsuitable critical value or interval width.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w17_er.pdf", "difficultyReportPages": [ 67, 70 ], "tagConfidence": "reviewed", "answerGuide": "Use the relevant standard error and the 99% normal critical value, then form and report both interval endpoints to the requested precision." }, { "id": "cie-2015-w-11-q03", "paperId": "cie-2015-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation and give both possible values of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Candidates who found the viable x² value often omitted the negative square root.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the inverse-sine relation to form a quadratic in x², solve it, then include both square roots of the viable value." }, { "id": "cie-2015-w-11-q07ii", "paperId": "cie-2015-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the requested shaded area in terms of r.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls the question very testing and describes the area decomposition needed for a successful solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the right-angle relation in the diagram, express the sector and semicircle areas in terms of r, and subtract the relevant segment area." }, { "id": "cie-2015-w-11-q09iii", "paperId": "cie-2015-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(iii)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the domain of the inverse function using the given value of m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part was often omitted, suggesting confusion about the domain concept.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the inverse expression and specified m to determine the allowed x-values, with the correct inequality." }, { "id": "cie-2015-w-11-q11ii", "paperId": "cie-2015-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(ii)", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact volume for the region rotated about the stated axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates using the curve equation found the integral and substitution very difficult to complete.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 1, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the y-axis volume method, square the required curve expression, integrate with the correct limits, and keep the answer exact." }, { "id": "cie-2015-w-12-q03i", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the tank diagram and context to obtain the stated volume relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many struggled with the context and diagram; those who found the triangle area were more successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Find the triangular end area from the diagram and multiply by the tank length to justify the given volume expression." }, { "id": "cie-2015-w-12-q03ii", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the water-height rate when volume changes at the given rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates often knew the chain rule but misused the rate notation and multiplied or divided terms incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the volume relation with respect to time by the chain rule, substitute the given rate and height, and solve for dh/dt." }, { "id": "cie-2015-w-12-q04i", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "proof", "trigonometry" ], "primaryTopicId": "proof", "summary": "Prove the stated trigonometric identity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many found the proof very difficult; starting from the right-hand side was rarely successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Start from the left-hand side, combine the fractions before squaring, use the cosine identity, and simplify." }, { "id": "cie-2015-w-12-q06i", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter values that make the two specified lengths equal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many equated gradients instead of recognising that the required condition was equality of lengths.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Write the squared-distance expressions for the specified point pairs, equate them, and solve the quadratic." }, { "id": "cie-2015-w-12-q07i", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the vectors to establish the stated collinearity relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Most were unsure how to start and often equated vectors instead of using a scalar multiple.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Form the relevant displacement vectors and show one is a scalar multiple of the other, keeping signs consistent." }, { "id": "cie-2015-w-12-q08ii", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find k from the stated relationship between the roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many misapplied the discriminant instead of using the stated roots as factors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Write the quadratic factors implied by the roots, expand them, and compare coefficients." }, { "id": "cie-2015-w-12-q08iii", "paperId": "cie-2015-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the parameter condition for the required behaviour of f(x+a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Many mishandled f(x+a), treating it as f(x)+f(a), or formed the wrong quadratic.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 4, 7 ], "tagConfidence": "reviewed", "answerGuide": "Expand f(x+a), form the resulting quadratic in x, and apply the discriminant condition to its correct coefficients." }, { "id": "cie-2015-w-13-q06", "paperId": "cie-2015-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Compare total travel for two bouncing-ball models through the required bounce.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The contextual question was very poorly done, with common errors in direction, first term, number of terms, and common difference or ratio.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 8, 11 ], "tagConfidence": "reviewed", "answerGuide": "Use an arithmetic sequence for fixed height loss and a geometric sequence for percentage loss; sum up-and-down travel with the correct starting height and term count." }, { "id": "cie-2015-w-13-q08i", "paperId": "cie-2015-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the greatest value of a satisfying the composite-function condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates found this very difficult and missed the restriction on x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 8, 11 ], "tagConfidence": "reviewed", "answerGuide": "Use the restriction on x together with the required expression, then solve the resulting inequality for its largest valid parameter." }, { "id": "cie-2015-w-13-q08iii", "paperId": "cie-2015-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(iii)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the composite-function inequality, accounting for the restriction on x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a small proportion reached the answer after the linked inequality and domain steps.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 8, 11 ], "tagConfidence": "reviewed", "answerGuide": "Form the composite, reverse inequalities when multiplying by a negative, choose the outside regions around the roots, then apply the domain restriction." }, { "id": "cie-2015-w-21-q04i", "paperId": "cie-2015-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "exponentials", "proof" ], "primaryTopicId": "exponentials", "summary": "Sketch the two functions on common axes to show the equation has only one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many omitted the sketch; candidates needed to state the single-intersection conclusion.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Sketch both curves on the same axes and state that their single intersection establishes one real root." }, { "id": "cie-2015-w-21-q06ii", "paperId": "cie-2015-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the quotient and remainder from part (i) to determine the required constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few related this part to the previous division; very few correct solutions were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Equate the quotient/remainder form to the stated factor form and compare coefficients, using the earlier division result rather than restarting the division." }, { "id": "cie-2015-w-21-q06iii", "paperId": "cie-2015-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Use the constants from the earlier parts to establish the stated root condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This was rarely attempted and very few gave correct statements about the two quadratic factors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Identify which quadratic factor has no real roots and which has equal roots, then deduce the requested repeated root." }, { "id": "cie-2015-w-21-q07i", "paperId": "cie-2015-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given parametric expressions and obtain the required derivative result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few obtained the given result; many missed the chain rule when differentiating x with respect to t.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule to x(t), differentiate y(t) term by term, and form dy/dx=(dy/dt)/(dx/dt)." }, { "id": "cie-2015-w-21-q07ii", "paperId": "cie-2015-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the parameter values of the two stationary points A and C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few attempted this part and most could not solve the cotangent equation validly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Set dy/dx=0 using the result in part (i), solve the resulting cotangent equation in 0 ≤ t < π, and give both values of t to 3 decimal places." }, { "id": "cie-2015-w-21-q07iii", "paperId": "cie-2015-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the curve at its x-axis crossing B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says very few candidates attempted this final part and gives −13/9 as the answer; its method comment appears inconsistent with the printed request for the gradient at B.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Use y=0 to identify the parameter value for B, then substitute it into dy/dx from part (i) and simplify." }, { "id": "cie-2015-w-22-q01ii", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation obtained in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few reached the required logarithmic statement needed to proceed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use logarithm laws and the relation from part (i) to isolate the unknown, then respect the domain of each logarithm." }, { "id": "cie-2015-w-22-q02ii", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Find the exact value requested from the iterative result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few could start; candidates often gave a rounded calculator answer despite the request for an exact value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the iteration’s limiting equation to derive the exact value; do not substitute the rounded numerical approximation from part (i)." }, { "id": "cie-2015-w-22-q03", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Determine parameters of the model from its straight-line transformation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few gained full marks; logarithm misconceptions and lack of context understanding were evident.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the gradient and intercept in the logarithmic linearisation, then recover the original model parameters and required accuracy." }, { "id": "cie-2015-w-22-q04iia", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Complete the exact factorisation of the polynomial.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many failed to show the full factorisation using all three linear factors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the quadratic factor obtained earlier, divide out the known factor, and present all three linear factors exactly." }, { "id": "cie-2015-w-22-q04iib", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)(b)", "marks": 2, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Use the factorisation to find valid solutions of the secant equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many were unsure how to deal with secant and often gave invalid solutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Solve each factor and check candidates in the original secant equation and its domain, rejecting invalid solutions." }, { "id": "cie-2015-w-22-q05i", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of the given product involving an exponential.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many failed to recognise the product-rule structure or to set the derivative equal to zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the product rule, differentiate the exponential factor correctly, set the derivative to zero, and substitute back as requested." }, { "id": "cie-2015-w-22-q05ii", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given quotient and solve for the requested values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Many applied the quotient rule incorrectly or made algebraic expansion errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Apply the quotient rule with brackets, simplify carefully, and solve the resulting equation without dropping terms." }, { "id": "cie-2015-w-22-q06ii", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the equation of the tangent to the parametric curve at the specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Very few found the required parameter value or a correct tangent equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Find the parameter at the point, evaluate dy/dx there from the parametric derivatives, and use point-gradient form." }, { "id": "cie-2015-w-22-q07i", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate both terms of the function in Question 7.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "Very few correct attempts were seen; candidates often misused the cosine identity for sec²x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity for the cosine term and the standard integral for sec²x, keeping the constants and limits correct." }, { "id": "cie-2015-w-22-q07ii", "paperId": "cie-2015-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the integral involving the square of the function from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 10, "difficultyEvidence": "Very few handled the required three-term expansion or linked this part to part (i).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Expand the square fully into three terms, integrate each term correctly, and apply the specified limits." }, { "id": "cie-2015-w-31-q03", "paperId": "cie-2015-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the simultaneous tangent conditions for the requested angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists trigonometry in Question 3 among the parts done least well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Use tangent difference and sum identities to form equations for the angles, then apply the restrictions to select the solutions." }, { "id": "cie-2015-w-31-q07", "paperId": "cie-2015-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 9, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection of a line with the plane through C perpendicular to AB, then find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies vector geometry in Question 7 among the parts done least well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Parameterise AB, substitute in the plane condition, solve for the intersection parameter, and use the resulting coordinates to calculate the distance." }, { "id": "cie-2015-w-31-q08", "paperId": "cie-2015-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 9, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the differential equation with the given initial condition and evaluate the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the differential-equation question as one done least well, with unsound attempts common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate sin²(2θ) using an identity, apply the initial condition, and substitute the required θ." }, { "id": "cie-2015-w-31-q10ii", "paperId": "cie-2015-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find p so that the area of the shaded region under the curve from x=1 to x=p is 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies integration in Question 10(ii) among the parts done least well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Integrate x²/(1+x³) from 1 to p using u=1+x³, set the result equal to 1, then solve for p to 3 significant figures." }, { "id": "cie-2015-w-32-q03", "paperId": "cie-2015-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the simultaneous tangent conditions for the requested angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists trigonometry in Question 3 among the parts done least well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Use tangent difference and sum identities to form equations for the angles, then apply the restrictions to select the solutions." }, { "id": "cie-2015-w-32-q07", "paperId": "cie-2015-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 9, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection of a line with the plane through C perpendicular to AB, then find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies vector geometry in Question 7 among the parts done least well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Parameterise AB, substitute in the plane condition, solve for the intersection parameter, and use the resulting coordinates to calculate the distance." }, { "id": "cie-2015-w-32-q08", "paperId": "cie-2015-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8", "marks": 9, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the differential equation with the given initial condition and evaluate the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the differential-equation question as one done least well, with unsound attempts common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate sin²(2θ) using an identity, apply the initial condition, and substitute the required θ." }, { "id": "cie-2015-w-32-q10ii", "paperId": "cie-2015-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find p so that the area of the shaded region under the curve from x=1 to x=p is 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies integration in Question 10(ii) among the parts done least well.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Integrate x²/(1+x³) from 1 to p using u=1+x³, set the result equal to 1, then solve for p to 3 significant figures." }, { "id": "cie-2015-w-33-q05", "paperId": "cie-2015-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 8, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the substitution in the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report notes that this integration part was challenging and candidates struggled with the substitution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 27, 31 ], "tagConfidence": "reviewed", "answerGuide": "Set u=4−3cos x, convert the differential and limits, and integrate the resulting expression exactly." }, { "id": "cie-2015-w-33-q08iii", "paperId": "cie-2015-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two possible coordinates of C on the line, given that CB is twice AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this position-vector part as challenging.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 27, 31 ], "tagConfidence": "reviewed", "answerGuide": "Parameterise the line through A and B, apply the distance condition in both directions from B, and give both coordinate triples." }, { "id": "cie-2015-w-33-q09b", "paperId": "cie-2015-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the common points of two Argand loci and express them in polar form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the straight-line locus was more difficult and many did not identify the equilateral-triangle geometry.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 27, 31 ], "tagConfidence": "reviewed", "answerGuide": "Interpret the modulus conditions as circles, find their intersections, and convert the common points to the required polar representation." }, { "id": "cie-2015-w-41-q07ii", "paperId": "cie-2015-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the speed when a car catches a cyclist after the cyclist has a 24-second start.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found the delayed overtaking part difficult and few fully correct answers were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 32, 34 ], "tagConfidence": "reviewed", "answerGuide": "Find the cyclist’s lead, equate the car’s distance with the cyclist’s total distance using the delayed start, and solve the resulting quadratic for the feasible speed." }, { "id": "cie-2015-w-42-q04", "paperId": "cie-2015-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to find the speed after motion down the rough incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Question 4 was the most difficult and describes the work-energy method and common sign errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 35, 37 ], "tagConfidence": "reviewed", "answerGuide": "Include the change in gravitational potential energy, the final kinetic energy, and work against friction over the rough section." }, { "id": "cie-2015-w-42-q05ii", "paperId": "cie-2015-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the original string length after it breaks during the particles’ motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few obtained the correct length; many failed to treat motion before and after the break separately.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 35, 37 ], "tagConfidence": "reviewed", "answerGuide": "Find the speed at the break, use the changed acceleration for the later motion to obtain its additional distance, then infer the original length." }, { "id": "cie-2015-w-43-q01", "paperId": "cie-2015-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension and normal reaction for the particle in equilibrium on a rough incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a challenging first question and notes that many ignored equilibrium or the inclined plane.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 38, 40 ], "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the plane, apply equilibrium, and account for friction consistently." }, { "id": "cie-2015-w-43-q03", "paperId": "cie-2015-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 7, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Resolve the forces to find the unknown force and its direction from the given resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this resultant-resolution question as difficult for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 38, 40 ], "tagConfidence": "reviewed", "answerGuide": "Resolve all forces into horizontal and vertical components, equate the resultant components to the given magnitude and direction, and solve." }, { "id": "cie-2015-w-43-q07i", "paperId": "cie-2015-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to find the driving force required for the car on a hill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says only a minority used the appropriate work-energy method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 38, 40 ], "tagConfidence": "reviewed", "answerGuide": "Account for kinetic-energy change, gain in gravitational potential energy and work against resistance over the distance, then solve for the driving force." }, { "id": "cie-2015-w-51-q04ii", "paperId": "cie-2015-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Calculate the string tension during the horizontal circular motion after the lower end is detached.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates found the required string length or radius.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "The original two-segment geometry fixes the total string length L. In the new motion use r=L sinθ, resolve vertically and horizontally, and eliminate θ to find T=mω²L." }, { "id": "cie-2015-w-51-q05ii", "paperId": "cie-2015-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the extension of the elastic string at the instant the particle leaves the surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates found the required extension.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Set the normal reaction to zero, resolve vertically to find the tension, and use Hooke’s law and the string geometry to obtain the extension." }, { "id": "cie-2015-w-51-q05iii", "paperId": "cie-2015-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle’s speed as it leaves the surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says a four-term energy equation was required for this more demanding part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Use a four-term energy equation relating initial elastic energy and kinetic energy to final kinetic and elastic energies." }, { "id": "cie-2015-w-51-q06", "paperId": "cie-2015-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass and equilibrium information for a uniform disc with two holes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies Question 6 as a harder centre-of-mass question involving a disc with holes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Treat the removed parts as negative masses, take moments to locate the composite centre of mass, and use the equilibrium condition." }, { "id": "cie-2015-w-51-q07ii", "paperId": "cie-2015-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile speed V in the delayed-collision situation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few candidates solved this part correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Use the collision time from part (i), write the vertical displacement equations with the one-second delay, equate heights, and solve for V." }, { "id": "cie-2015-w-51-q07iii", "paperId": "cie-2015-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the height difference between the collision point and the projectile’s maximum height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Only a handful found the height difference correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the projectile’s maximum height and its height at the collision time, then subtract." }, { "id": "cie-2015-w-52-q04ii", "paperId": "cie-2015-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Calculate the string tension during the horizontal circular motion after the lower end is detached.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates found the required string length or radius.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "The original two-segment geometry fixes the total string length L. In the new motion use r=L sinθ, resolve vertically and horizontally, and eliminate θ to find T=mω²L." }, { "id": "cie-2015-w-52-q05ii", "paperId": "cie-2015-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the extension of the elastic string at the instant the particle leaves the surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates found the required extension.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Set the normal reaction to zero, resolve vertically to find the tension, and use Hooke’s law and the string geometry to obtain the extension." }, { "id": "cie-2015-w-52-q05iii", "paperId": "cie-2015-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle’s speed as it leaves the surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says a four-term energy equation was required for this more demanding part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Use a four-term energy equation relating initial elastic energy and kinetic energy to final kinetic and elastic energies." }, { "id": "cie-2015-w-52-q06", "paperId": "cie-2015-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass and equilibrium information for a uniform disc with two holes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies Question 6 as a harder centre-of-mass question involving a disc with holes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Treat the removed parts as negative masses, take moments to locate the composite centre of mass, and use the equilibrium condition." }, { "id": "cie-2015-w-52-q07ii", "paperId": "cie-2015-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile speed V in the delayed-collision situation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few candidates solved this part correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Use the collision time from part (i), write the vertical displacement equations with the one-second delay, equate heights, and solve for V." }, { "id": "cie-2015-w-52-q07iii", "paperId": "cie-2015-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the height difference between the collision point and the projectile’s maximum height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Only a handful found the height difference correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the projectile’s maximum height and its height at the collision time, then subtract." }, { "id": "cie-2015-w-53-q04iib", "paperId": "cie-2015-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the angular speed when the specified reaction force acts on the particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this subpart was more difficult and required Newton’s second law towards the centre.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 45, 47 ], "tagConfidence": "reviewed", "answerGuide": "Resolve radially, use the reaction-force condition to determine speed at the position, then divide by the radius to find angular speed." }, { "id": "cie-2015-w-53-q05ii", "paperId": "cie-2015-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the greatest height reached under quadratic resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Question 5 among the harder questions on the paper.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 45, 47 ], "tagConfidence": "reviewed", "answerGuide": "Use Newton’s second law with resistance proportional to v², relate speed to height, and integrate to the turning point." }, { "id": "cie-2015-w-53-q05iii", "paperId": "cie-2015-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the particle’s speed during its descent under quadratic resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies Question 5 among the harder questions on the paper.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 45, 47 ], "tagConfidence": "reviewed", "answerGuide": "Use the motion equation with resistance acting opposite the downward motion and apply the condition at maximum height." }, { "id": "cie-2015-w-53-q06", "paperId": "cie-2015-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of the composite shell and cone and its toppling condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this was one of the hardest questions and describes the centre-of-mass derivation and moments method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 45, 47 ], "tagConfidence": "reviewed", "answerGuide": "Combine the masses and component centre positions to locate the composite centre, then apply the geometry of toppling about the edge." }, { "id": "cie-2015-w-53-q07ii", "paperId": "cie-2015-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the elastic-string energy equation to find the unknown mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this three-term energy equation was too difficult for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 45, 47 ], "tagConfidence": "reviewed", "answerGuide": "Set up the three-term balance between gravitational potential energy, kinetic energy and elastic energy, then substitute the extension relation and solve." }, { "id": "cie-2015-w-53-q07iii", "paperId": "cie-2015-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance AB using the stated energy relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says the energy relation in this final part was too difficult for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 45, 47 ], "tagConfidence": "reviewed", "answerGuide": "Equate gravitational potential energy at A to elastic energy at B, then solve the resulting quadratic for the physical distance." }, { "id": "cie-2015-w-62-q04i", "paperId": "cie-2015-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the ways to assign the friends to two taxis under the group-size condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The taxi grouping question stretched most candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Choose the required group for one taxi and account for whether swapping the taxi assignments creates a distinct arrangement." }, { "id": "cie-2015-w-62-q04ii", "paperId": "cie-2015-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the seating arrangements for the friends in the two taxis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This part was difficult for most; very few gave the correct count and common errors used addition instead of multiplication.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Count valid group assignments, multiply by the seat arrangements within each taxi, and account for the possible seat for the fourth person." }, { "id": "cie-2015-w-62-q06iii", "paperId": "cie-2015-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate the mean and variance of the random variable defined by the spinner.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a small percentage evaluated the mean and sum of squared outcomes correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Build the probability distribution, calculate E(X) and E(X²), then use Var(X)=E(X²)−[E(X)]²." }, { "id": "cie-2015-w-62-q06iv", "paperId": "cie-2015-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iv)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the conditional probability requested for the spinner experiment.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few found the conditional probability, with confusion about the meanings of even and positive.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 50, 52 ], "tagConfidence": "reviewed", "answerGuide": "Use the table to count outcomes satisfying both events and divide by the number or probability of the conditioning event; distinguish zero from positive values." }, { "id": "cie-2015-w-73-q06ii", "paperId": "cie-2015-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error after the population mean changes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates needed to establish the acceptance region first and many did not.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w15_er.pdf", "difficultyReportPages": [ 63, 67 ], "tagConfidence": "reviewed", "answerGuide": "Establish the null acceptance region first, then standardise its boundary under the changed mean and calculate the probability of remaining in that region." }, { "id": "cie-2015-s-11-q04ii", "paperId": "cie-2015-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the required scalar value using the vector condition in the geometry of the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says finding the expression for OC was challenging and notes that some candidates incorrectly treated equality of moduli as equality of vectors.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the required vector expression, use the equal-modulus condition stated in the question, and solve for the admissible scalar values." }, { "id": "cie-2015-s-11-q09iii", "paperId": "cie-2015-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(iii)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use the discriminant condition to find the allowed range of the parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a few candidates correctly applied the discriminant after differentiating.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, form the resulting quadratic condition, and apply the discriminant requirement for the stated number of solutions." }, { "id": "cie-2015-s-12-q02", "paperId": "cie-2015-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the area of the composite region shown in the diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls the question challenging for many candidates and says only stronger candidates found the semicircle area after recognising the radius relation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the triangle, sector and semicircle areas, express each using the geometry in the diagram, and combine them with the correct radius relations." }, { "id": "cie-2015-s-12-q06i", "paperId": "cie-2015-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Find the maximum height described by the trigonometric model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often failed to use the cosine range, some differentiated after seeing the word maximum, and others gave a value contradicted by a later part.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the range of the cosine term to identify the maximum value of the height function; do not differentiate a periodic model unnecessarily." }, { "id": "cie-2015-s-12-q06iii", "paperId": "cie-2015-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Find when the modelled height is above the stated threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was challenging and only the most able candidates recognised that there was more than one interval above the threshold.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the trigonometric inequality over the specified time interval and include every interval that satisfies the height condition." }, { "id": "cie-2015-s-13-q09b", "paperId": "cie-2015-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the range of angles for which the geometric series converges.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies these five marks as the most difficult on the paper; few candidates formed the correct convergence inequality and many reversed the interval because cosine is decreasing.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the convergence condition for the common ratio, solve the resulting trigonometric inequality on the stated domain, and preserve the correct interval endpoints and order." }, { "id": "cie-2015-s-13-q10ii", "paperId": "cie-2015-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2015, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the required area between the curve and the line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part discriminated well and describes frequent errors from combining equations before integration or using incorrect limits.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the triangle area and subtract the curve area using the correct limits; keep the two regions' limits separate if integrating both boundaries." }, { "id": "cie-2015-s-21-q02", "paperId": "cie-2015-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Determine the constants in the exponential model represented by the straight-line graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates completed this question successfully, often because they misread ln y on the vertical axis or could not form the logarithmic equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Interpret the vertical axis as ln y, write the transformed model in linear form, and use the graph's gradient and intercept to find the constants." }, { "id": "cie-2015-s-21-q06iib", "paperId": "cie-2015-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact integral using the trigonometric result established earlier.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few candidates used the earlier trigonometric result to help integrate, and fewer still obtained the correct exact integral.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (i) to rewrite the integrand, integrate in the required variable, and retain an exact result with the correct limits." }, { "id": "cie-2015-s-21-q07iii", "paperId": "cie-2015-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the required point where the implicit curve has a tangent parallel to the vertical axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says recognising that the denominator must be zero was rare, and complete correct solutions were a rarity.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the implicit derivative, set its denominator to zero for a vertical tangent, and substitute the resulting relation back into the curve equation." }, { "id": "cie-2015-s-22-q01ii", "paperId": "cie-2015-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Count the integers satisfying the logarithmic inequality.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few correct solutions were seen; many gave the inequality itself instead of counting integers, and others forgot zero.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the inequality for the integer variable, then count every integer in the resulting interval, including zero where applicable." }, { "id": "cie-2015-s-22-q05i", "paperId": "cie-2015-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "functions", "numerical-methods" ], "primaryTopicId": "functions", "summary": "Sketch the two graphs and use their intersections to show that the equation has two real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few completely correct pairs of graphs were seen and lists incorrect curves, domains and omitted intersection evidence as common errors.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw both functions on the same axes with the correct domains and identify two distinct intersections to establish two real roots." }, { "id": "cie-2015-s-22-q06i", "paperId": "cie-2015-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation in the required interval using radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates reached an intermediate sine value but then gave incorrect degree answers; the correct radian result was rare.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the equation for all angles in the stated interval, using radians as required by the calculus context and checking each solution against the interval." }, { "id": "cie-2015-s-23-q01ii", "paperId": "cie-2015-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Count the integers satisfying the logarithmic inequality.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few correct solutions were seen; many gave the inequality itself instead of counting integers, and others forgot zero.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the inequality for the integer variable, then count every integer in the resulting interval, including zero where applicable." }, { "id": "cie-2015-s-23-q05i", "paperId": "cie-2015-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "functions", "numerical-methods" ], "primaryTopicId": "functions", "summary": "Sketch the two graphs and use their intersections to show that the equation has two real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few completely correct pairs of graphs were seen and lists incorrect curves, domains and omitted intersection evidence as common errors.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw both functions on the same axes with the correct domains and identify two distinct intersections to establish two real roots." }, { "id": "cie-2015-s-23-q06i", "paperId": "cie-2015-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2015, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation in the required interval using radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates reached an intermediate sine value but then gave incorrect degree answers; the correct radian result was rare.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the equation for all angles in the stated interval, using radians as required by the calculus context and checking each solution against the interval." }, { "id": "cie-2015-s-31-q05a", "paperId": "cie-2015-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate the expression involving tan squared using a trigonometric identity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates recognised the identity needed to make progress, and even those who did often integrated it incorrectly.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Replace tan squared with sec squared minus one, integrate each term, and include the constant of integration." }, { "id": "cie-2015-s-31-q08iii", "paperId": "cie-2015-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the circle parameter from its centre, radius and the point it passes through.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only the odd candidate recognised the centre-radius relation needed to determine the single parameter.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the sketch and circle conditions to relate the radius to the centre coordinate, then impose the condition that the circle passes through the given complex point." }, { "id": "cie-2015-s-32-q01", "paperId": "cie-2015-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Apply the trapezium rule with the given ordinates and interval width.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question was poorly answered and lists frequent errors in the ordinates, weights, number of values and interval width.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct equally spaced ordinates, apply half weight to the two endpoints and full weight to interior ordinates, then multiply by the interval width." }, { "id": "cie-2015-s-32-q06i", "paperId": "cie-2015-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral after making the stated substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says fully correct solutions were not common and describes candidates discarding the minus sign without understanding the effect of reversing limits.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the substitution carefully, account for the reversed limits or negative differential, and keep the sign consistent through the transformed integral." }, { "id": "cie-2015-s-32-q07ii", "paperId": "cie-2015-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the complex-number locus in an Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes many incorrect circles, including wrong centres, wrong radii and two-circle sketches.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot the correct centre and radius using equal scales on the real and imaginary axes, and draw only the locus satisfying the stated condition." }, { "id": "cie-2015-s-33-q02", "paperId": "cie-2015-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the modular inequality and state the complete solution set.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a minority of candidates using purely algebraic methods completed the solution; squaring introduced an extra critical point that had to be interpreted.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the modulus condition into the relevant regions, locate the critical value, and test which intervals satisfy the original inequality." }, { "id": "cie-2015-s-33-q07i", "paperId": "cie-2015-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the differential equation by separation of variables and integration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often separated correctly but then made common integration errors in both the rational and trigonometric terms.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the dependent and independent variables, integrate each side correctly, include the constant, and use the initial condition to determine it." }, { "id": "cie-2015-s-33-q08ii", "paperId": "cie-2015-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2015, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the specified complex-number region in an Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often used the wrong circle centre or radius, and that the next part depended on a correct sketch.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the complex-modulus conditions to locate each circle accurately, keep the diagram's scale consistent, and show the required intersection region." }, { "id": "cie-2015-s-41-q05", "paperId": "cie-2015-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 8, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use power, speed and Newton's second law in both directions of travel to determine the unknowns.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this question particularly difficult and says candidates often missed the power-over-speed driving force, used constant-acceleration equations, or omitted a force.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the driving force as power divided by speed, resolve all forces along the plane for uphill and downhill motion, and solve the simultaneous equations." }, { "id": "cie-2015-s-42-q07", "paperId": "cie-2015-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 12, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse the forces and limiting equilibrium for the ring, then determine the added mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was the most difficult question on the paper and that very few candidates completed the final mass calculation successfully.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the forces on the ring in the correct directions, use the limiting friction relation, and then balance vertical forces after the extra particle is added." }, { "id": "cie-2015-s-43-q02i", "paperId": "cie-2015-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Explain the motion after one particle reaches the floor and the other continues to the pulley.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates answered successfully and many incorrectly assumed both particles stopped when the hanging particle reached the floor.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Distinguish the motion before and after the hanging particle reaches the floor, and explain how the loss of tension changes the subsequent motion of the other particle." }, { "id": "cie-2015-s-43-q06ii", "paperId": "cie-2015-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the distance moved after the pulling force is removed, accounting for friction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was more challenging and describes candidates omitting friction or leaving out terms from the energy equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Determine the post-pull deceleration including both gravity and friction, then use the appropriate motion equation or a complete work-energy balance." }, { "id": "cie-2015-s-43-q07ii", "paperId": "cie-2015-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2015, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "kinematics", "functions" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph using the turning points found from zero acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates were challenged by this graph and often joined the points with straight segments instead of drawing the curve.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the velocity values at the stationary points, place them accurately with the end points, and draw the smooth curve with the correct maxima and minima." }, { "id": "cie-2015-s-51-q05iib", "paperId": "cie-2015-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use an energy equation to find the speed when the elastic string breaks.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was very difficult and required a five-term energy equation; many candidates found the extension but stopped before finding the height or speed.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Account for all kinetic, gravitational potential and elastic energy changes in the five-term energy balance, then solve for the speed." }, { "id": "cie-2015-s-51-q06", "paperId": "cie-2015-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "mechanics", "differentiation", "integration" ], "primaryTopicId": "mechanics", "summary": "Derive and solve the differential equation for the particle's velocity, then find its limiting speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the question was very difficult for many candidates and required forming, solving and then applying limits to the differential equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton's second law to form the differential equation, solve it with the stated initial condition, and take the required limit after substituting the correct bounds." }, { "id": "cie-2015-s-51-q07", "paperId": "cie-2015-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 13, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Analyse the prism's centre of mass, toppling condition and friction condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls the question extremely difficult and lays out the linked centre-of-mass, toppling and friction stages that candidates had to complete.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass, take moments about the toppling point, resolve parallel and perpendicular to the plane, and compare actual and limiting friction." }, { "id": "cie-2015-s-52-q05ii", "paperId": "cie-2015-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the weight of the cube at the point of toppling.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was far too difficult for many candidates, particularly because they struggled to find the correct moment arm for the weight.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the point of toppling and use the correct perpendicular distances, or resolve the weight into components before taking moments." }, { "id": "cie-2015-s-52-q06", "paperId": "cie-2015-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Use horizontal and vertical motion to solve the two-stage projectile problem.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as the hardest question on the paper and specifies the paired motion equations needed for both parts.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the initial velocity into components, apply the given velocity condition, and then express the later horizontal and vertical displacements in terms of the required speed." }, { "id": "cie-2015-s-52-q07ii", "paperId": "cie-2015-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "mechanics", "differentiation" ], "primaryTopicId": "mechanics", "summary": "Form the differential equation for the particle's speed while the string remains in motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was not well done by many candidates and gives the differential equation needed to model the motion.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the driving force and normal reaction correctly, then apply Newton's second law with the changing speed and friction term." }, { "id": "cie-2015-s-52-q07iii", "paperId": "cie-2015-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Solve the differential equation using the correct initial and final limits to find the speed at loss of contact.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says incorrect limits were often used and only a few candidates reached the correct answer.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the differential equation, apply the limits associated with the start and loss of contact, and solve for the speed." }, { "id": "cie-2015-s-53-q02", "paperId": "cie-2015-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Resolve the projectile's velocity and displacement into horizontal and vertical components.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says too many candidates failed to score on the question and identifies the component-velocity and displacement calculations required.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stated direction of the velocity at the given time to relate its components, then apply the constant-acceleration equations for both displacements." }, { "id": "cie-2015-s-53-q03ii", "paperId": "cie-2015-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of restitution using the full energy balance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was far too difficult and that too many candidates omitted one term from the five-term energy equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Include every kinetic and elastic energy term before and after the motion, then solve the resulting equation for the coefficient of restitution." }, { "id": "cie-2015-s-53-q05ii", "paperId": "cie-2015-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the particle's speed when the prism topples.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was too difficult for many candidates and required the extension at the instant of toppling followed by a three-term energy equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the elastic extension at the toppling point, then use the three-term energy equation to calculate the speed." }, { "id": "cie-2015-s-53-q07ii", "paperId": "cie-2015-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2015, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Resolve the conical-motion forces to find the string tension, angle and height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was extremely difficult and required two simultaneous equations in the tension and angle before the height could be found.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the vertical force balance and the horizontal centripetal-force equation, solve for the tension and angle, then use the geometry to find the height." }, { "id": "cie-2015-s-61-q01", "paperId": "cie-2015-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the normal distribution tables in reverse to find the unknown value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates found the probability and used the tables in reverse correctly; many omitted the negative sign.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the stated probability into the correct tail area, identify the corresponding signed standard normal value, and transform back to the original scale." }, { "id": "cie-2015-s-61-q03", "paperId": "cie-2015-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Test whether two events are independent and mutually exclusive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says there were very few reasonable attempts and little evidence that candidates understood independence and mutual exclusion.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the marginal and joint probabilities, compare the product with the intersection for independence, and check whether the intersection probability is zero for mutual exclusion." }, { "id": "cie-2015-s-61-q06ii", "paperId": "cie-2015-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to the binomial distribution with the correct continuity correction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many centres were unfamiliar with the normal approximation and candidates who used it often omitted the continuity correction.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, calculate the binomial mean and variance, apply the correct continuity correction, then standardise and evaluate the probability." }, { "id": "cie-2015-s-61-q07aii", "paperId": "cie-2015-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the admissible numbers by separating the possible final digits into cases.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates recognised that the possible even and odd final digits had to be considered as separate events.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the even endings and odd endings as separate cases, apply the restrictions for each ending, and combine the valid counts without double-counting." }, { "id": "cie-2015-s-62-q05i", "paperId": "cie-2015-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of the required pattern when three discs are selected without replacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question discriminated well and some candidates found one ordering but did not include the other possible orders.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the probability for one valid ordering, then include every distinct ordering that satisfies the event." }, { "id": "cie-2015-s-63-q06i", "paperId": "cie-2015-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw the cumulative frequency graph using a suitable scale and fully labelled axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few fully correct solutions were seen and describes inaccurate scales, plotting and axis labels as common problems.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot the cumulative totals at the class boundaries with a readable linear scale, label both axes and draw a smooth increasing curve through the points." }, { "id": "cie-2015-s-63-q06iv", "paperId": "cie-2015-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iv)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Construct the histogram using frequency density and an appropriate scale.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not attempt this part or drew a frequency graph instead of a frequency-density histogram.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert each class frequency to frequency density, use the correct class boundaries and plot bars with heights proportional to those densities." }, { "id": "cie-2015-s-63-q07iii", "paperId": "cie-2015-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2015, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count all arrangements that satisfy the alternating ornament pattern.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates gained full credit because many counted only one of the two valid alternating patterns.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "List both possible alternating patterns, count the valid choices in each position, and add the two distinct arrangement counts." }, { "id": "cie-2015-s-71-q04ii", "paperId": "cie-2015-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Interpret a Type II error in the context of the stated test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says a significant number of candidates described a Type I error or gave no response.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "State the test's conclusion under the null hypothesis and the contrasting situation that is actually true, using the context of the question." }, { "id": "cie-2015-s-71-q07iii", "paperId": "cie-2015-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability by enumerating the valid pairs of values from the two distributions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the required two cases were not identified by the majority of candidates who attempted this part.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify every integer pair satisfying the stated linear relation, calculate each independent joint probability and add them." }, { "id": "cie-2015-s-72-q03", "paperId": "cie-2015-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the expectation and variance for the scaled total over the stated number of days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a small number successfully calculated the expectation and variance after finding the five-day rate.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Scale the daily mean and variance correctly for the time period, apply the multiplier to the mean and its square to the variance, and use independence where stated." }, { "id": "cie-2015-s-72-q07iii", "paperId": "cie-2015-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate the stated conditional probability for the Poisson variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says conditional probability was poorly attempted and few candidates used the correct conditional-probability formula.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the probability of the conditioning event, find the intersection event from the Poisson probabilities, and divide the intersection by the conditioning probability." }, { "id": "cie-2015-s-73-q05iii", "paperId": "cie-2015-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Decide whether the central limit theorem is needed for the test and justify the decision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part the most demanding and says full credit required referring to the unknown population distribution.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the choice to whether the population distribution is known, and state why the sample size alone does or does not settle the need for the theorem." }, { "id": "cie-2015-s-73-q06iii", "paperId": "cie-2015-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2015, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability for the combined checkout rate over the two-minute period.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s15_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this the most challenging part and says many candidates used only the second checkout's rate instead of combining both rates.", "difficultySourceUrl": "https://www.cambridgeinternational.org/images/348170-june-2015-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the rates for both checkouts over the full interval, use that Poisson mean, and calculate the requested tail probability." }, { "id": "ial-legacy-2018-january-WMA01-q02b", "paperId": "ial-legacy-2018-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2018, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Evaluate a finite sum defined by a recurrence sequence.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_que_20180109.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_rms_20180308.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that Q2(b) caused more difficulty: common errors included summing the wrong terms or omitting the integer sum, and some candidates incorrectly treated the values as an arithmetic series. (PEF pp. 3–4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Generate the needed recurrence terms, then calculate the finite sum by subtracting the sum of the sequence terms from the sum of the corresponding integers; the result is 18.", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-legacy-2018-january-WMA01-q10ii", "paperId": "ial-legacy-2018-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2018, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation using algebra and give the exact value in logarithmic form.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_que_20180109.pdf#page=30", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_rms_20180308.pdf#page=16", "qpPages": [ 30, 33 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson notes that many candidates earned the marks in part (i) but none in part (ii), often because they applied logarithms to the separate terms. Candidates who formed a quadratic usually proceeded well, with some losing the final mark by retaining the invalid negative root. (PEF p. 10)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Set t = 3^y to obtain t² + 3t = 10. Keep the valid positive root t = 2, reject t = −5, and state y = log₃2.", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-legacy-2018-january-WMA01-q11c", "paperId": "ial-legacy-2018-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2018, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of a triangle formed by a circle centre, an external point, and a tangent point.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_que_20180109.pdf#page=34", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_rms_20180308.pdf#page=18", "qpPages": [ 34, 37 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson describes Q11 as challenging and says many candidates omitted part (c). A common error was using the wrong base and perpendicular height, while another was leaving a decimal approximation instead of the requested exact surd. (PEF pp. 10–11)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Use that a radius is perpendicular to the tangent at the contact point. Apply Pythagoras to find the needed triangle side, then use half base times perpendicular height to obtain 5√70.", "sourceEvidencePages": { "examinerReport": [ 10, 11 ] } }, { "id": "ial-legacy-2018-january-WMA01-q13c", "paperId": "ial-legacy-2018-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2018, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(c)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Determine the bearing of boat B from boat A to the nearest degree.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_que_20180109.pdf#page=42", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_rms_20180308.pdf#page=20", "qpPages": [ 42, 45 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson reports that interpreting the direction bearing in part (c) troubled many candidates; this part often had no attempt or only partial work. Many found a triangle angle but did not convert it to the required bearing correctly. (PEF p. 12)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the cosine or sine rule to find the needed angle in the triangle, then combine it with the given north bearings and measure clockwise from north; the accepted result is 113° or 114°.", "sourceEvidencePages": { "examinerReport": [ 12 ] } }, { "id": "ial-legacy-2018-january-WMA01-q14b", "paperId": "ial-legacy-2018-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2018, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(b)", "marks": 8, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Calculate the exact area of a shaded region bounded by coordinate axes, a line, and a quadratic curve.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_que_20180109.pdf#page=46", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_rms_20180308.pdf#page=23", "qpPages": [ 46, 50 ], "msPages": [ 23, 26 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson describes Q14 as a challenging problem with full marks earned by a minority. In part (b), common difficulties included choosing limits that matched the shaded region and recognizing where algebraic integration had to be combined with a geometric area. (PEF pp. 12–13)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the relevant intersection at x = 3.5, integrate the curve over the correct interval, and combine the result with the line or trapezium area needed to match the shaded region; the exact area is 499/24.", "sourceEvidencePages": { "examinerReport": [ 12, 13 ] } }, { "id": "ial-legacy-2018-january-WMA01-q15", "paperId": "ial-legacy-2018-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2018, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 15, "questionPart": null, "marks": 10, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use coefficients from a binomial expansion to determine its parameters and a later coefficient.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_que_20180109.pdf#page=50", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA01_01_rms_20180308.pdf#page=27", "qpPages": [ 50, 52 ], "msPages": [ 27, 27 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson reports that Q15 was often not attempted and appeared to be a less familiar binomial-expansion problem. It notes that candidates needed to compare coefficients systematically, then use the recovered parameters to form the cubic coefficient. (PEF pp. 13–14)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the x and x² coefficients to obtain nk = 36 and nk(n − 1) = 252, giving n = 8 and k = 4.5. Substitute these values into the binomial coefficient for x³ to obtain 5103.", "sourceEvidencePages": { "examinerReport": [ 13, 14 ] } }, { "id": "cie-2014-w-11-q11i", "paperId": "cie-2014-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(i)", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the angle between the tangents to two curves at their intersection Q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates used both tangent gradients to find their angles with the x-axis and reached the final answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 1, 4 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate both curves, evaluate the gradients at Q, find each tangent angle to the x-axis, and take their difference." }, { "id": "cie-2014-w-11-q03", "paperId": "cie-2014-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation for every angle in the stated degree interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates often omitted the negative cosine solution or answered in radians despite a degree interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 1, 4 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity to form an equation in cos θ, solve it, and include both valid angles, including the one from the negative cosine." }, { "id": "cie-2014-w-11-q05", "paperId": "cie-2014-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the values of k for which a line meets a quadratic curve at two distinct points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many solutions stopped after eliminating y; candidates who found the set of k values rarely showed the correct inequalities or diagram.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 1, 4 ], "tagConfidence": "reviewed", "answerGuide": "Equate the line and curve, form a quadratic in x, require its discriminant to be strictly positive, then solve the inequality in k." }, { "id": "cie-2014-w-11-q07ii", "paperId": "cie-2014-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find n from the nth and (3n)th terms of an arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates often overlooked direct elimination of the common difference and made algebraic errors in a longer substitution route.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 1, 4 ], "tagConfidence": "reviewed", "answerGuide": "Write both terms using the first term and common difference, eliminate the difference, and solve for the positive integer n." }, { "id": "cie-2014-w-12-q09iii", "paperId": "cie-2014-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(iii)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find BE when ABCE is a parallelogram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The first two parts were more accessible, but only a few candidates completed the final part successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 5, 8 ], "tagConfidence": "reviewed", "answerGuide": "Use the coordinates of A, B and C to obtain vector BE, then calculate its magnitude." }, { "id": "cie-2014-w-12-q02ii", "paperId": "cie-2014-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Calculate the area of the shaded region bounded by two arcs and a triangle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only the more able candidates combined sector and triangle areas in the correct configuration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 5, 8 ], "tagConfidence": "reviewed", "answerGuide": "Find the relevant angles, calculate sector areas in radians, and combine them with the triangle area using the correct signs." }, { "id": "cie-2014-w-12-q01", "paperId": "cie-2014-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the shaded region about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates found an area or rotated about the x-axis; others used incorrect limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 5, 8 ], "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of y, use the disc or washer method with respect to y, and apply the shaded region limits." }, { "id": "cie-2014-w-12-q03ii", "paperId": "cie-2014-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the earlier expansion to find the positive constant p from a coefficient condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The second part was more challenging; candidates often omitted terms or mishandled the squared p term.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 5, 8 ], "tagConfidence": "reviewed", "answerGuide": "Use the expansion from part (i), identify every contribution to the x² coefficient, and solve for positive p." }, { "id": "cie-2014-w-12-q08a", "paperId": "cie-2014-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the first term and common difference from the arithmetic-series sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Candidates often treated S₂ as the second term or abandoned coefficient comparison after expanding the formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 5, 8 ], "tagConfidence": "reviewed", "answerGuide": "Substitute small n values or compare coefficients with the AP sum formula; distinguish the first term from S₂." }, { "id": "cie-2014-w-12-q08b", "paperId": "cie-2014-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the first term of a positive GP from its infinite sum and first-two-term sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Candidates confused the two-term sum with the second term, used an incorrect infinite-sum formula, or stopped after finding r.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 5, 8 ], "tagConfidence": "reviewed", "answerGuide": "Use S∞=a/(1−r) and a+ar=12.8, solve for the positive ratio r, then substitute to find a." }, { "id": "cie-2014-w-13-q02", "paperId": "cie-2014-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter and area of a shaded region formed by tangents and a circular sector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The question was often left until last or omitted, and a significant proportion could not find the tangent length and angles.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 9, 11 ], "tagConfidence": "reviewed", "answerGuide": "Use tangent-radius geometry to determine the sector angle and lengths, then combine straight edges, arc length and sector/triangle areas." }, { "id": "cie-2014-w-13-q08ii", "paperId": "cie-2014-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 7, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Find f′(x) and f(x) from the second derivative and stationary-point conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many omitted constants at both integrations or used f′(3)=7 instead of the stationary condition f′(3)=0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 9, 11 ], "tagConfidence": "reviewed", "answerGuide": "Integrate twice with two separate constants, using f′(3)=0 for the first and f(3)=7 for the second." }, { "id": "cie-2014-w-13-q03ii", "paperId": "cie-2014-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Determine whether the cubic is increasing, decreasing or neither.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Relatively few connected part (i) to the derivative; many used the second derivative or substituted isolated values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 9, 11 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the completed-square form from part (i), then use the derivative sign over the domain to justify monotonicity." }, { "id": "cie-2014-w-13-q04i", "paperId": "cie-2014-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the infinite sum of progression R when the three sums form an arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates often assumed R followed P and Q’s term pattern, ignoring that the infinite sums form an AP; the question was often omitted.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 9, 11 ], "tagConfidence": "reviewed", "answerGuide": "Find the sums to infinity of P and Q, use them as the first two AP terms, and continue to its third term." }, { "id": "cie-2014-w-13-q04ii", "paperId": "cie-2014-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the sum of R’s first three terms given its first term and infinite sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A common error was writing the first three terms without calculating their sum.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 9, 11 ], "tagConfidence": "reviewed", "answerGuide": "Use the first term and infinite sum to determine the common ratio, then add the first three terms." }, { "id": "cie-2014-w-13-q10aii", "paperId": "cie-2014-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2014, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)(ii)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the cube-root function and state its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report notes errors handling the cube root and says inverse-domain notation was often unsuccessful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 9, 11 ], "tagConfidence": "reviewed", "answerGuide": "Set y=f(x), cube both sides, solve for x, and use the original function range as the inverse domain." }, { "id": "cie-2014-w-21-q01", "paperId": "cie-2014-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use the trapezium rule to estimate the integral containing an absolute-value expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few answers were correct; candidates often ignored the modulus, squared the expression, or integrated instead of using the rule.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the integrand at all required ordinates, take the absolute value at each point, apply the four-interval weights and multiply by the interval width." }, { "id": "cie-2014-w-21-q02", "paperId": "cie-2014-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Find a and b from a straight-line plot of ln y against x for an exponential model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates related the plotted ln y values to the model correctly; premature rounding also caused inaccurate results.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Use the graph’s gradient and intercept to determine b and ln a, then exponentiate the intercept and round as requested." }, { "id": "cie-2014-w-21-q07ii", "paperId": "cie-2014-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the given tangent information to find an exact compound-angle value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few correct solutions were seen; candidates rarely substituted the exact tan α result into the compound-angle formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Apply the tangent addition formula, substitute tan α from part (i), and simplify to an exact result." }, { "id": "cie-2014-w-21-q03a", "paperId": "cie-2014-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the indefinite trigonometric integral in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates recognised the required double-angle identity before integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 12, 14 ], "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity to rewrite the integrand, integrate each term with respect to θ, and include the integration constant." }, { "id": "cie-2014-w-22-q07iii", "paperId": "cie-2014-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Deduce the greatest value of the trigonometric expression and the first angle where it occurs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct solutions were seen; many did not use part (i) or tried an unnecessary calculus method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the amplitude form from part (i), apply the angle restriction, and identify the smallest positive angle attaining the maximum." }, { "id": "cie-2014-w-22-q02ii", "paperId": "cie-2014-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Deduce the improper integral over the positive half-line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few correct responses were seen; many left terms involving e^(−∞) unevaluated.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the antiderivative from part (i), evaluate its limit as x tends to infinity, and use e^(−mx)→0 for positive m." }, { "id": "cie-2014-w-22-q04a", "paperId": "cie-2014-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation for x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates split logarithms incorrectly, and some retained an invalid solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Combine logarithms using valid laws, preserve domain restrictions, solve, and reject extraneous values." }, { "id": "cie-2014-w-22-q05ii", "paperId": "cie-2014-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the exact area bounded by the curve, axes and vertical line through its maximum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Poor integration was common, and many otherwise correct responses lost the final mark by not giving an exact answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 15, 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the maximum-point x-coordinate from part (i) as the upper limit, integrate the curve, and simplify exactly." }, { "id": "cie-2014-w-23-q01", "paperId": "cie-2014-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use the trapezium rule to estimate the integral containing an absolute-value expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few answers were correct; candidates often ignored the modulus, squared the expression, or integrated instead of using the rule.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 18, 20 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the integrand at all required ordinates, take the absolute value at each point, apply the four-interval weights and multiply by the interval width." }, { "id": "cie-2014-w-23-q02", "paperId": "cie-2014-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Find a and b from a straight-line plot of ln y against x for an exponential model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates related the plotted ln y values to the model correctly; premature rounding also caused inaccurate results.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 18, 20 ], "tagConfidence": "reviewed", "answerGuide": "Use the graph’s gradient and intercept to determine b and ln a, then exponentiate the intercept and round as requested." }, { "id": "cie-2014-w-23-q07ii", "paperId": "cie-2014-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the given tangent information to find an exact compound-angle value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few correct solutions were seen; candidates rarely substituted the exact tan α result into the compound-angle formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 18, 20 ], "tagConfidence": "reviewed", "answerGuide": "Apply the tangent addition formula, substitute tan α from part (i), and simplify to an exact result." }, { "id": "cie-2014-w-23-q03a", "paperId": "cie-2014-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2014, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the indefinite trigonometric integral in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates recognised the required double-angle identity before integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 18, 20 ], "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity to rewrite the integrand, integrate each term with respect to θ, and include the integration constant." }, { "id": "cie-2014-w-31-q05ii", "paperId": "cie-2014-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find z from the given relation when w=z and the real part of z is negative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Most candidates lost all possible marks after substituting a guessed z; only a minority used the quadratic approach successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Substitute w=z, rearrange to a quadratic in z, solve using complex arithmetic, and select the root with negative real part." }, { "id": "cie-2014-w-31-q08iii", "paperId": "cie-2014-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iii)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the cubic after using the trigonometric substitution from the earlier parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates often stopped after solving for the angle instead of calculating x; the negative root was especially elusive.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Solve the corresponding sine equation over its required range, convert each angle to x, and include the negative root." }, { "id": "cie-2014-w-31-q07ii", "paperId": "cie-2014-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the maximum annual tax revenue predicted by the differential-equation model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Part (ii) was poorly done; very few candidates set dR/dx to zero to find the maximum.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Use R(x) from part (i), differentiate, find the stationary tax rate, and evaluate R there." }, { "id": "cie-2014-w-31-q02i", "paperId": "cie-2014-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the cosecant integral using the trapezium rule with three intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Question 2 was generally poorly answered; candidates used abscissae, incorrect weights or interval widths.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Use four equally spaced ordinates, apply weights 1,2,2,1, multiply by the interval width, and round to two decimal places." }, { "id": "cie-2014-w-31-q02ii", "paperId": "cie-2014-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Explain whether the trapezium estimate is above or below the true cosecant integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Only a few candidates adequately explained why the trapezium rule gives an overestimate.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 21, 23 ], "tagConfidence": "reviewed", "answerGuide": "Sketch y=cosec x on the interval and use its curvature to show how each trapezium compares with the curve." }, { "id": "cie-2014-w-32-q05ii", "paperId": "cie-2014-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find z from the given relation when w=z and the real part of z is negative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Most candidates lost all possible marks after substituting a guessed z; only a minority used the quadratic approach successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Substitute w=z, rearrange to a quadratic in z, solve using complex arithmetic, and select the root with negative real part." }, { "id": "cie-2014-w-32-q08iii", "paperId": "cie-2014-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the cubic after using the trigonometric substitution from the earlier parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates often stopped after solving for the angle instead of calculating x; the negative root was especially elusive.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Solve the corresponding sine equation over its required range, convert each angle to x, and include the negative root." }, { "id": "cie-2014-w-32-q07ii", "paperId": "cie-2014-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the maximum annual tax revenue predicted by the differential-equation model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Part (ii) was poorly done; very few candidates set dR/dx to zero to find the maximum.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Use R(x) from part (i), differentiate, find the stationary tax rate, and evaluate R there." }, { "id": "cie-2014-w-32-q02i", "paperId": "cie-2014-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the cosecant integral using the trapezium rule with three intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Question 2 was generally poorly answered; candidates used abscissae, incorrect weights or interval widths.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Use four equally spaced ordinates, apply weights 1,2,2,1, multiply by the interval width, and round to two decimal places." }, { "id": "cie-2014-w-32-q02ii", "paperId": "cie-2014-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Explain whether the trapezium estimate is above or below the true cosecant integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Only a few candidates adequately explained why the trapezium rule gives an overestimate.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 24, 26 ], "tagConfidence": "reviewed", "answerGuide": "Sketch y=cosec x on the interval and use its curvature to show how each trapezium compares with the curve." }, { "id": "cie-2014-w-33-q09i", "paperId": "cie-2014-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Sketch two functions to show that the stated equation has exactly one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Responses were weak; candidates often drew inaccurate curves or failed to show what happens outside the first quadrant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 27, 30 ], "tagConfidence": "reviewed", "answerGuide": "Sketch y=ln(x+1) and y=40−x³ with correct domains and behaviour outside the first quadrant, then state the second curve equation and show the unique intersection." }, { "id": "cie-2014-w-33-q08ii", "paperId": "cie-2014-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 3, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Use the initial condition to find the particular solution value at x=25.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A fully correct answer to part (ii) was comparatively rare.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 27, 30 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the initial condition into the general solution, determine the integration constant, and evaluate at x=25." }, { "id": "cie-2014-w-33-q03ii", "paperId": "cie-2014-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2014, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the remainder when the cubic polynomial is divided by x²+1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates struggled to keep quadratic division aligned or use a linear remainder; some confused quotient and remainder.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 27, 30 ], "tagConfidence": "reviewed", "answerGuide": "Use polynomial division or x²=−1 to reduce the cubic to a linear expression, then identify the remainder." }, { "id": "cie-2014-w-41-q03ii", "paperId": "cie-2014-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the friction inequality when the block slides down the inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates achieved full marks on Question 3 because they did not clearly explain the inequalities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 31, 33 ], "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the plane, use limiting friction F=μR, then apply the positive down-slope resultant condition." }, { "id": "cie-2014-w-41-q03i", "paperId": "cie-2014-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the lower bound on the coefficient of friction while the block remains at rest with the force acting up the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many found the equality value of μ but did not explain the required inequality; full marks on the question were uncommon.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 31, 33 ], "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the plane to obtain F and R, then use the static-friction condition F≤μR." }, { "id": "cie-2014-w-42-q06ii", "paperId": "cie-2014-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find AB when the particle has two different constant accelerations on the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 6 was called the most difficult; only a few candidates completed the distance calculation successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 34, 37 ], "tagConfidence": "reviewed", "answerGuide": "Use the accelerations for AB and BC with v²=u²+2as, then apply AB+BC=5 m and the final-speed condition." }, { "id": "cie-2014-w-42-q02iii", "paperId": "cie-2014-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(iii)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "State the force replacing 20 N while reversing the resultant direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few completed Question 2 correctly; the replacement force was often stated as −5 N without its direction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 34, 37 ], "tagConfidence": "reviewed", "answerGuide": "Use the unchanged resultant magnitude from part (ii), reverse its direction, and resolve the replacement force from the other two forces." }, { "id": "cie-2014-w-43-q07", "paperId": "cie-2014-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7", "marks": 11, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Analyse the block’s motion up and then back down the rough plane after the pulling force is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Question 7 the most challenging and describes recurring errors in the forces, normal reaction and subsequent distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 38, 40 ], "tagConfidence": "reviewed", "answerGuide": "For each stage include the weight component, normal reaction and friction; use work-energy or Newton’s second law consistently to find work, friction coefficient and distance." }, { "id": "cie-2014-w-43-q04iii", "paperId": "cie-2014-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find displacement after 18 seconds when velocity changes from constant acceleration to a quadratic model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many integrated over the whole interval using one formula or used the wrong time for the integration constant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 38, 40 ], "tagConfidence": "reviewed", "answerGuide": "Integrate velocity separately over 0–8 and 8–18 seconds, using the continuity value and constant found earlier." }, { "id": "cie-2014-w-43-q06i", "paperId": "cie-2014-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 1", "year": 2014, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the initial acceleration magnitude and each particle’s mass from the velocity-time graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates were unclear how to find the two masses and sometimes gave signed accelerations instead of the required magnitude.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 38, 40 ], "tagConfidence": "reviewed", "answerGuide": "Read the initial slopes of both velocity-time segments, apply Newton’s second law to the connected particles, and use total mass 1 kg." }, { "id": "cie-2014-w-51-q05ii", "paperId": "cie-2014-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the ball’s height and speed when its direction is 60° below horizontal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates scored any marks on this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Use the trajectory to relate slope to velocity components, find vertical velocity and displacement, then combine horizontal and vertical components for speed." }, { "id": "cie-2014-w-51-q01", "paperId": "cie-2014-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the launch speed from the angle of the position vector after two seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The question was challenging; candidates often confused the position-vector slope with the velocity direction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Use projectile displacement equations at t=2, set the ratio of vertical to horizontal displacement to tan 15°, and solve for V." }, { "id": "cie-2014-w-51-q04ii", "paperId": "cie-2014-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest and least horizontal forces that keep the prism in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Most found the greatest force, but only a few continued to find the least by taking moments about F.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Use limiting friction and moments about the appropriate contact point for each extreme, including the centre-of-mass distance." }, { "id": "cie-2014-w-52-q05ii", "paperId": "cie-2014-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the ball’s height and speed when its direction is 60° below horizontal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates scored any marks on this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Use the trajectory to relate slope to velocity components, find vertical velocity and displacement, then combine horizontal and vertical components for speed." }, { "id": "cie-2014-w-52-q01", "paperId": "cie-2014-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the launch speed from the angle of the position vector after two seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The question was challenging; candidates often confused the position-vector slope with the velocity direction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Use projectile displacement equations at t=2, set the ratio of vertical to horizontal displacement to tan 15°, and solve for V." }, { "id": "cie-2014-w-52-q04ii", "paperId": "cie-2014-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest and least horizontal forces that keep the prism in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Most found the greatest force, but only a few continued to find the least by taking moments about F.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Use limiting friction and moments about the appropriate contact point for each extreme, including the centre-of-mass distance." }, { "id": "cie-2014-w-53-q06ii", "paperId": "cie-2014-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest and least angular speeds that allow the particle to rotate without slipping.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many reused the friction from part (i), and many found only one of the two extreme angular speeds.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Resolve tension and weight to obtain the normal reaction, use the friction range, and apply the radial force equation at both limiting cases." }, { "id": "cie-2014-w-53-q05ii", "paperId": "cie-2014-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed after the block is raised and released between two elastic strings.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Too many candidates failed to set up the required five-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Use the work-energy equation with all energy terms as the block moves, then find the position at which speed is greatest." }, { "id": "cie-2014-w-53-q03i", "paperId": "cie-2014-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2014, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show the differential relation between velocity and height while the ball moves upward against resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a small number of candidates arrived at the required result in this show part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Resolve forces to find acceleration, use a=v(dv/dx), and substitute the given resisting force." }, { "id": "cie-2014-w-61-q05iii", "paperId": "cie-2014-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that exactly seven packets each contain at least one faulty screw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only the best candidates found the single-packet probability before using a binomial model for eight packets.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 47, 48 ], "tagConfidence": "reviewed", "answerGuide": "Find the probability a packet has at least one faulty screw, then use a binomial model for eight independent packets." }, { "id": "cie-2014-w-61-q02iii", "paperId": "cie-2014-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(iii)", "marks": 3, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Find the probability that the daily call count exceeds its mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only the better candidates found the mean before selecting the relevant call-count probabilities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 47, 48 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the mean from the probability table, then sum the probabilities for integer call counts strictly above it." }, { "id": "cie-2014-w-62-q06ii", "paperId": "cie-2014-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate h when 28% of daffodils are at least that tall.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This was the worst-attempted part; only about 5% converted 28% above h into 72% below h.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Use the cumulative frequency graph at the 72nd percentile, since 72% are at or below h, then read the height." }, { "id": "cie-2014-w-62-q02ii", "paperId": "cie-2014-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements in which no two girls stand next to each other.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates commonly excluded only the arrangement with all four girls together, missing runs of two or three.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Arrange the six boys first, then choose four of the seven gaps for the girls and arrange the four girls in them." }, { "id": "cie-2014-w-62-q04ii", "paperId": "cie-2014-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Complete the distribution of the number of attempts and find its expectation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates formed the table but did not complete the final instruction to find E(X).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Find P(X=1), P(X=2), and P(X=3) from the answer process, then calculate E(X)=ΣxP(X=x)." }, { "id": "cie-2014-w-63-q06a", "paperId": "cie-2014-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count ordered arrangements of seven dice whose end values sum to four.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few fully correct solutions were seen; candidates used factorials, probabilities or incorrect powers for the five middle dice.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 51, 53 ], "tagConfidence": "reviewed", "answerGuide": "List the possible ordered end pairs (3,1), (1,3), and (2,2), then multiply by the six choices for each of the five middle dice." }, { "id": "cie-2014-w-63-q02ii", "paperId": "cie-2014-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Evaluate the sum of squared deviations from 62 for the eight car speeds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few correct solutions were seen; most interpreted the expression as squaring their answer to part (i).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w14_er.pdf", "difficultyReportPages": [ 51, 53 ], "tagConfidence": "reviewed", "answerGuide": "Subtract 62 from each speed, square each difference, and add all eight values without confusing this with the square of the sum." }, { "id": "cie-2014-w-63-q05ii", "paperId": "cie-2014-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2014, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find k such that P(k 3.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA02_01_que_20180116.pdf#page=28", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA02_01_rms_20180308.pdf#page=21", "qpPages": [ 28, 31 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "Pearson says this range question caused problems for many; candidates often identified 1 but gave an incorrect inequality or range.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite g(x) as 1 + 3/(x − 3). Since x > 3, the second term is positive, so the range is (1, ∞).", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-legacy-2018-january-WMA02-q3c", "paperId": "ial-legacy-2018-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2018, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Justify that the root of f(x) = x²/4 + ln(2x) rounds to 0.473 to three decimal places.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA02_01_que_20180116.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WMA02_01_rms_20180308.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 10, "difficultyEvidence": "Pearson identifies this as the least well-answered part of Q3; common omissions were a reason or conclusion, and some candidates misread values in standard form.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Evaluate f at suitable bounds around 0.473 (for example, 0.4725 and 0.4735), show the sign change, and state the rounding conclusion.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "cie-2016-m-12-q09b", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Use the sector result from part (a) to find the exact shaded area inside the equilateral triangle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls part (b) a significant challenge, including for many good candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the common radius and sector angle from the tangent geometry. Subtract the three equal circular segments from the area of the equilateral triangle, keeping the exact π and √3 terms." }, { "id": "cie-2016-m-12-q07i", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the vector from C to P is 2.4i − 1.8k using the stated pyramid geometry.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says Question 7 was one of the more difficult questions and that many candidates struggled to find CA and use CP = 3/5 of CA.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write CA from the given unit-vector directions and lengths, then use the fact that CP is three-fifths of CA. Preserve the direction from C to A." }, { "id": "cie-2016-m-12-q07ii", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Express OP and BP in terms of the unit vectors i, j and k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was similarly difficult after many candidates struggled with part (i).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the known position vectors of O, B and C together with CP from part (i). Form OP from O to P, then subtract OB from OP to obtain BP." }, { "id": "cie-2016-m-12-q04b", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Factorise and solve the trigonometric equation for 0 ≤ θ ≤ π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says part (b) proved more challenging than expected; many could not factorise it, and some lost solutions or used degrees instead of radians.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Factor the expression as (2 cos θ − 1)(sin θ − 1) = 0. Solve both factors in the full radian interval, including both endpoint checks." }, { "id": "cie-2016-m-12-q09aii", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)(ii)", "marks": 2, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the area of the shaded circular segment in terms of r and α.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that a surprisingly large number of candidates did not use the triangle-area formula for this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Subtract the isosceles triangle area from the sector area: use the included angle 2α, sector area r²α, and triangle area ½r² sin(2α)." }, { "id": "cie-2016-m-12-q08ii", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the specified quadratic function and state its domain when a = 1 and b = −1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many did not complete the square; only the strongest candidates selected the negative square-root branch required by the given domain.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Complete the square, interchange x and y, and solve for y. Use the original restriction x ≤ 1/2 to choose the correct branch, then state the inverse domain from the quadratic's minimum value." }, { "id": "cie-2016-m-12-q06ii", "paperId": "cie-2016-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2016, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Find the stationary radius for the flask's surface-area function and verify it gives a minimum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says a greater number of candidates were unsure how to confirm efficiency at their stationary value; many found an area value instead of proving it was a minimum.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate A = 2πr² + 2000/r, solve dA/dr = 0 for the positive radius, and use the second derivative (or a clear sign test) to establish a minimum." }, { "id": "cie-2016-m-22-q04ii", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Use the equation satisfied by the converged iteration to find the exact value of α.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few correct solutions were seen; candidates often reused the decimal from part (i) instead of solving for the exact limit.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the limiting value equal to its image under the iteration, solve the resulting quadratic exactly, and select the root consistent with the stated convergence." }, { "id": "cie-2016-m-22-q03", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation for x in terms of the positive constant k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few candidates used the addition rule for the two logarithms, so many did not obtain the required quadratic.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the right-hand logarithms into one log, equate arguments, and solve the resulting quadratic. Apply the original logarithm conditions so every argument is positive." }, { "id": "cie-2016-m-22-q02", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the inequality involving |x − 5| and |2x + 3| and give the complete range for x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the critical values were often found but the correct range for x was seldom stated.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Because both sides are non-negative, compare their squares or split at the zeros x = 5 and x = −3/2. Test the resulting intervals and state both valid ranges." }, { "id": "cie-2016-m-22-q07i", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Differentiate 2x³ + y³ = 24 implicitly and show that the gradient is never positive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates obtained a relationship but were unable to justify why the gradient could never be positive.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate with respect to x, keeping dy/dx on terms involving y. Rearrange to a quotient and use the squared quantities and the curve restriction to justify its sign." }, { "id": "cie-2016-m-22-q07ii", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the two points on the curve where the gradient is −2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates commonly found only one solution and no attempts at the second solution were seen.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the derivative from part (i) equal to −2, solve the resulting relation between x and y, and substitute every branch into the original curve equation." }, { "id": "cie-2016-m-22-q08iia", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)(a)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the least possible value of the trigonometric expression using the identity from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says candidates often reached a simplified form but failed to recognize the least possible value of its trigonometric term.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use sin 2x cot x = 2 cos²x and cos 2x = 2 cos²x − 1 to rewrite the expression as a constant plus a non-negative multiple of cos²x. Use cos²x ≥ 0." }, { "id": "cie-2016-m-22-q05", "paperId": "cie-2016-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2016, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find a from the definite integral involving 6e^(2x+1).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says most candidates integrated correctly, but few rearranged the resulting equation successfully; logarithm work caused most incorrect answers.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate 6e^(2x+1), substitute both limits, isolate the exponential term, then take natural logarithms and solve for a." }, { "id": "cie-2016-m-32-q08ii", "paperId": "cie-2016-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector equation for the line in the plane that passes through (5, 3, 1) and is perpendicular to the given line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates made much progress; many used an incorrect vector, while successful solutions formed a vector product from the line direction and plane normal.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find a direction vector perpendicular to both the given line direction and the plane normal, for example using their vector product. Use (5, 3, 1) as the position vector of the line." }, { "id": "cie-2016-m-32-q10a", "paperId": "cie-2016-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the conjugate equation for z and give the result in Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates made any progress after substituting z and its conjugate; many did not separate real and imaginary parts.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write z = x + iy and z* = x − iy. Substitute both, equate real and imaginary parts to form two real equations, then solve for x and y." }, { "id": "cie-2016-m-32-q07i", "paperId": "cie-2016-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 7, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "exponentials", "summary": "Solve the separable differential equation with y = 0 when x = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates who separated variables often struggled to integrate e^(−x), while integration by parts for xe^x was also required.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the y- and x-dependent factors, integrate both sides, use integration by parts for xe^x, and apply the initial condition to determine the constant." }, { "id": "cie-2016-m-32-q10bi", "paperId": "cie-2016-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(i)", "marks": 4, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch and shade the Argand-plane region satisfying the circle and imaginary-part inequalities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the circle was often assigned the wrong centre or radius, and Im z ≥ 3 was often mistaken for a circle centred at the origin.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the circle with centre (−1, 3) and radius 1, then the horizontal line y = 3. Shade only the part of the disk on or above that line." }, { "id": "cie-2016-m-32-q10bii", "paperId": "cie-2016-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 2, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the difference between the greatest and least arguments of points in the shaded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says only candidates with the correct diagram could attempt this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the two extreme boundary points of the upper semicircle, (−2, 3) and (0, 3), to obtain the endpoint arguments, then subtract them in the correct order." }, { "id": "cie-2016-m-32-q09ii", "paperId": "cie-2016-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2016, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of f(x) from 1 to 2 exactly after using the decomposition in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says the logarithmic integration proved difficult when a decomposition term was wrong and stresses that candidates needed to show all limit-substitution and logarithm steps.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate each decomposed term, substitute both limits, and show the subsequent logarithm simplification explicitly rather than jumping to the stated exact answer." }, { "id": "cie-2016-m-42-q07iv", "paperId": "cie-2016-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled over 0 ≤ t ≤ 20 from the piecewise velocity function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says almost all candidates found this difficult: integrating from 0 to 4 gives displacement, not distance, because velocity is negative before t = 2.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the first interval at the zero of velocity t = 2, add the absolute areas on 0–2 and 2–4, then add the areas under the positive later sections." }, { "id": "cie-2016-m-42-q05ii", "paperId": "cie-2016-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find how long the car–trailer system takes to stop after the driving force is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates incorrectly assumed the acceleration remained unchanged when the driving force became zero.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recalculate the system acceleration with no driving force, including resistance and the component of weight, then use v = u + at with final speed zero." }, { "id": "cie-2016-m-42-q05i", "paperId": "cie-2016-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of the car–trailer system and the tension in the cable while travelling uphill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates were confused about which forces act on each body and recommends separate force diagrams for the car and trailer.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First apply Newton's second law to the combined system to find acceleration. Then isolate either the car or trailer and use its equation of motion to find cable tension." }, { "id": "cie-2016-m-42-q06ii", "paperId": "cie-2016-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2016, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed of A immediately before it reaches the pulley when the surface is rough.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says a frequent error was reusing the tension from the smooth-surface case instead of finding the new acceleration.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the new friction force from the coefficient and normal reaction. Apply Newton's second law to both connected particles to find acceleration, then use v² = u² + 2as." }, { "id": "cie-2016-m-52-q07iiib", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest possible tension in the string and the corresponding speed of the particle on the sphere.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Question 7 as the hardest on the paper and says only a few candidates recognized that maximum tension occurs when the normal reaction is zero.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the two equations from the earlier parts and set R = 0 at the limiting position for greatest tension. Then use the radial equation to find angular speed and convert to linear speed using radius 0.6 sin 60°." }, { "id": "cie-2016-m-52-q05iii", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle's speed when it is 0.5 m above its equilibrium position.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a few candidates scored any marks on this part; a four-term energy balance was required.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy between the equilibrium position and the stated point, including initial elastic energy, kinetic energy, gravitational potential-energy change and final elastic energy." }, { "id": "cie-2016-m-52-q05ii", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find AP when the particle is below equilibrium and its speed is 3.5 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says only a few candidates solved this part; a five-term energy equation was needed and terms were often omitted.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write a complete energy balance from the equilibrium position to the lower point, including kinetic, gravitational and elastic-energy terms at both positions. Relate the extension to AP." }, { "id": "cie-2016-m-52-q07iiia", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)(a)", "marks": 2, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction R when the angular speed is 2 rad s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates recognized that the two simultaneous equations were needed, but the equations were often incorrect.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute ω = 2 into the horizontal and vertical equations from parts (i) and (ii), then eliminate T or R and solve the resulting simultaneous equations." }, { "id": "cie-2016-m-52-q07ii", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Form the equation involving R, T and the angular speed for motion around the horizontal circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates often used the wrong radius, 0.6 instead of the horizontal radius 0.6 sin 60°.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally and equate the resultant inward force to mω²r, using the circle's horizontal radius 0.6 sin 60° and the correct components of T and R." }, { "id": "cie-2016-m-52-q07i", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve vertically to obtain the relationship between R and T.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says too many candidates mixed up the trigonometric ratios in the vertical resolution.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the contact and tension forces vertically, using cos 60° for R and cos 30° for T, and set their sum equal to the particle's weight." }, { "id": "cie-2016-m-52-q02i", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "mechanics", "proof" ], "primaryTopicId": "mechanics", "summary": "Show that the force P keeps the hemisphere in limiting equilibrium at the stated angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says few candidates could do this part; the key step was taking moments about the contact point and finding P's perpendicular distance.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the point where the hemisphere touches the plane. Use the perpendicular distance 0.8 − 0.8 cos θ for P and the given centre-of-mass geometry." }, { "id": "cie-2016-m-52-q03", "paperId": "cie-2016-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2016, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the horizontal distance between the two points where the projectile's path is at 45° to the horizontal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report describes the question as quite difficult and gives a two-stage resolution and constant-acceleration method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use constant horizontal velocity to find the speed at each 45° point, then use vertical motion to find the time between them. Multiply that time by the unchanged horizontal velocity." }, { "id": "cie-2016-m-62-q05i", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Given that a randomly chosen person goes camping, find the probability that the person goes abroad.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says weaker candidates did not recognize that this is a conditional-probability question and gave a joint probability instead.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find P(abroad and camping) and divide by P(camping), or use a tree/table with the stated country and activity percentages. Keep the condition in the denominator." }, { "id": "cie-2016-m-62-q03i", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the two throws of the eight-sided die produce scores differing by exactly 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says some candidates ignored the 7 and 8, while weaker solutions failed to include both orders of each valid pair.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "List all ordered outcomes from two eight-sided throws that differ by 3. Count both possible orders for every eligible pair and divide by 64." }, { "id": "cie-2016-m-62-q03ii", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the product of the two scores is greater than 19.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often counted the qualifying outcomes inaccurately, even when they understood the product condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use an 8 by 8 outcome space, list or count each ordered pair whose product exceeds 19, and divide the total by 64." }, { "id": "cie-2016-m-62-q03iii", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(iii)", "marks": 3, "topicIds": [ "probability", "proof" ], "primaryTopicId": "probability", "summary": "Determine whether the two events are independent and justify the conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes circular independence arguments and says candidates needed to compare the intersection probability and state a conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate P(R ∩ S) and P(R)P(S) using the counts from the earlier parts. Compare the values and explicitly state whether R and S are independent." }, { "id": "cie-2016-m-62-q07ii", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that none of four cars takes more than 2 hours to have a tow bar fitted.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many candidates could not gain full credit because they did not keep sufficient accuracy and needed a more precise mean from part (i).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardize 2 hours using the normal model and precise mean from part (i). Find the single-car probability of a time below 2, then raise it to the fourth power." }, { "id": "cie-2016-m-62-q07iii", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the probability a second garage takes longer than 0.6μ hours when its model has μ = 3σ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says weaker candidates made algebra errors when substituting μ = 3σ into the standardization formula.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute μ = 3σ consistently into (0.6μ − μ)/σ, simplify the z-score, and use the upper tail of the standard normal distribution." }, { "id": "cie-2016-m-62-q01ii", "paperId": "cie-2016-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2016, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the coded sum and mean to find the constant a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says the coded mean caused difficulty and that weaker solutions failed to replace Σa with 10a.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the mean to find Σx, then expand Σ(x − a) as Σx − 10a for the ten observations and solve for a." }, { "id": "cie-2016-m-72-q01", "paperId": "cie-2016-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of the sum of two independent binomial counts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a poorly attempted question; candidates often failed to combine the variances correctly or left the final answer as a variance rather than a standard deviation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the mean and variance of each independent binomial variable. Add the means and add the variances, then take the positive square root for the standard deviation." }, { "id": "cie-2016-m-72-q02", "paperId": "cie-2016-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2016, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a 1% significance test to assess whether Jill's hit rate improved from 65% using 18 hits in 25 shots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m16_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says Questions 1 and 2 were more demanding; candidates often omitted hypotheses, the binomial tail probability or a cautious conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_m16_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State H₀: p = 0.65 and H₁: p > 0.65. Under H₀ use X ~ Bin(25, 0.65), calculate P(X ≥ 18), compare it with 0.01, and give the conclusion in context." }, { "id": "cie-2013-s-11-q01", "paperId": "cie-2013-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "Show that a polynomial function is increasing for all real x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates substituted only a few x-values instead of proving the derivative was positive for all x, so the central proof method was often missing.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the function and show its derivative is positive throughout the stated domain." }, { "id": "cie-2013-s-11-q05ii", "paperId": "cie-2013-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over a full revolution after deriving a squared identity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few candidates remembered the ± when taking square roots, so most found only two of the four solutions.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (i), isolate the squared trigonometric term, and remember both signs when taking square roots before checking all angles in the interval." }, { "id": "cie-2013-s-11-q06ii", "paperId": "cie-2013-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the magnitude of a vector difference in terms of parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says part (ii) was not well done: many made errors finding the vector difference, and candidates who reached it often failed to recognise that two components were zero before finding its magnitude.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Subtract the position vectors to form the difference, simplify its components, then use the magnitude formula." }, { "id": "cie-2013-s-11-q07ii", "paperId": "cie-2013-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent line parameter and its point of contact with a parabola.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says that few correct answers were seen for this part despite its standard method of equating the equations and applying the equal-root condition.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the line equal to the curve, impose the repeated-root condition, solve for the non-zero parameter, then find the point of contact." }, { "id": "cie-2013-s-11-q10iii", "paperId": "cie-2013-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the area between a curve and its tangent over a specified region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls part (iii) far more challenging: many used identical limits for the curve and line or ignored the tangent, and few scored more than one mark.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct intersection limits for the curve and tangent, integrate their difference, and interpret the shaded region." }, { "id": "cie-2013-s-12-q02ii", "paperId": "cie-2013-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find a coefficient after multiplying a binomial expansion by a polynomial factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part caused considerable difficulty because many candidates did not realise they needed the constant term of the binomial expansion.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify which terms of the expansion contribute to x squared after multiplication, including the constant term, and combine their coefficients." }, { "id": "cie-2013-s-12-q05i", "paperId": "cie-2013-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Show that a squared trigonometric expression has a constant value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the squaring was poor, with common sign errors, and many candidates failed to recognise that the resulting sum had a constant value.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand both squares carefully, collect the sine-squared and cosine-squared terms, and use their identity." }, { "id": "cie-2013-s-12-q06i", "paperId": "cie-2013-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find parameter values that make two vectors parallel.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Of the three parts, the report says part (i) caused the most difficulty because many candidates did not use the requirement that all vector components have the same ratio.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare all three corresponding components in the same ratio and solve for the parameters." }, { "id": "cie-2013-s-12-q07", "paperId": "cie-2013-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Calculate the reflection of a point in a straight line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the majority had problems: many treated reflection as a function inverse and made little progress, while others confused the foot of the perpendicular with the reflected point.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the perpendicular through the point, locate its intersection with the mirror line, then reflect through that foot to obtain the image." }, { "id": "cie-2013-s-12-q08i", "paperId": "cie-2013-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 2, "topicIds": [ "algebra", "differentiation" ], "primaryTopicId": "algebra", "summary": "Express a cylinder's surface area in terms of its radius using a fixed volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was badly answered: many made no attempt or manipulated formulas incorrectly to force the supplied result.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the volume condition to write the height in terms of the radius, then substitute into the total surface-area formula." }, { "id": "cie-2013-s-12-q11ii", "paperId": "cie-2013-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(ii)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the area of a region bounded by a curve, axes and a normal line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says the final area caused difficulty: candidates often used an upper limit of 2 instead of 0 and rarely found the triangle area correctly.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve over the correct limits and subtract the triangular area below the normal." }, { "id": "cie-2013-s-13-q02i", "paperId": "cie-2013-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine a sector angle from an equality between shaded and circular areas.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says some candidates used 6 cm instead of 9 cm for the larger circle's radius, giving a wrong result that was sometimes carried into the next part.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the shaded area using the correct radii, set it equal to the small circle's area, and solve for the angle in radians." }, { "id": "cie-2013-s-13-q03ii", "paperId": "cie-2013-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation whose rearrangement gives a quartic in cosine.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies the quartic reduction as the main problem and notes candidates missed the upper limit, failed to take square roots, or did not give answers in terms of pi.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Reduce the quartic to a quadratic in cos squared, take square roots where needed, and include every solution in the required interval and form." }, { "id": "cie-2013-s-13-q06", "paperId": "cie-2013-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find and classify a stationary value under a linear constraint between two variables.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many omitted the required value, and determining the maximum or minimum proved difficult; some ignored the linear constraint and treated a variable as constant.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the constraint to express the objective in one variable, differentiate, solve for the stationary point, and justify whether it is a maximum or minimum." }, { "id": "cie-2013-s-13-q11i", "paperId": "cie-2013-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(i)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find where a tangent to a fractional-power curve meets a vertical line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says some could not start because they missed the tangent-gradient link; common problems included handling a negative fractional power and evaluating the gradient at x=1 rather than x=4.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, evaluate the gradient at the specified point, and use the tangent equation with the correct x-coordinate." }, { "id": "cie-2013-s-13-q11ii", "paperId": "cie-2013-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2013, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(ii)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the area between a curve and its tangent line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says integration of the curve troubled several candidates, although many did succeed by subtracting the line's area from the area under the curve.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve and subtract the area below the tangent line over the bounded interval." }, { "id": "cie-2013-s-21-q02", "paperId": "cie-2013-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Solve a logarithmic equation with a restricted logarithm domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the least well-done question on the paper and says logarithm laws were seldom applied correctly, with many candidates unable to earn marks.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply logarithm laws correctly, solve the resulting equation, then reject any root outside the domain of the original logarithms." }, { "id": "cie-2013-s-21-q03i", "paperId": "cie-2013-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity using double-angle relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates established the relationship because they did not start with double angles; sign errors also prevented full marks.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Start from a suitable double-angle identity and manage the signs consistently through the algebra." }, { "id": "cie-2013-s-21-q06i", "paperId": "cie-2013-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Show graphically that a trigonometric equation has exactly one root in an interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates could draw the cotangent graph needed for this uniqueness argument.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch both functions on the stated interval and show they intersect exactly once." }, { "id": "cie-2013-s-21-q06iii", "paperId": "cie-2013-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the root equation into the form required for iteration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was misunderstood by many candidates, who attempted numerical substitution instead of rewriting the equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange the original equation algebraically into the stated fixed-point form without substituting numerical values." }, { "id": "cie-2013-s-21-q07ii", "paperId": "cie-2013-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a compound-angle trigonometric equation over a full angular range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says very few candidates obtained every solution in the given range, with most providing only the first two.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the amplitude-phase form from part (i), solve each valid angle, and convert all values of the doubled angle to the requested interval for theta." }, { "id": "cie-2013-s-21-q07iii", "paperId": "cie-2013-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the least value of a squared trigonometric expression using a previous amplitude result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report calls this the paper's discriminator and says very few correct solutions were seen because it required insight and deduction.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the maximum amplitude obtained in part (i) to identify the least value of the squared expression." }, { "id": "cie-2013-s-22-q01", "paperId": "cie-2013-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Integrate a rational function to find the curve through a given point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates failed to recognise the logarithmic integral and common errors involved its coefficient.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise the logarithmic integral, integrate with the correct coefficient, and use the point to find the constant." }, { "id": "cie-2013-s-22-q06iii", "paperId": "cie-2013-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "exponentials", "numerical-methods" ], "primaryTopicId": "exponentials", "summary": "Rearrange an equation into the fixed-point form used by an iteration formula.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many misunderstood this part and tried numerical substitution rather than rewriting the original equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential expression and take logarithms to obtain the specified iteration equation." }, { "id": "cie-2013-s-22-q07a", "paperId": "cie-2013-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find an exact area involving an exponential curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often found an expression of the right form but the constant was less successful, and many lost the final accuracy mark by giving a decimal instead of an exact answer.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the exponential correctly, apply both limits, and keep the exact form rather than a decimal approximation." }, { "id": "cie-2013-s-23-q02", "paperId": "cie-2013-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Solve a logarithmic equation with a restricted logarithm domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the least well-done question on the paper; logarithm laws were seldom applied correctly and many candidates gained no marks.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply logarithm laws correctly, solve the resulting equation, and reject any root outside the original domain." }, { "id": "cie-2013-s-23-q06i", "paperId": "cie-2013-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Show graphically that a trigonometric equation has exactly one root in an interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates could draw the cotangent graph needed to establish the unique root.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch both functions on the stated interval and demonstrate their single intersection." }, { "id": "cie-2013-s-23-q07ii", "paperId": "cie-2013-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a compound-angle trigonometric equation over the full stated range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few candidates found all solutions in the interval and most gave only the first two.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the amplitude-phase form from part (i), solve every permitted angle, and convert the doubled angles back to theta." }, { "id": "cie-2013-s-23-q07iii", "paperId": "cie-2013-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2013, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find a least value using the amplitude of a squared trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes this as the discriminator and says very few correct solutions were seen because it required insight and deduction.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the maximum amplitude from part (i) to infer the minimum of the squared expression." }, { "id": "cie-2013-s-31-q06i", "paperId": "cie-2013-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a plane perpendicular to a line through the midpoint of two points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this question very difficult for nearly all candidates; many used the wrong vector as the midpoint or as the plane normal.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the line direction as the plane normal, find the midpoint, and substitute it into the plane equation." }, { "id": "cie-2013-s-31-q07a", "paperId": "cie-2013-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a complex-number equation involving a conjugate without a calculator.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part proved difficult because few candidates realised they needed to use the conjugate equation and take real and imaginary parts.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the unknown and its conjugate in real and imaginary parts, then equate those parts to solve for the real coefficients." }, { "id": "cie-2013-s-31-q10i", "paperId": "cie-2013-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 7, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Form and solve a differential equation for a tank with constant inflow and volume-dependent leakage.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says it was rare for candidates to write the correct differential equation; even some who did omitted the integration constant or initial condition.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Balance inflow and outflow rates, separate variables, integrate, and use the initial condition to determine the integration constant." }, { "id": "cie-2013-s-32-q03", "paperId": "cie-2013-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find parameters in an exponential model from points on a transformed graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was difficult for many: most misread transformed-graph coordinates as ordinary x-y values, and a common sign error was using k instead of −k as the gradient.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use ln y = ln A − kx squared, identify the transformed coordinates, then solve for A and k." }, { "id": "cie-2013-s-32-q06ii", "paperId": "cie-2013-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an integral using the trigonometric substitution specified in the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The examiner report calls the substitution attempts mostly very poor and lists errors in the differential, the tangent expression, and use of x instead of theta. Although the report calls this part (iii), those details match the QP’s substitution integral in part (ii), so the QP numbering is used.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the stated tangent expression, transform the differential and limits consistently, and simplify the result to one logarithm." }, { "id": "cie-2013-s-32-q09b", "paperId": "cie-2013-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the maximum modulus within a shaded complex-locus region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says a correct maximum-modulus calculation was rarely seen and many diagrams were poor. The report labels this as Q9(ii), while the QP labels the matching modulus-and-region task Q9(b); the task content is exact and the QP label is used.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Interpret both locus constraints on an Argand diagram, identify the farthest permitted point from the origin, and calculate its modulus." }, { "id": "cie-2013-s-32-q10ii", "paperId": "cie-2013-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 7, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a plane containing a line and making a specified acute angle with another plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls this a challenging test; a substantial number made no useful progress and errors in the normal-vector modulus were common.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the line condition and the angle between plane normals to form equations for the unknown plane coefficients, then solve consistently." }, { "id": "cie-2013-s-33-q04ii", "paperId": "cie-2013-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact trigonometric integral using a result from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates wasted time in heavy algebra when they failed to use part (i); few completed the intended tangent-form integration and some omitted showing limits.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the expression using its amplitude-phase form, square it, integrate with the correct limits, and show the substitutions." }, { "id": "cie-2013-s-33-q07ii", "paperId": "cie-2013-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use complex conjugates to transform a locus equation and identify the resulting circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says it was rare to see anything approaching a correct solution; most did not use part (i), and expansion errors were common.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand the modulus equation by squaring both sides, use conjugate identities from part (i), then complete the locus argument." }, { "id": "cie-2013-s-33-q08i", "paperId": "cie-2013-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 9, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve a separable differential equation with two boundary conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says integration with respect to x proved much more difficult; logarithm laws were often muddled while finding the constant and expression for x.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate t and x, integrate both sides, use the conditions to find the constant, and solve for x in terms of t." }, { "id": "cie-2013-s-33-q09i", "paperId": "cie-2013-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the coordinate of a maximum of a trigonometric product.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says about half of candidates failed to differentiate sin squared 2x correctly, most often using a coefficient of 2 instead of 4.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate with the product and chain rules, then solve the resulting equation for the maximum in the specified interval." }, { "id": "cie-2013-s-33-q09ii", "paperId": "cie-2013-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2013, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find an area by integrating after a sine substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates often made early errors in finding or substituting du/dx, causing the cosine factors to reinforce instead of cancel.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u equal to sin x, transform du and the limits correctly, and simplify the resulting integrand." }, { "id": "cie-2013-s-41-q07iv", "paperId": "cie-2013-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2013, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph for a car journey from rest, through a maximum speed, back to rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says attempts were generally disappointing: many joined key points with straight lines and very few showed zero acceleration at the origin through a tangent to the time axis.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Mark both endpoints and the maximum point, then use the acceleration information to shape a smooth curve with a horizontal tangent at the start." }, { "id": "cie-2013-s-42-q02ii", "paperId": "cie-2013-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2013, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work-energy principles to analyse the reverse motion of a block along an incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this difficult question was often missing energy terms; in part (ii), candidates commonly assumed an unstated initial speed or constant acceleration.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Include work by the pulling force, work against resistance, kinetic-energy change, and potential-energy change; use the initial speed stated by the first motion rather than assuming a value." }, { "id": "cie-2013-s-42-q04i", "paperId": "cie-2013-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2013, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Use the plane's acceleration and distance to find the distance and speed at a midpoint.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says a very common error was reading the denominator of sin(alpha)=16/65 as the plane length, and another common error was taking acceleration to be g.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use sin(alpha) as the incline ratio to determine the acceleration, then apply constant-acceleration relations to the stated distance." }, { "id": "cie-2013-s-43-q01", "paperId": "cie-2013-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2013, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the final speed of a sledge travelling down a rough inclined track.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a challenging opening question and says common approaches omitted or mis-signed friction, giving an incorrect final speed.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use Newton's second law or work-energy with gravity and friction, ensuring friction contributes work rather than being omitted or added with the wrong sign." }, { "id": "cie-2013-s-43-q02i", "paperId": "cie-2013-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2013, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Calculate the loss of gravitational potential energy of a two-particle system.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was a difficult question; the best answers compared the initial and final potential energies, while other candidates sought the supplied result without showing what it represented.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Account for the vertical displacement and mass of each particle, then calculate the net change in potential energy as one descends and the other rises." }, { "id": "cie-2013-s-43-q05ii", "paperId": "cie-2013-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2013, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find both particles' velocities when their heights differ by a fixed amount after a delayed launch.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls the question very challenging and notes that many used equal times instead of the two-second launch offset when applying the height difference.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use displacement equations for both particles, account for the launch delay in the time variables, and solve the height-difference condition before finding velocities." }, { "id": "cie-2013-s-51-q03", "paperId": "cie-2013-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Solve a composite rigid-body equilibrium question involving centre of mass, weight and friction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found the whole question difficult, with errors in moment equations and in the force and reaction used for friction.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about suitable points for the joined rods, then apply translational equilibrium and the limiting-friction relation to find the requested values." }, { "id": "cie-2013-s-51-q06ii", "paperId": "cie-2013-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Determine the limiting angular speed and greatest kinetic energy for a particle moving on a horizontal circle on a conical string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the second part was rather difficult and records frequent mistakes in the force balance and in using the horizontal circular radius.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve tension vertically and horizontally, use the horizontal radius of the circle in the centripetal equation, then calculate the greatest speed and kinetic energy." }, { "id": "cie-2013-s-51-q07", "paperId": "cie-2013-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 9, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Solve an equilibrium problem for a uniform sector-shaped prism hinged on an inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this a very difficult question and says many candidates scored few marks.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the hinge, include the weight and applied force with their correct moment arms, and apply the equilibrium condition to the angle." }, { "id": "cie-2013-s-52-q05i", "paperId": "cie-2013-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use energy to find a particle's speed when a block reaches limiting equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report notes that the 0.3 m extension was often used and that quite a number of candidates included too many terms in the energy equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Track the elastic extension and include the gravitational, kinetic, and elastic-energy changes with consistent signs." }, { "id": "cie-2013-s-52-q06i", "paperId": "cie-2013-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a cone-particle system from moments.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says too many candidates used incorrect distances in their moment equation for the centre-of-mass part.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the cone's and particle's moment contributions about the reference plane using their correct distances, then divide by total mass." }, { "id": "cie-2013-s-52-q06ii", "paperId": "cie-2013-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on a particle moving on the smooth surface of a cone.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates used horizontal and vertical resolutions, with sign or angle errors and mistakes solving the resulting equations.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct geometry to resolve tension, weight and contact force along suitable directions, then solve the simultaneous equations." }, { "id": "cie-2013-s-53-q05ii", "paperId": "cie-2013-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the tension in a second inextensible string for several specified lengths while an elastic string remains attached.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates caused themselves problems by failing to distinguish the elastic string from the inextensible one.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "For each length, distinguish whether the elastic string is slack or stretched, then apply equilibrium and compatibility of the two strings." }, { "id": "cie-2013-s-53-q05iii", "paperId": "cie-2013-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed of a particle using a multi-term energy equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this required a four-term energy equation and proved too difficult for many candidates.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the full energy balance, including gravitational potential change and the change in elastic energy, then solve for the speed." }, { "id": "cie-2013-s-53-q06", "paperId": "cie-2013-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse toppling and sliding conditions for two joined conical sections on a horizontal surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls it difficult and specifies that candidates needed the correct moment equation for toppling and the friction relation for sliding.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the centre of mass, apply moments about the base for toppling, and use limiting friction where required." }, { "id": "cie-2013-s-53-q07", "paperId": "cie-2013-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2013, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7", "marks": 10, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed of a ball moving in a tube under a position-dependent resisting force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this was difficult; it identifies setting up Newton's second law and using the correct limits in integration as necessary starting steps.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Start from Newton's second law, replace acceleration with v dv/dx, separate variables and integrate using the correct displacement limits." }, { "id": "cie-2013-s-61-q01", "paperId": "cie-2013-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the standard deviation and coded mean from summary statistics using a working mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates understood the working-mean method and very few answers to either part were correct; common errors included dividing the coded sum of squares directly by n.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the coded sums to recover the original mean and variance, applying the working-mean adjustment correctly." }, { "id": "cie-2013-s-61-q04", "paperId": "cie-2013-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use Normal distributions to find a probability and infer an unknown standard deviation from sample counts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few attempts were made in either part; many confused a probability with a z-value or rounded to decimal places instead of significant figures.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardise the negative-value event for part (a), then use the observed proportion and the appropriate Normal tail to solve for the unknown spread in part (b)." }, { "id": "cie-2013-s-61-q05i", "paperId": "cie-2013-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Identify the binomial model and its parameters for a count of multiples of five.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates were unfamiliar with the binomial distribution and very few recognised the required model and parameters in part (a).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the success probability among the integers 7 through 21 and specify n and p in the binomial distribution." }, { "id": "cie-2013-s-61-q05iii", "paperId": "cie-2013-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the least sample size so the probability of no multiples of five falls below a threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates recognised the need to compare 0.8 raised to n with 0.01 or 0.99 to find the least n.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the zero-success binomial probability below 0.01 and solve for the least integer n." }, { "id": "cie-2013-s-62-q04ii", "paperId": "cie-2013-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the parameters of a binomial model from its mean and variance and a restricted probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report names this the paper's most difficult part question: many formed the mean and variance equations but could not solve them or connect p to k/9.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use np=96 and npq=32 to find n and p, then use p=k/9 to obtain the integer k." }, { "id": "cie-2013-s-62-q05i", "paperId": "cie-2013-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Construct a stem-and-leaf display for three-figure data rounded to tens.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates struggled with three-figure values and the single-digit leaf convention; only about one in ten remembered the dollar sign in the key.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Keep stems and single-digit leaves consistent with the tens scale and include a key that identifies the dollar units." }, { "id": "cie-2013-s-63-q06ii", "paperId": "cie-2013-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Read a value from a cumulative frequency graph for a specified number of people.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates used the correct cumulative frequency and read the graph accurately for the stated count.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert the stated count into the appropriate cumulative frequency, then read the corresponding weight from the graph." }, { "id": "cie-2013-s-63-q06iii", "paperId": "cie-2013-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 6, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean and standard deviation from grouped data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says fully correct solutions were uncommon; frequent errors used cumulative instead of class frequencies or class limits instead of midpoints.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use class midpoints and class frequencies to estimate the mean and variance, then obtain the standard deviation." }, { "id": "cie-2013-s-63-q07ii", "paperId": "cie-2013-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2013, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements of selected dogs with two breeds required to remain together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says using combinations was a common error and many treated each pair as a single unit without accounting for internal permutations.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat each adjacent breed pair as a block, account for the internal permutations, and arrange the remaining dogs as distinct individuals." }, { "id": "cie-2013-s-71-q04iii", "paperId": "cie-2013-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Explain whether the Central Limit Theorem was needed for a sample-mean probability calculation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates showed a good understanding of the Central Limit Theorem and its application in this part.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the theorem's requirement to the unknown population distribution and the sample size used." }, { "id": "cie-2013-s-71-q05i", "paperId": "cie-2013-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Describe a binomial count and a suitable Poisson approximation with justification.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the question was not well attempted and very few candidates justified the choice of an approximating distribution.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "State both distributions and their parameters, then justify the approximation using the large n and small p conditions." }, { "id": "cie-2013-s-71-q06ii", "paperId": "cie-2013-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the expectation and variance of a continuous random variable from its density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says weaker candidates struggled with integration and commonly omitted subtracting E(T) squared when finding the variance.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate t f(t) and t squared f(t), then subtract the square of the mean when computing the variance." }, { "id": "cie-2013-s-71-q07i", "paperId": "cie-2013-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Interpret and calculate a Type I error for a die-bias test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says Type I error probabilities were not well attempted and stresses that definitions must be tied to the die-test context.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Describe the false rejection of the fair-die null in context and calculate the probability of at least three sixes under p=1/6." }, { "id": "cie-2013-s-71-q07ii", "paperId": "cie-2013-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the Type II error probability for a specified biased die.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says the calculation of Type II error was not well attempted.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the stated alternative p=2/3, find the probability of fewer than three sixes in five throws." }, { "id": "cie-2013-s-72-q01i", "paperId": "cie-2013-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Identify the distribution and parameters for the number of bent rods in a sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates could not fully describe the distribution and often gave only the word Binomial without n and p.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the fixed sample size and probability of a bent rod to specify a binomial model." }, { "id": "cie-2013-s-72-q01ii", "paperId": "cie-2013-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Choose and justify an approximation for the count of bent rods.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the approximation was not always identified as Poisson and even candidates who did identify it often failed to give the reasons.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Check the large sample and small event-probability conditions and give the Poisson mean." }, { "id": "cie-2013-s-72-q03i", "paperId": "cie-2013-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the sample size from a one-sample Normal test statistic and state an assumption.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few candidates stated the necessary assumption that the standard deviation remained the same for the region.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange the test-statistic formula for n and state that the population standard deviation is unchanged in the region." }, { "id": "cie-2013-s-72-q04iii", "paperId": "cie-2013-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability that neither of two confidence intervals contains the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates mishandled the required 0.02 squared and thought twice 0.02 was the probability.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the independent miss probability for one interval and square it for two intervals." }, { "id": "cie-2013-s-72-q06i", "paperId": "cie-2013-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the lower-tail critical region for a Poisson test and carry out the test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says parts (i) and (ii) caused problems and specifically required both P(X≤1) and P(X≤2) to justify the critical region.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the cumulative probabilities at the boundary counts to justify the critical region before deciding whether the observed count lies in it." }, { "id": "cie-2013-s-73-q03ii", "paperId": "cie-2013-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Explain the meaning of a 92 percent confidence interval for a population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies interpretation as the most challenging part and says many failed to state that 92 percent of such intervals capture the true mean.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "State that over repeated sampling, 92 percent of intervals constructed in this way contain the true population mean." }, { "id": "cie-2013-s-73-q04i", "paperId": "cie-2013-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2013, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a conditional probability involving two independent Poisson variables given their sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s13_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was the paper's most difficult part: many multiplied the two Poisson probabilities but did not divide by the probability of the given sum.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/191416-june-2013-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the joint probability for the specified values and divide by the probability that the sum equals five." }, { "id": "ial-legacy-2017-january-WMA01-q5", "paperId": "ial-legacy-2017-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Rewrite logarithmic expressions in terms of y = log base 3 of x, then solve the resulting equation for x.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_que_20170110.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_rms_20170301.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson lists Question 5 among the paper's most challenging questions and says many candidates answered it poorly. Common problems included leaving logarithms unsimplified, misusing powers, and stopping before finding x.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use quotient, power and change-of-base rules to express each logarithm in y. Substitute into the equation, solve for y, then recover x using x = 3^y.", "sourceEvidencePages": { "examinerReport": [ 3, 5 ] } }, { "id": "ial-legacy-2017-january-WMA01-q8d", "paperId": "ial-legacy-2017-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation by substituting 3^t into the factorised cubic expression.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_que_20170110.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_rms_20170301.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson lists 8(d) among the paper's most challenging parts and reports that fewer than half completed it successfully. Many omitted the part or did not handle the logarithmic solution and positive-value restriction correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = 3^t, substitute into the factorised equation and solve for u. Since u must be positive, reject any negative root, then use logarithms and round t to three decimal places.", "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-legacy-2017-january-WMA01-q9b", "paperId": "ial-legacy-2017-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State roots and a turning point after vertical and horizontal transformations of a curve.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_que_20170110.pdf#page=26", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_rms_20170301.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson lists 9(b) among the paper's most challenging parts. Its question-specific comments say many lost marks by giving full coordinates when only roots were requested and by setting up unnecessary equations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "A nonzero vertical scale leaves the roots unchanged; adding 2 shifts the turning point up by 2; for f(4x) = 0, divide the original roots by 4.", "sourceEvidencePages": { "examinerReport": [ 3, 7 ] } }, { "id": "ial-legacy-2017-january-WMA01-q14a", "paperId": "ial-legacy-2017-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(a)", "marks": 4, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "proof", "summary": "Prove the finite geometric-series sum formula.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_que_20170110.pdf#page=46", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_rms_20170301.pdf#page=29", "qpPages": [ 46, 49 ], "msPages": [ 29, 30 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson lists Question 14 among the paper's most challenging questions. For part (a), many students omitted the proof or used an invalid reversed-series argument instead of subtracting the multiplied series.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Write S_n and rS_n with corresponding terms aligned, subtract the equations, then rearrange for S_n. Keep both end terms and show the subtraction clearly.", "sourceEvidencePages": { "examinerReport": [ 3, 11 ] } }, { "id": "ial-legacy-2017-january-WMA01-q14c", "paperId": "ial-legacy-2017-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(c)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the total volume remaining across 20 barrels after different numbers of years of evaporation.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_que_20170110.pdf#page=46", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_rms_20170301.pdf#page=29", "qpPages": [ 46, 49 ], "msPages": [ 29, 30 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson lists Question 14 among the paper's most challenging questions. It says most candidates who attempted part (c) did not fully understand the cumulative total; many earned only the first method mark.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Represent each barrel's end-of-year amount as 180 times a power of 0.93, from the first through the twentieth power, then sum the geometric series and round to the nearest litre.", "sourceEvidencePages": { "examinerReport": [ 3, 11 ] } }, { "id": "ial-legacy-2017-january-WMA01-q15", "paperId": "ial-legacy-2017-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 15, "questionPart": "15", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the exact area enclosed by three circular arcs in a rounded-triangle logo.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_que_20170110.pdf#page=50", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA01_01_rms_20170301.pdf#page=31", "qpPages": [ 50, 52 ], "msPages": [ 31, 31 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson lists Question 15 among the paper's most challenging questions. Many candidates did not attempt it and only a small proportion of attempts earned full marks; choosing an area strategy, using radians and simplifying the exact result caused difficulty.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the three equal 60-degree sectors and the central equilateral triangle to form an exact area expression, then factor out r squared and simplify the constant without decimals.", "sourceEvidencePages": { "examinerReport": [ 3, 11 ] } }, { "id": "ial-legacy-2017-january-WMA02-q4a", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the range of the given rational function on its stated domain.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson says part 4(a) caused the most difficulty in Question 4: most candidates scored only zero or one of its two marks, often omitting the required positive-domain condition.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the domain restriction to determine the sign and limiting values of f(x); state the range with correct open or closed endpoints.", "sourceEvidencePages": { "examinerReport": [ 4, 4 ] } }, { "id": "ial-legacy-2017-january-WMA02-q7", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Identify an absolute-value linear function from its graph and sketch a transformed version.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=13", "qpPages": [ 20, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson calls Question 7 more testing and says only very able students scored highly; the mean was 4/7 and one fifth earned full marks. Candidates often missed a possible coefficient value or misplaced the transformed vertex.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the vertex and y-intercept to find all compatible coefficient pairs, write the absolute-value rule, then apply the horizontal stretch and upward translation to the new graph.", "sourceEvidencePages": { "examinerReport": [ 5, 5 ] } }, { "id": "ial-legacy-2017-january-WMA02-q9", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9", "marks": 9, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate a logarithmic integral using a prescribed substitution, then apply it to a volume of revolution.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=26", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=15", "qpPages": [ 26, 29 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson calls Question 9 discriminating: grade E students averaged 2 marks while grade A students averaged more than 8. The report notes errors in the substitution, logarithmic integration, and applying the result to the volume.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = 2x + 3, rewrite the integrand and limits in u, and integrate the resulting terms carefully. Then use the disk method for the solid and substitute the exact integral result.", "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-legacy-2017-january-WMA02-q11ci", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(c)(i)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "functions", "summary": "Find the minimum of a rational function whose denominator contains a sinusoidal expression.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=34", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=18", "qpPages": [ 34, 37 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson says 11(c)(i) caused more difficulty. Many candidates mishandled the reciprocal relationship between the denominator's maximum and the function's minimum.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the linear sine-cosine expression as R sin(x - alpha). To minimize the reciprocal function, first maximize its positive denominator, then substitute that value.", "sourceEvidencePages": { "examinerReport": [ 7, 7 ] } }, { "id": "ial-legacy-2017-january-WMA02-q12a", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Use a constant-rate ice-growth model to express elapsed time in terms of ice thickness.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=38", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=19", "qpPages": [ 38, 41 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says Question 12, particularly part (a), was difficult at all levels. Many candidates used the wrong variables or failed to rearrange the proportional relationship to make t the subject.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat thickness as directly proportional to elapsed time. Use the observation that 1.5 mm forms in 2 hours to find the rate, then rearrange to make t the subject.", "sourceEvidencePages": { "examinerReport": [ 8, 8 ] } }, { "id": "ial-legacy-2017-january-WMA02-q13d", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(d)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "functions", "summary": "Eliminate the parameter from a trigonometric parametric curve and give its Cartesian equation in simplified surd form.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=42", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=20", "qpPages": [ 42, 45 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies part 13(d) as the most challenging part for many and says a significant proportion scored no marks. Common issues were eliminating the parameter and presenting the constant in the required exact surd form.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the expressions for tan(theta) and sec(theta) in terms of x and y into 1 + tan squared theta = sec squared theta; rearrange and simplify the required constant.", "sourceEvidencePages": { "examinerReport": [ 8, 9 ] } }, { "id": "ial-legacy-2017-january-WMA02-q14", "paperId": "ial-legacy-2017-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14", "marks": 11, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use three-dimensional vectors to find a parallelogram's fourth vertex, angle and area, then find a related trapezium area.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_que_20170117.pdf#page=46", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WMA02_01_rms_20170301.pdf#page=22", "qpPages": [ 46, 48 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes Question 14 as a serious challenge for most students and reports a mean of 4.9/11. The report highlights confused vector work, choosing the correct vectors for the interior angle, and failure to use the 1.5 area relationship in part (d).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use d = a - b + c for the fourth vertex. Choose the correctly directed adjacent side vectors for the interior angle, find the parallelogram area, then use the midpoint relation to obtain the trapezium area as 1.5 times that area.", "sourceEvidencePages": { "examinerReport": [ 9, 9 ] } }, { "id": "ial-legacy-2017-june-WMA01-q10c", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the first two terms of a binomial series to determine the remaining constant in a function expansion.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=32", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=17", "qpPages": [ 32, 35 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says only a small proportion succeeded on 10(c). Many formed a two-term equation instead of accounting for all three contributions, or left x in an equation that needed to involve only a and b.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the known constant to determine a, then match the coefficient of x in the product expansion to the stated 352x coefficient and solve for b. Keep all three contributions in the coefficient equation.", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-legacy-2017-june-WMA01-q5d", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use the graph and roots to state all x-values for which the cubic is non-positive.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that few candidates earned both marks. Many recognized 0.5 and 3 but did not use the diagram to determine the complete set of x-values; some treated it as an ordinary quadratic inequality.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "The repeated root is x=0.5, where the graph touches the axis, and the curve crosses at x=3. Therefore f(x)≤0 when x=0.5 or x≥3.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2017-june-WMA01-q15c", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 15, "questionPart": "15(c)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a sinusoidal height equation for the first two times and express them as clock times.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=52", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=25", "qpPages": [ 52, 56 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson notes that candidates often struggled to interpret the negative inverse-sine value and select the correct quadrants. Some obtained hour values but failed to convert them into 24-hour or am/pm times.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Set H=3, rearrange to a sine equation, find both solutions in the required interval using the correct quadrants, convert radians to hours, and express the results as actual times to the nearest minute.", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-legacy-2017-june-WMA01-q12c", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(c)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use calculus to find the exact area of a region bounded by a cubic curve, the x-axis and a normal.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=40", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=20", "qpPages": [ 40, 43 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson calls Question 12 a good discriminator. In part (c), many candidates did not understand how to form the compound area; they treated it as a simple line-minus-curve area, confused which triangle to use, or struggled to combine the final areas.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the compound region at the relevant intersections, integrate the curve over the appropriate intervals, find the straight-line triangle area, and combine the signed areas exactly.", "sourceEvidencePages": { "examinerReport": [ 9 ] } }, { "id": "ial-legacy-2017-june-WMA01-q14c", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equations of two distinct circles centred at the origin that each touch the given circle.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=48", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=24", "qpPages": [ 48, 51 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson states that most candidates found part (c) difficult. Common errors included reusing earlier coordinates instead of finding the centre-to-origin distance, and not recognizing how a second circle could touch externally.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "The given circle has centre (4,6) and radius √13; its centre is 2√13 from the origin. The two possible radii are √13 and 3√13, so the equations are x^2+y^2=13 and x^2+y^2=117.", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-legacy-2017-june-WMA01-q5c", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the factorisation from earlier parts to fully factorise a cubic polynomial.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson says many candidates struggled to factor the quadratic remainder with a negative leading coefficient; taking out -1 first would have helped.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide by the known repeated factor, take out the negative leading coefficient if helpful, and factor the remaining quadratic completely. Check that the factors reproduce the cubic.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2017-june-WMA01-q4b", "paperId": "ial-legacy-2017-june-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2017, "session": "June", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of a stationary point after differentiating an expression with fractional indices.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_que_20170524.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA01_01_rms_20170816.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson says weaker candidates found the index work challenging. Many struggled to manipulate negative fractional indices when solving for x, and others omitted the final substitution needed to find the y-coordinate.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the first derivative from part (a) equal to zero, solve the resulting fractional-index equation carefully for x, and substitute that value into the original curve to obtain y.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2017-june-WMA02-q14d", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(d)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the parametric region’s area integral exactly.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=48", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=30", "qpPages": [ 48, 52 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says correct solutions to 14(d) were very rare; most candidates made no attempt, and integration proved too challenging for many of those who tried.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the given trigonometric expression over its parameter limits, evaluate the exact antiderivative at both bounds, and combine it with the stated area term.", "sourceEvidencePages": { "examinerReport": [ 3, 13 ] } }, { "id": "ial-legacy-2017-june-WMA02-q5c", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove that the given rational function is decreasing on its domain.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=14", "qpPages": [ 12, 15 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson lists 5(c) among the paper’s most challenging parts and says it was very poorly answered, with many candidates leaving it blank or offering an unsupported numerical argument rather than a proof.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative from part (b), show both terms are negative for x>2 because their squared denominators are positive, and conclude f' (x)<0 throughout the domain.", "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-legacy-2017-june-WMA02-q14c", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(c)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Show the area of the parametric region is represented by a trigonometric integral.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=48", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=30", "qpPages": [ 48, 52 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson identifies Question 14 as the most demanding and says many could not make headway with parts (c) and (d). For (c), many candidates failed to differentiate the composite trigonometric terms correctly or form an integrable expression.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the parametric area formula, differentiate y and x with respect to theta, simplify using sin^2(theta)=1-cos^2(theta), and add the trapezium area with the correct orientation and limits.", "sourceEvidencePages": { "examinerReport": [ 3, 13 ] } }, { "id": "ial-legacy-2017-june-WMA02-q12a", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(a)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Derive the differential equation for the water height in a cylindrical tank with inflow and outflow.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=40", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=27", "qpPages": [ 40, 43 ], "msPages": [ 27, 28 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says Question 12 caused general difficulty and part (a) was demanding. Candidates often failed to combine inflow and outflow into a net rate or to use the chain rule correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the net rate of volume change as inflow minus outflow, express cylinder volume in terms of h, use dV/dt=(dV/dh)(dh/dt), and rearrange to the required differential equation.", "sourceEvidencePages": { "examinerReport": [ 3, 11 ] } }, { "id": "ial-legacy-2017-june-WMA02-q12c", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(c)", "marks": 7, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use a substitution and definite integration to find the tank filling time.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=40", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=27", "qpPages": [ 40, 43 ], "msPages": [ 27, 28 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson describes widespread difficulty with Question 12. In part (c), many candidates made little progress beyond a substitution, simplified the denominator incorrectly, or struggled to integrate and evaluate the definite expression.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute h=(2-x)^2 into the integral, change the bounds consistently, simplify to a rational expression in x, integrate logarithmic and linear terms, and evaluate the definite result before rounding to minutes.", "sourceEvidencePages": { "examinerReport": [ 3, 11 ] } }, { "id": "ial-legacy-2017-june-WMA02-q13c", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(c)", "marks": 4, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find the time at which the ant population reaches its minimum.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=44", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=29", "qpPages": [ 44, 47 ], "msPages": [ 29, 29 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson says part 13(c) was considerably more demanding. Candidates often struggled to collect the exponential terms and made errors with logarithms, including taking logs before isolating the exponential.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the numerator of dP/dt from part (b) equal to zero, collect exponential terms to obtain an equation for e^(0.8t), then take logarithms and solve for t to two decimal places.", "sourceEvidencePages": { "examinerReport": [ 3, 12 ] } }, { "id": "ial-legacy-2017-june-WMA02-q10cii", "paperId": "ial-legacy-2017-june-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2017, "session": "June", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(c)(ii)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the latest time in the 24-hour period when the warehouse temperature reaches its maximum.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_que_20170621.pdf#page=32", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 June/WMA02_01_rms_20170816.pdf#page=24", "qpPages": [ 32, 35 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson includes 10(c) among the most challenging parts. For (c)(ii), many candidates who attempted it gave only one maximum time instead of checking the later branch; most earned only one of the two marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20170816.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the phase-shifted sine form from part (a), set its angle to each maximum phase within the interval, solve for t, and choose the greatest valid time below 24 hours to one decimal place.", "sourceEvidencePages": { "examinerReport": [ 3, 9, 10 ] } }, { "id": "cie-2012-s-11-q01", "paperId": "cie-2012-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report identifies this equation as a challenge for many candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identities to reduce the equation, then list every solution in the interval." }, { "id": "cie-2012-s-11-q04", "paperId": "cie-2012-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Use a changing radius to find the rate of change of a spherical object's mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found k but then did not differentiate or use the chain rule to relate mass change to radius change.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First use the given mass and radius to find k. Differentiate M = kr³ with respect to time using the chain rule." }, { "id": "cie-2012-s-11-q07bi", "paperId": "cie-2012-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 2, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the angle range for which the geometric progression converges.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates knew the convergence condition or used it to find the required θ range.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the common ratio in terms of θ and apply the convergence condition 0 < r < 1, respecting the given θ interval." }, { "id": "cie-2012-s-11-q08iv", "paperId": "cie-2012-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iv)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse function and its domain after restricting the original quadratic function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates included the ± when taking the square root while finding the inverse.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange y = f(x), include both square-root branches initially, then choose the branch consistent with the restricted domain and state the inverse domain." }, { "id": "cie-2012-s-12-q02ii", "paperId": "cie-2012-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find a related rate along a curve when x increases at a fixed rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Fewer than half of candidates obtained this rate correctly despite many recognising the chain rule.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative from part (i), apply the chain rule dy/dt = (dy/dx)(dx/dt), then substitute x = 4." }, { "id": "cie-2012-s-12-q07a", "paperId": "cie-2012-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Recover the first term and common difference from a formula for the sum of n terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this part a challenging question and says correct answers were rare.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use S₁ for the first term and form the difference S₂ − S₁ for the common difference." }, { "id": "cie-2012-s-12-q07b", "paperId": "cie-2012-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Determine the first term of a positive geometric progression from two conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The examiner report says this part was again a challenge.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Let the first term be a and ratio r. Translate the second-term difference and the sum of terms two and three into equations, then use positivity to select the solution." }, { "id": "cie-2012-s-12-q08iii", "paperId": "cie-2012-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find a unit vector in the direction of a vector joining two points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes that many candidates had difficulty with the concept of a unit vector.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Subtract the position vectors to obtain AB, find its magnitude, and divide every component by that magnitude." }, { "id": "cie-2012-s-12-q09i", "paperId": "cie-2012-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 7, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find both points where a quadratic curve has a maximum and the joining line has a stated gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says finding B challenged the majority, including confusion between the curve gradient and the line gradient.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate to locate the maximum point, then use the given gradient of AB and the curve equation to locate B." }, { "id": "cie-2012-s-12-q10iii", "paperId": "cie-2012-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find parameter values for which an equation involving composite functions has no real solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says the resulting quadratic inequality was rarely solved correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the resulting quadratic equation and use its discriminant and the direction of the inequality to determine the parameter range." }, { "id": "cie-2012-s-13-q01ii", "paperId": "cie-2012-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 1, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Use a trigonometric identity to explain a strict inequality over an acute-angle interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few candidates answered this part correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite tan²θ − sin²θ using the identity from part (i), then show the resulting factors have the required signs throughout the interval." }, { "id": "cie-2012-s-13-q02i", "paperId": "cie-2012-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find a unit vector in the direction of AB from position vectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says some candidates found AB but did not calculate its modulus or combine the result correctly to form a unit vector.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Subtract the coordinates of A from B, compute the magnitude, and divide the vector by that magnitude." }, { "id": "cie-2012-s-13-q06ii", "paperId": "cie-2012-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use terms of an arithmetic progression to determine a geometric ratio and an index.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates made progress with this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic progression's first, ninth and nth terms as the first three geometric terms, then solve the resulting ratio and index conditions." }, { "id": "cie-2012-s-13-q11iv", "paperId": "cie-2012-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(iv)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch a restricted quadratic function and its inverse on the same axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Most candidates ignored the restricted domain and sketched the unrestricted curve and its reflection instead.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the restricted curve, reflect it in y = x, and preserve the domain restriction when deciding the inverse branch." }, { "id": "cie-2012-s-21-q02", "paperId": "cie-2012-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find exponential-model constants from two points on a graph of ln y against x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report identifies this as a challenge for many candidates, especially the need to interpret the ln y axis.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use ln y = ln A + x ln b, read the intercept and gradient from the plotted points, then recover A and b." }, { "id": "cie-2012-s-21-q06i", "paperId": "cie-2012-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Derive the equation satisfied by the parameter at a point with a specified curve gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the algebraic manipulation for the final two marks proved too challenging for many candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find dy/dx from the parametric equations, set the gradient to −1, and rearrange carefully to the required equation." }, { "id": "cie-2012-s-21-q07ii", "paperId": "cie-2012-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate an exact definite integral after rewriting a squared trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies sign and multiple errors in the integration, and notes that many candidates lost the exact-value mark through calculator approximations.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity established in part (i), integrate each term exactly, and substitute both limits without decimal approximation." }, { "id": "cie-2012-s-22-q02i", "paperId": "cie-2012-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Make a substitution that reduces an exponential equation to a quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Question 2 the paper's most problematic question and says many candidates tried to solve for x at this first step.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = 5ˣ, rewrite 5²ˣ as u², and solve the resulting quadratic for the valid positive value of u." }, { "id": "cie-2012-s-22-q07i", "paperId": "cie-2012-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 7, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Show a trigonometric identity involving tan²x and cos²x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates could not obtain the given result because of difficulties with double-angle formulae and squaring.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert tan²x to sec²x − 1 and use the double-angle identity for cos²x before simplifying." }, { "id": "cie-2012-s-23-q02", "paperId": "cie-2012-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find exponential-model constants from two points on a graph of ln y against x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report identifies this as a challenge for many candidates, especially the need to interpret the ln y axis.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use ln y = ln A + x ln b, read the intercept and gradient from the plotted points, then recover A and b." }, { "id": "cie-2012-s-23-q06i", "paperId": "cie-2012-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Derive the equation satisfied by the parameter at a point with a specified curve gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the algebraic manipulation for the final two marks proved too challenging for many candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find dy/dx from the parametric equations, set the gradient to −1, and rearrange carefully to the required equation." }, { "id": "cie-2012-s-23-q07ii", "paperId": "cie-2012-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate an exact definite integral after rewriting a squared trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies sign and multiple errors in the integration, and notes that many candidates lost the exact-value mark through calculator approximations.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity established in part (i), integrate each term exactly, and substitute both limits without decimal approximation." }, { "id": "cie-2012-s-31-q01", "paperId": "cie-2012-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve a modulus equation and give the result to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates expanded the squared expression instead of removing the modulus, and many considered only one case.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the absolute-value equation into its positive and negative cases, solve both, and retain valid solutions." }, { "id": "cie-2012-s-31-q05i", "paperId": "cie-2012-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show that a curve reaches its maximum at a specified x-coordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_31.pdf", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The examiner report says differentiating the tangent of a square-root expression proved difficult.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate each trigonometric term using the chain rule, set the derivative to zero, and identify the maximum in the stated interval." }, { "id": "cie-2012-s-31-q05ii", "paperId": "cie-2012-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of a region bounded by a trigonometric curve and the axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_31.pdf", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the logarithmic antiderivative was rarely obtained correctly after candidates differentiated the curve.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the given curve between the specified limits and keep the logarithmic and trigonometric values exact." }, { "id": "cie-2012-s-31-q08ii", "paperId": "cie-2012-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a plane through a point and a line in three dimensions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates often assumed the two given position vectors already lay in the required plane instead of first constructing another in-plane vector.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find two independent direction vectors in the plane, take their cross product for a normal, and use the given point to determine the plane equation." }, { "id": "cie-2012-s-31-q10iv", "paperId": "cie-2012-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iv)", "marks": 3, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation across the full interval from −π to π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says most candidates used only the first-quadrant solution when solving the final equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use all previously found tangent values and list every corresponding angle in the full interval, checking all quadrants." }, { "id": "cie-2012-s-32-q05", "paperId": "cie-2012-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "exponentials", "summary": "Solve a differential equation involving an exponential of x and y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates did not recognise that the exponential expression first had to be separated into x and y factors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables by rewriting e^(2x+y) as e^(2x)e^y, integrate both sides, and use the initial condition." }, { "id": "cie-2012-s-32-q07iii", "paperId": "cie-2012-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Use arguments of two complex numbers to prove an exact trigonometric relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a few candidates recognised the argument-difference method that yields the stated result.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the argument of the quotient as the difference of the two arguments, then apply the tangent addition formula and account for the quadrant." }, { "id": "cie-2012-s-32-q08i", "paperId": "cie-2012-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 4, "topicIds": [ "integration", "integration" ], "primaryTopicId": "integration", "summary": "Transform a radical integral using a substitution and partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates had considerable difficulty and the key transformed-integral step was rarely seen.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = √(6 − x), change the limits, simplify dx and the integrand, then factor the denominator for partial fractions." }, { "id": "cie-2012-s-32-q10iii", "paperId": "cie-2012-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find two points on a line that are equidistant from two planes and calculate their separation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates completed this part correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Parametrize a point on the line, equate the absolute perpendicular-distance formulas, solve both resulting cases, then find the distance between the points." }, { "id": "cie-2012-s-33-q04i", "paperId": "cie-2012-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of a curve that is a product of an exponential and a power of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part proved difficult and notes that many candidates applied the quotient rule to an incorrectly formed expression.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product rule or logarithmic differentiation, set dy/dx to zero, and solve for x." }, { "id": "cie-2012-s-33-q05i", "paperId": "cie-2012-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve a differential equation for a reacting substance with a time-varying input mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says determining the integration constant was difficult; many treated e⁰ as zero and then exponentiated incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute x = 5e^(−3t) into dy/dt, separate y and t, integrate, and use y(0) = 70 to determine the constant." }, { "id": "cie-2012-s-33-q07i", "paperId": "cie-2012-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "integration", "integration" ], "primaryTopicId": "integration", "summary": "Use a substitution to evaluate a trigonometric integral and establish the given expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often mishandled du/dx, and the final step was challenging because the original x-limits were used as u-limits.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set x = u², transform dx and the limits, then integrate the resulting expression and use the stated area condition." }, { "id": "cie-2012-s-33-q08i", "paperId": "cie-2012-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Express a rational function as partial fractions before integrating.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not give a correct general partial-fraction form, which prevented them from evaluating the integral.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a constant over each linear factor and a linear numerator over the repeated quadratic factor, then solve for the coefficients." }, { "id": "cie-2012-s-41-q02", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find an angle and a component from two forces and their resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a challenge and says few candidates showed the correct relation between the forces and resultant.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the cosine rule or vector resolution to relate the force magnitudes and resultant, then project the resultant onto the 14 N direction." }, { "id": "cie-2012-s-41-q05i", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find a particle's speed after it descends a smooth curved section of a slide.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates assumed vertical fall; finding the vertical displacement from O to A was the main difficulty.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy from O to A and calculate the vertical drop from the diagram rather than treating the path as vertical." }, { "id": "cie-2012-s-41-q05ii", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find acceleration on the rough section of a slide inclined at a given angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was rarely solved successfully because candidates omitted forces or used incorrect trigonometric components.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve along and perpendicular to the slope, calculate the normal reaction and friction, then apply Newton's second law along the slope." }, { "id": "cie-2012-s-41-q06i", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the tension and acceleration for particles on two non-vertical smooth slopes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the non-vertical pulley arrangement proved very difficult; many used full weights instead of slope components and then made algebra errors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve each weight component along its slope, write Newton's second law for both particles, and solve the simultaneous equations." }, { "id": "cie-2012-s-41-q06ii", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total time until the lighter particle reaches its highest point after the heavier particle stops.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates rarely found and added both time intervals, often assuming gravity alone immediately after P stopped.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the time for P to reach the ground, then the separate upward-motion time for Q, and add the two intervals." }, { "id": "cie-2012-s-41-q07i", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find both string tensions in an equilibrium system with a ring on a vertical rod.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Only the strongest candidates made substantial progress; most did not form the correct force equations at B.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces at B in directions parallel to BA and BC, then solve the two equations for the tensions." }, { "id": "cie-2012-s-41-q07ii", "paperId": "cie-2012-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of limiting friction for the ring on the rod.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says the main difficulty was identifying all forces on the ring and their directions.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces on the ring vertically and horizontally, calculate the normal reaction, and use limiting friction F = μR." }, { "id": "cie-2012-s-42-q04i", "paperId": "cie-2012-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal and vertical components of the force exerted by a rod on a ring.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates knew the mechanics but could not find the force components on the ring.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the string tension into components, then use equilibrium in each direction to find the rod force components." }, { "id": "cie-2012-s-42-q06ii", "paperId": "cie-2012-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find engine work when a car accelerates uphill with changing driving force and power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report explicitly says this part was more challenging than part (i).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the work-energy principle with the rise in gravitational potential energy and resistance, then use the stated force and power ratios to relate the endpoint speeds." }, { "id": "cie-2012-s-42-q07i", "paperId": "cie-2012-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 9, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Sketch the velocity-time graph for a block that rebounds after losing kinetic energy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates drew a valid velocity-time graph showing negative velocity after the rebound.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate velocity before and after impact with signs relative to the original direction, then use constant frictional acceleration on each interval." }, { "id": "cie-2012-s-42-q07ii", "paperId": "cie-2012-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Sketch displacement against time through a collision and rebound.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only a few candidates produced a valid displacement-time graph.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the signed velocities and time intervals to locate the turning point at the wall and the final stopping position, then join the segments with the correct slopes." }, { "id": "cie-2012-s-43-q05ii", "paperId": "cie-2012-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find a lorry's speed after descending a hill with work done by driving and resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report says candidates found this question more challenging, especially part (ii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply work-energy from the top to the bottom, including gravitational potential energy, driving work, resistance work and the initial kinetic energy." }, { "id": "cie-2012-s-43-q07iii", "paperId": "cie-2012-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Track the motion of two particles after one reaches the ground and the string becomes slack.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often misread the time as covering two rather than three stages of motion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the stages of A's motion, use the appropriate acceleration in each stage, and solve for the second return to the stated height and the total time." }, { "id": "cie-2012-s-43-q07iv", "paperId": "cie-2012-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total distance travelled across successive stages of a particle's motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the later stages of Question 7 as more problematic and notes the repeated confusion about the motion stages.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the distance for each motion stage separately and add the magnitudes, taking care not to cancel motion in opposite directions." }, { "id": "cie-2012-s-51-q02ii", "paperId": "cie-2012-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the suspension angle and decide where the lowest point of a composite circular object lies.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the required inverse-tangent angle was rarely seen.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the centre-of-mass offset from part (i) and the radius to form the tangent ratio for the angle, then compare the two sides of the diameter." }, { "id": "cie-2012-s-51-q03i", "paperId": "cie-2012-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the speed of a sphere moving in a horizontal circle inside a smooth bowl.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a few candidates earned all four marks on this part; errors arose when eliminating the contact force.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the normal reactions and weight vertically, and use the horizontal resultant as the centripetal force." }, { "id": "cie-2012-s-51-q04i", "paperId": "cie-2012-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use elastic-string energy to find a particle's mass at an instantaneous rest point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the correct three-term energy equation was rarely seen because candidates calculated the extension incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate both string extensions at the endpoints and equate the change in kinetic, gravitational potential and elastic energy." }, { "id": "cie-2012-s-51-q05i", "paperId": "cie-2012-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the greatest speed of a particle moving down a smooth slope against a position-dependent force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls this question a particular challenge and gives the variable-force energy calculation required for the maximum speed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton's second law along the slope, integrate with respect to distance, then set acceleration to zero to locate the maximum speed." }, { "id": "cie-2012-s-51-q06ii", "paperId": "cie-2012-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the range of forces that keeps a prism in equilibrium on a rough surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was more difficult and required moments about two separate points.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take limiting moments about each relevant contact point to obtain the lower and upper bounds for F." }, { "id": "cie-2012-s-52-q02ii", "paperId": "cie-2012-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the force needed to keep a non-uniform sphere in equilibrium on a horizontal surface.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report names this among the harder parts and says a common error was using a lever arm half the correct length.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the point of contact and use the correct perpendicular lever arm for the applied force." }, { "id": "cie-2012-s-52-q06i", "paperId": "cie-2012-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Derive the centre-of-mass position of a lamina formed by subtracting two semicircles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report names this part among the paper's harder questions and notes errors in the centre-of-mass formula and moment signs.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use signed areas or masses for the full and removed semicircles, then take moments about the common centre." }, { "id": "cie-2012-s-52-q07ii", "paperId": "cie-2012-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the total string length for a particle moving in a horizontal circle on a cone.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the final three marks were lost by quite a number of candidates on this harder part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the tension from vertical equilibrium, use horizontal components for centripetal acceleration, and relate the geometry to the string length." }, { "id": "cie-2012-s-53-q05ii", "paperId": "cie-2012-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the maximum speed of a particle attached to an elastic string on a smooth slope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Question 5 was more challenging and specifically requires a five-term energy equation in part (ii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write an energy equation with all five kinetic, gravitational and elastic-energy terms for the release and maximum-speed positions." }, { "id": "cie-2012-s-53-q06iii", "paperId": "cie-2012-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "mechanics", "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a lamina dimension from its equilibrium orientation when suspended freely.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this component part as more challenging and gives the centre-of-mass geometry needed to determine the dimension.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the centre-of-mass offset to the geometry and use the tangent ratio for the stated horizontal side." }, { "id": "cie-2012-s-53-q07ii", "paperId": "cie-2012-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find a projectile's speed and direction on its second passage through a specified height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies part (ii) as more challenging and works through the second-time velocity components and direction.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the vertical displacement equation for the second time, then combine horizontal and vertical velocity components with the correct signs." }, { "id": "cie-2012-s-61-q04i", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that fewer than five days in a week exceed a snowfall threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates were challenged by both parts of Question 4.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Model the number of qualifying days with a binomial distribution using n = 7 and p = 0.21, and sum probabilities for 0 through 4." }, { "id": "cie-2012-s-61-q04ii", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly three of four weeks contain a qualifying snowfall day.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The examiner report says both parts of this question challenged candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First find the probability a seven-day period has at least one qualifying day, then use a binomial model for four periods." }, { "id": "cie-2012-s-61-q05iii", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate the mean and standard deviation of a small data set of flat-screen TV diagonals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says there were very few high marks across this question and specifically notes that few obtained the standard-deviation mark because of formula errors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the nine listed values to calculate the mean and population standard deviation, taking care to use the correct standard-deviation formula." }, { "id": "cie-2012-s-61-q06i", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Estimate the mean and standard deviation of feather lengths from two tail counts in a normal sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates could not progress with either part of this question.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert both sample proportions to normal z-values, solve the two equations for μ and σ, and check which tail each count represents." }, { "id": "cie-2012-s-61-q06ii", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Estimate how many feathers lie more than one standard deviation from the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many candidates could not progress with either part and notes that incorrect regions were often evaluated.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the standard normal probability outside −1 < Z < 1, then multiply by the sample size of 1000." }, { "id": "cie-2012-s-61-q07ai", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count line arrangements when each of seven couples must stand together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says very few candidates realised that each couple could be arranged in two ways.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat each couple as one block, arrange the seven blocks, then multiply by two internal orders per couple." }, { "id": "cie-2012-s-61-q07biii", "paperId": "cie-2012-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)(iii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count teams of four that contain at least two girls.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says candidates often omitted a possible outcome by combining the selections incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the cases with exactly two and exactly three girls, then choose the remaining team members from boys and adults." }, { "id": "cie-2012-s-62-q02i", "paperId": "cie-2012-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Construct the probability distribution of the difference between two independent values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates misread this problem and describes it as a useful discriminator.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "List the ordered pairs of outcomes, map equal pairs to zero and other pairs to their absolute difference, and combine equal outcomes." }, { "id": "cie-2012-s-62-q04i", "paperId": "cie-2012-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the median and interquartile range from a back-to-back stem-and-leaf diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found the upper-quartile position difficult because it lay midway between two observations.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the values for A, locate the median and quartile positions, then interpolate correctly when a position lies halfway between data values." }, { "id": "cie-2012-s-71-q04iii", "paperId": "cie-2012-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Use a probability density graph to explain whether its median is below, equal to or above one half.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this explanation was not well attempted; many knew the comparison but could not justify it.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the area under the density on either side of x = 0.5 and link the larger side to where the cumulative probability reaches one half." }, { "id": "cie-2012-s-71-q06iv", "paperId": "cie-2012-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iv)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State a Type II error region in a contextual test about the mean number of computers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few candidates could give the set of test-statistic values that would lead to a Type II error.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Describe a false acceptance of the unchanged-mean claim, then use the test's two-sided critical values to state the non-rejection interval." }, { "id": "cie-2012-s-72-q05ii", "paperId": "cie-2012-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that one independent normal variable is at least twice another.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates misread the event as 2F − J > 0 and that variance calculation also caused problems.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange to F − 2J ≥ 0, find the mean and variance of this linear combination, and standardise." }, { "id": "cie-2012-s-72-q07iib", "paperId": "cie-2012-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)(b)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for a two-sided normal test when the true mean is specified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Question 7, particularly part (ii), a challenge and says many candidates did not know how to calculate the Type II probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the acceptance interval from part (i), standardise both endpoints under the stated true mean, and add the two acceptance-tail probabilities." }, { "id": "cie-2012-s-73-q01ii", "paperId": "cie-2012-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State the probability that a whole 99 percent confidence interval lies below the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few candidates correctly found this probability; many did not use symmetry to identify the required tail.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use symmetry of the sampling distribution: the interval's two tails are equal, so the probability below the mean is half of the 1 percent outside the interval." }, { "id": "cie-2012-s-73-q05i", "paperId": "cie-2012-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Calculate the significance level for a one-tailed coin-bias test with 12 tosses.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Question 5 the most challenging on the paper and identifies incorrect approaches to the 12-toss significance level.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the unbiased null model use X ∼ Bin(12, 0.5) and calculate P(X ≥ 10), then express it as a percentage." }, { "id": "cie-2012-s-73-q05ii", "paperId": "cie-2012-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Find the rejection region for a one-tailed coin-bias test with 100 tosses.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s12_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this high-demand part required a normal approximation, standardisation and continuity correction, with many candidate errors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a normal approximation to the binomial, include the continuity correction, and solve the standardised boundary at the 5 percent level." }, { "id": "cie-2012-w-11-q07ii", "paperId": "cie-2012-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the given smallest positive solution to find n, then find the largest solution for θ in 0° to 360°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says relatively few candidates succeeded; the key step was identifying n = 3 and finding the largest solution for 3θ up to 1080°.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve first for nθ, use the 10° solution to identify n, then select the largest allowed solution of nθ and divide by n." }, { "id": "cie-2012-w-11-q03", "paperId": "cie-2012-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the instantaneous rate at which the area of the circular oil patch is increasing.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates found the contextual setup difficult, especially defining the variables before applying the chain rule.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate A = πr² with respect to time, substitute r = 50 and dr/dt = 3, and evaluate dA/dt." }, { "id": "cie-2012-w-11-q11iii", "paperId": "cie-2012-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection of OA and BC, then decide whether that point is the midpoint of OA.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the equation of BC was often wrong or omitted, leaving only a minority able to find E correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the equation of BC, solve it simultaneously with OA, and compare the intersection coordinates with the midpoint of OA." }, { "id": "cie-2012-w-11-q08ii", "paperId": "cie-2012-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 6, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of the shaded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes numerous integration and arithmetic errors in this area calculation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve between the stated limits, subtract the triangle area, and keep the exact terms in π and √3." }, { "id": "cie-2012-w-11-q09ii", "paperId": "cie-2012-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the midpoint information to find angle EOD.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report describes a very common mistake: treating OE as one half of CD rather than accounting for the position vector to the midpoint.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form OE from the position vectors of C and D, then use the dot product with OD to obtain the angle." }, { "id": "cie-2012-w-11-q10iii", "paperId": "cie-2012-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the inverse function and state its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says few candidates remembered the required ± when taking a square root and selected the negative branch required here.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange the function equation for x, choose the branch consistent with the original domain, and use the original range as the inverse domain." }, { "id": "cie-2012-w-12-q11i", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(i)", "marks": 4, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Show that x = 7.218 to 3 decimal places using the geometry in the diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says all candidates found this part difficult and only the most able candidates made progress using the trigonometric relation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the sine ratio from the right-angled triangle to form the given relation between x and 20 − x, solve it, and round to 3 decimal places." }, { "id": "cie-2012-w-12-q10ii", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Show that the gradient has a minimum at x = 2 and find that minimum value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part presented difficulty for all candidates and only a handful of correct solutions were seen.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the gradient, show its derivative is zero at x = 2, verify the minimum using the next derivative or an equivalent argument, and evaluate the gradient there." }, { "id": "cie-2012-w-12-q07ii", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the values of k for which the vector AB has unit length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was badly answered, with few correct responses; only a small handful equated the vector's magnitude to 1.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write AB in terms of k, set its squared magnitude equal to 1, and solve for the permitted values of k." }, { "id": "cie-2012-w-12-q02ii", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the domain of the inverse function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was poorly answered and only a small proportion recognised that the inverse domain is the range of the original function.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the range of the original function on its stated domain; that range is the domain of the inverse." }, { "id": "cie-2012-w-12-q08b", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find how many sectors are in the circle when their angles form an arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says only a small proportion earned full marks; candidates often used the common difference but did not equate the sum of n terms to 360°.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the first term 3°, common difference 2°, and sum 360° in the arithmetic-series formula, then select the positive integer n." }, { "id": "cie-2012-w-12-q03i", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the playground area is A = x² − 30x + 1200.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says at least a third of attempts used Pythagoras for side lengths and made no further progress instead of subtracting triangle areas.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Subtract the three triangle areas from the park area and simplify the remaining expression." }, { "id": "cie-2012-w-12-q04i", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the value of k for which the two curves meet in the required repeated-root case.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says simple algebra errors affected many solutions and only about a third led to the required cubic condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Eliminate y to obtain a quadratic in x, set its discriminant to zero, and solve the resulting equation for k." }, { "id": "cie-2012-w-12-q06ii", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 2 cos 2y = 3 tan 2y for 0° ≤ y ≤ 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says at least a quarter did not connect this part to part (i), and many others missed the required 2y solutions before dividing by 2.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the result from part (i), find every allowed value of 2y first, then divide those angles by 2 and check the interval." }, { "id": "cie-2012-w-12-q09ii", "paperId": "cie-2012-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 1, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Decide whether point C is nearer to B or to O.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "The report says this part was poorly answered because many candidates did not use the tangent's x-intercept/value to make the distance comparison.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the y-intercept of the tangent, then compare its position with the midpoint level between B and O." }, { "id": "cie-2012-w-13-q11iv", "paperId": "cie-2012-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(iv)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the minimum value of the curve's gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates did not realise the minimum of the gradient was required and some merely repeated part (ii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate dy/dx, find where d²y/dx² is zero, and substitute that x-value into dy/dx." }, { "id": "cie-2012-w-13-q07ii", "paperId": "cie-2012-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Write the inverse of the piecewise function in the same piecewise form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates earned all five marks; many omitted the inverse domains or gave copied domains and incorrect square-root signs.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Invert each branch separately, determine the image interval of each branch, and state the matching domains for the inverse branches." }, { "id": "cie-2012-w-13-q08ii", "paperId": "cie-2012-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Verify that the curve has a stationary point at x = −1 and determine its nature.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this part a challenge and notes that candidates often failed to justify the nature or show sufficient working.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Show dy/dx is zero at −1, then use the second derivative or an equivalent test to establish the type of stationary point." }, { "id": "cie-2012-w-13-q09i", "paperId": "cie-2012-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find p and the unit vector in the direction of OA when OAB is a straight line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies this part among the more challenging questions and says a significant number could not form the unit vector.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the parallel-vector condition to determine p, then divide OA by its magnitude to obtain the unit vector." }, { "id": "cie-2012-w-13-q06ii", "paperId": "cie-2012-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2012, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the compound trigonometric equation to 2 decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report identifies Question 6 as more challenging and notes common missing sign branches and premature rounding in its solution.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the trigonometric identities in the stated range, retain both positive and negative branches where required, and round only at the end." }, { "id": "cie-2012-w-21-q06b", "paperId": "cie-2012-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the integral involving (eˣ − 2)²e²ˣ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part one of the most difficult on the paper; very few used the correct expansion-and-integration method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand the product into exponential terms, integrate each term, and apply the limits and constant factors carefully." }, { "id": "cie-2012-w-21-q07ii", "paperId": "cie-2012-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder when p(x) is divided by x² − 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was more problematic, with algebraic and arithmetic errors common in the long division and remainder.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division with the values of a and b from part (i), and state both the quotient and remainder." }, { "id": "cie-2012-w-22-q02", "paperId": "cie-2012-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "differentiation", "trigonometry", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinate of the stationary point of y = e²ˣ sin 2x in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates solved the resulting expression set equal to zero, and reducing it to the required tangent equation was not often seen.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the product rule, set the derivative to zero, divide by the nonzero exponential factor, and solve the resulting trigonometric equation in the interval." }, { "id": "cie-2012-w-22-q07ii", "paperId": "cie-2012-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates where the tangent to the curve is parallel to the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates wrongly interpreted a horizontal tangent as y = 0 and could not progress further.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, set dy/dx = 0, solve together with the curve equation, and substitute to find both coordinates." }, { "id": "cie-2012-w-22-q06b", "paperId": "cie-2012-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Show that the given definite integral equals ln 16.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says some candidates could not recognise the logarithmic form and made no meaningful progress; numerical factors were a common issue.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise the logarithmic integral form, identify the required numerical factor, and apply the limits using logarithm laws." }, { "id": "cie-2012-w-23-q06b", "paperId": "cie-2012-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the integral involving (eˣ − 2)²e²ˣ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part one of the most difficult on the paper; very few used the correct expansion-and-integration method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand the product into exponential terms, integrate each term, and apply the limits and constant factors carefully." }, { "id": "cie-2012-w-23-q07ii", "paperId": "cie-2012-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2012, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder when p(x) is divided by x² − 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was more problematic, with algebraic and arithmetic errors common in the long division and remainder.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division with the values of a and b from part (i), and state both the quotient and remainder." }, { "id": "cie-2012-w-31-q06", "paperId": "cie-2012-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the differential equation for y in terms of x using the given condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the question was poorly answered by most candidates; most failed after separating variables because they integrated 1/(1 − y²) incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, factor 1 − y² and use partial fractions, integrate both sides, then apply y = 0 when x = 2 and solve for y." }, { "id": "cie-2012-w-31-q09ii", "paperId": "cie-2012-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the other two roots of p(x) = 0 after the given complex roots are known.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only the strongest candidates completed this successfully.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the conjugate pair to form a quadratic factor, divide the quartic by it, and solve the remaining quadratic." }, { "id": "cie-2012-w-31-q07ii", "paperId": "cie-2012-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates where the tangent is parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was difficult and the majority misinterpreted a tangent parallel to the y-axis as a horizontal-gradient condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the denominator of dy/dx to zero (or set dx/dy = 0), substitute the resulting relation into the curve, and solve for the coordinates." }, { "id": "cie-2012-w-31-q10i", "paperId": "cie-2012-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane described by the vector geometry.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the question was poorly answered overall and many candidates applied routine methods to the wrong pairs of vectors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find a normal perpendicular to both AB and OC, for example by a vector product, then use a point on AB to form the plane equation." }, { "id": "cie-2012-w-31-q05iv", "paperId": "cie-2012-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iv)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Show the exact value of the definite integral involving (sec x − tan x)².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says most candidates did not use the result from part (i) to write down the required integral, and the overall question challenged most candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (ii), expand the square, rewrite the integrand using sec x and tan x, then integrate over the stated limits." }, { "id": "cie-2012-w-31-q05ii", "paperId": "cie-2012-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 1, "topicIds": [ "proof", "trigonometry" ], "primaryTopicId": "proof", "summary": "Show that 1/(sec x − tan x) is identically sec x + tan x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates lacked a sound strategy for this identity and often stopped after rewriting the left-hand side.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Multiply numerator and denominator by sec x + tan x and use sec²x − tan²x = 1." }, { "id": "cie-2012-w-32-q06", "paperId": "cie-2012-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the differential equation for y in terms of x using the given condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the question was poorly answered by most candidates; most failed after separating variables because they integrated 1/(1 − y²) incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, factor 1 − y² and use partial fractions, integrate both sides, then apply y = 0 when x = 2 and solve for y." }, { "id": "cie-2012-w-32-q09ii", "paperId": "cie-2012-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the other two roots of p(x) = 0 after the given complex roots are known.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only the strongest candidates completed this successfully.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the conjugate pair to form a quadratic factor, divide the quartic by it, and solve the remaining quadratic." }, { "id": "cie-2012-w-32-q07ii", "paperId": "cie-2012-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates where the tangent is parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was difficult and the majority misinterpreted a tangent parallel to the y-axis as a horizontal-gradient condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the denominator of dy/dx to zero (or set dx/dy = 0), substitute the resulting relation into the curve, and solve for the coordinates." }, { "id": "cie-2012-w-32-q10i", "paperId": "cie-2012-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane described by the vector geometry.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the question was poorly answered overall and many candidates applied routine methods to the wrong pairs of vectors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find a normal perpendicular to both AB and OC, for example by a vector product, then use a point on AB to form the plane equation." }, { "id": "cie-2012-w-32-q05iv", "paperId": "cie-2012-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iv)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Show the exact value of the definite integral involving (sec x − tan x)².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says most candidates did not use the result from part (i) to write down the required integral, and the overall question challenged most candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (ii), expand the square, rewrite the integrand using sec x and tan x, then integrate over the stated limits." }, { "id": "cie-2012-w-32-q05ii", "paperId": "cie-2012-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 1, "topicIds": [ "proof", "trigonometry" ], "primaryTopicId": "proof", "summary": "Show that 1/(sec x − tan x) is identically sec x + tan x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates lacked a sound strategy for this identity and often stopped after rewriting the left-hand side.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Multiply numerator and denominator by sec x + tan x and use sec²x − tan²x = 1." }, { "id": "cie-2012-w-33-q07i", "paperId": "cie-2012-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use u = sin 2x in a suitable integral to find A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part very challenging; only a minority progressed beyond finding du/dx, and many did not obtain the correct limits.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute u = sin 2x, use the double-angle identity to rewrite the integrand, transform the limits, and integrate in u." }, { "id": "cie-2012-w-33-q10bii", "paperId": "cie-2012-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 6, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Find p, q, α and β for the moduli and arguments of the complex-number region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a minority showed clear understanding of the geometry and many made incorrect assumptions about the tangent contact points.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read the circle and angular boundaries from the Argand diagram, find the extreme moduli and argument limits, and give the requested precision." }, { "id": "cie-2012-w-33-q06i", "paperId": "cie-2012-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the negative integer α that is a root of the polynomial.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates could not find a starting method and that many algebraic approaches led nowhere.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem and test the plausible negative integers in the polynomial until the root is found." }, { "id": "cie-2012-w-33-q10a", "paperId": "cie-2012-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2012, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the complex-number equation without a calculator.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says large numbers of candidates found one square root but overlooked the second solution.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange to isolate w², then take both square roots of the resulting complex number and check both solutions." }, { "id": "cie-2012-w-41-q06ii", "paperId": "cie-2012-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle's speed at the midpoint of CD after the specified frictional motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates made progress on this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the work–energy relation over the relevant section, accounting for the constant friction force and the fact that the particle comes to rest at D." }, { "id": "cie-2012-w-41-q07iii", "paperId": "cie-2012-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the requested distance AC in the linked-particle motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few candidates made progress in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the motion equations for the particles and use the result from the preceding parts to find the required displacement." }, { "id": "cie-2012-w-41-q06i", "paperId": "cie-2012-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the decrease in P's speed as it moves from B to C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few candidates obtained the correct decrease in speed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy between B and C, accounting for the change in height and the work done by friction, and solve for the speed change." }, { "id": "cie-2012-w-41-q05ii", "paperId": "cie-2012-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find k and the speed of Q at B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says very few candidates made progress in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the motion information from part (i) to form the appropriate displacement and acceleration relations, then solve for k and Q's speed." }, { "id": "cie-2012-w-42-q07iv", "paperId": "cie-2012-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find P's speed when it next passes through O.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most difficult part and says very few candidates integrated v(t) to identify t = 80.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the return time by integrating the velocity to obtain the required displacement, then substitute that time into v(t)." }, { "id": "cie-2012-w-42-q06i", "paperId": "cie-2012-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the car's gain in kinetic energy as it moves up the hill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says Question 6 was significantly harder than earlier questions and a large proportion omitted forces when applying Newton's second law.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use Newton's second law and P = Fv to find the speeds at the bottom and top, including both the weight component and resistance, then calculate the kinetic-energy change." }, { "id": "cie-2012-w-43-q07iii", "paperId": "cie-2012-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the time for B to reach the pulley after the string becomes slack.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates incorrectly assumed acceleration throughout both stages after the string became slack.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the motion at the instant the string becomes slack and use the correct constant-speed or accelerated motion for each stage." }, { "id": "cie-2012-w-43-q06", "paperId": "cie-2012-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the full range of P for which the particle remains in equilibrium on the inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says most assumed friction acted in only one direction and only a few understood that P must lie in a range.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Consider friction acting both up and down the plane and apply |F| ≤ μR to obtain the lower and upper possible values of P." }, { "id": "cie-2012-w-43-q05iii", "paperId": "cie-2012-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the horizontal force in the setup.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies part (iii) as one of the harder parts and says the horizontal force was often misinterpreted as acting up the plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the horizontal force into components parallel and perpendicular to the plane, then apply the force or energy relations established in earlier parts." }, { "id": "cie-2012-w-43-q01ii", "paperId": "cie-2012-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2012, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find P's speed as it passes through B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this was one of the less well-answered questions and notes confusion about the semicircular path and the energy terms.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the energy balance from the starting point to B, including the potential-energy change and work done against resistance, then solve for speed." }, { "id": "cie-2012-w-51-q07iii", "paperId": "cie-2012-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest magnitude of P's acceleration during the motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Question 7 the hardest on the paper and says this part was very rarely attempted; candidates often found only the acceleration at the top.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the accelerations at both the top and bottom of the motion and compare their magnitudes." }, { "id": "cie-2012-w-51-q07i", "paperId": "cie-2012-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find how far P moves before first coming to rest above B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says Question 7 was the hardest on the paper and many candidates used the wrong number of terms in the energy equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use an energy equation with the kinetic, gravitational and elastic-energy changes, and account for the slack string at the turning point." }, { "id": "cie-2012-w-51-q07ii", "paperId": "cie-2012-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show that maximum speed occurs 2.1 m below A and find that speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the equilibrium position and a second energy equation were necessary to complete this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the tension balance to locate the equilibrium position, then apply an energy equation at that position to find the greatest speed." }, { "id": "cie-2012-w-52-q07iii", "paperId": "cie-2012-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest magnitude of P's acceleration during the motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Question 7 the hardest on the paper and says this part was very rarely attempted; candidates often found only the acceleration at the top.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the accelerations at both the top and bottom of the motion and compare their magnitudes." }, { "id": "cie-2012-w-52-q07i", "paperId": "cie-2012-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find how far P moves before first coming to rest above B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says Question 7 was the hardest on the paper and many candidates used the wrong number of terms in the energy equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use an energy equation with the kinetic, gravitational and elastic-energy changes, and account for the slack string at the turning point." }, { "id": "cie-2012-w-52-q07ii", "paperId": "cie-2012-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show that maximum speed occurs 2.1 m below A and find that speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the equilibrium position and a second energy equation were necessary to complete this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the tension balance to locate the equilibrium position, then apply an energy equation at that position to find the greatest speed." }, { "id": "cie-2012-w-53-q04iii", "paperId": "cie-2012-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use the centre-of-mass condition to calculate h.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the required tangent relation was rarely seen and very little progress was made in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the expression for OG into the condition relating OG to the radius, then solve the resulting equation for h." }, { "id": "cie-2012-w-53-q02ii", "paperId": "cie-2012-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find where P has greatest kinetic energy and calculate that kinetic energy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a minority used the correct tension balance to find the equilibrium position and then make progress.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the equilibrium position by balancing the two spring tensions and the particle's weight, then use an energy balance to find the kinetic energy there." }, { "id": "cie-2012-w-53-q06ii", "paperId": "cie-2012-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 7, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find when the particles have speed 2.5 m/s and how far they have travelled by then.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes repeated errors integrating the exponential function and candidates often assumed the integration constant was zero.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use v = 5e⁻³ᵗ to find the time, then integrate the velocity function with the correct constant and limits to obtain distance." }, { "id": "cie-2012-w-53-q02i", "paperId": "cie-2012-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2012, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Calculate the speed of projection of P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says Question 2 was difficult for many candidates and energy equations often omitted a term or included an extra one.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy between the starting position and the point described, including the elastic and gravitational energy terms." }, { "id": "cie-2012-w-61-q07c", "paperId": "cie-2012-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the ways to divide nine distinct pies among three people so each gets an odd number.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only the very best candidates earned full marks and describes the required partition cases and ordering counts.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "List the odd partitions 1+1+7, 1+3+5 and 3+3+3, then count the assignments of people and distinct pies for each case." }, { "id": "cie-2012-w-61-q04iii", "paperId": "cie-2012-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that among three caravan prices, two exceed the median and one is below the lower quartile.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only the best candidates recognised that the selected categories could be ordered in 3C2 ways.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the relevant proportions for prices above the median and below the lower quartile, and count the possible assignments of these categories to the three selections." }, { "id": "cie-2012-w-61-q05ii", "paperId": "cie-2012-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Calculate the expected profit for the company.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says only the best candidates scored full marks on this expected-profit part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "List the possible outcomes, calculate each profit and probability, then sum profit × probability over all outcomes." }, { "id": "cie-2012-w-61-q02", "paperId": "cie-2012-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Recover Σx and Σx² from the coded sums for 24 people's pocket money.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls this question a challenge and says many candidates mixed coded and raw means or variances.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand the coded expressions, use the number of observations and the supplied sums to isolate Σx and Σx², keeping raw and coded values consistent." }, { "id": "cie-2012-w-62-q06iii", "paperId": "cie-2012-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether the events R and S are independent, showing your working.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only the best candidates earned full marks and many confused independence with mutual exclusivity.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find P(R), P(S) and P(R ∩ S), then compare P(R ∩ S) with P(R)P(S) and state the conclusion." }, { "id": "cie-2012-w-62-q05biii", "paperId": "cie-2012-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements in which all eight people sit together with no empty seats between them.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part rarely earned full marks; candidates often counted arrangements of the five empty seats rather than their five possible positions.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Arrange the eight people as one contiguous block and count the five possible starting positions for the block in the row of twelve seats." }, { "id": "cie-2012-w-62-q05bii", "paperId": "cie-2012-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements where Mary and Frances are not in adjacent seats.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this a discriminating part and gives the paired-block count that candidates needed to subtract.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count all arrangements, subtract arrangements where Mary and Frances sit together, and account for the two possible orders within the pair." }, { "id": "cie-2012-w-62-q03iii", "paperId": "cie-2012-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(iii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean travel time from the grouped data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only about half the candidates estimated this grouped mean correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Multiply each class midpoint by its frequency, add the products, and divide by 112." }, { "id": "cie-2012-w-63-q06c", "paperId": "cie-2012-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that no two neighbouring tiles have the same colour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most difficult part for most candidates and describes common attempts that counted the complement incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "There are 3 choices for the first tile and 2 for each later tile, so use 3 × 2⁷ favourable rows out of 3⁸ total rows." }, { "id": "cie-2012-w-63-q06bi", "paperId": "cie-2012-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the even seven-digit numbers formed by rearranging the digits of 1 244 687.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates found this part very difficult and most correct solutions separated the final-digit cases.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate cases by the final digit (2, 4, 6 or 8), then count distinct arrangements of the remaining digits while accounting for repeated 4s." }, { "id": "cie-2012-w-63-q03ii", "paperId": "cie-2012-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of a low birth rate, given that the country does not have a medium GDP.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says only a minority correctly identified both the denominator group and the required numerator.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Restrict the sample space to the low- and high-GDP countries, then divide the low-birth-rate count in that group by its total." }, { "id": "cie-2012-w-63-q05i", "paperId": "cie-2012-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2012, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a normally distributed value rounds to 84.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most candidates standardised at 84 instead of using the rounding interval from 83.5 to 84.5.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the continuity interval 83.5 ≤ X < 84.5, standardise both bounds, and subtract the corresponding cumulative probabilities." }, { "id": "cie-2012-w-71-q04i", "paperId": "cie-2012-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region for the binomial test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found this question challenging and needed P(X ≤ 1) and P(X ≤ 2) to establish the critical region.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For X ~ B(20, 0.25), calculate cumulative probabilities and identify the largest lower-tail region that satisfies the significance level." }, { "id": "cie-2012-w-71-q03ii", "paperId": "cie-2012-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of the total amount paid by six independent bicycle hires.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the question was challenging and that standard deviations were less well attempted than the means.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Add the six expected values and add the variances using the stated scaled distributions; take the square root for the standard deviation." }, { "id": "cie-2012-w-71-q06iii", "paperId": "cie-2012-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find x for a confidence interval of width 0.1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the final step from z = 1.802 to the confidence percentage was not often understood.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the confidence-interval width formula equal to 0.1, solve for the corresponding z-value, and convert that central probability to a confidence percentage." }, { "id": "cie-2012-w-71-q07ii", "paperId": "cie-2012-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find n from the stated probability relation for the sum S.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was slightly more challenging and identifies missing brackets around powers as a recurring source of wrong answers.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the probabilities for S under the sum's distribution, substitute the given ratio, and solve the resulting equation in n with brackets kept around powers." }, { "id": "cie-2012-w-72-q04i", "paperId": "cie-2012-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region for the binomial test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found this question challenging and needed P(X ≤ 1) and P(X ≤ 2) to establish the critical region.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For X ~ B(20, 0.25), calculate cumulative probabilities and identify the largest lower-tail region that satisfies the significance level." }, { "id": "cie-2012-w-72-q03ii", "paperId": "cie-2012-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of the total amount paid by six independent bicycle hires.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the question was challenging and that standard deviations were less well attempted than the means.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Add the six expected values and add the variances using the stated scaled distributions; take the square root for the standard deviation." }, { "id": "cie-2012-w-72-q06iii", "paperId": "cie-2012-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find x for a confidence interval of width 0.1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the final step from z = 1.802 to the confidence percentage was not often understood.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the confidence-interval width formula equal to 0.1, solve for the corresponding z-value, and convert that central probability to a confidence percentage." }, { "id": "cie-2012-w-72-q07ii", "paperId": "cie-2012-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find n from the stated probability relation for the sum S.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was slightly more challenging and identifies missing brackets around powers as a recurring source of wrong answers.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the probabilities for S under the sum's distribution, substitute the given ratio, and solve the resulting equation in n with brackets kept around powers." }, { "id": "cie-2012-w-73-q04", "paperId": "cie-2012-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2012, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that one of two potato masses is at least twice the other.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w12_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates doubled the one-tail probability to include either potato being at least twice as heavy as the other.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w12_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Let D = X − 2Y for independent normal masses; find its mean and variance, calculate P(D ≥ 0), and double it for the symmetric event." }, { "id": "ial-legacy-2015-january-WMA01-q14i", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a sine-cosine equation throughout a full 0° to 360° interval.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=36", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=25", "qpPages": [ 36, 36 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says part 14(i) was very demanding and that a large number of candidates did not attempt it. Common errors included incorrect trigonometric identities and mishandling the algebra before finding both angles.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the left side as a single shifted sine or use tan x = -7/3. Find both solutions in the correct quadrants, then give each to one decimal place.", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-legacy-2015-january-WMA01-q1b", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Simplify a quotient of fractional powers, surds and powers of x.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that 48% gained no marks or only one mark on part 1(b), describing it as very poorly done. Candidates often could not manipulate the surds and fractions or combined the powers of x incorrectly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the numerical coefficients and the powers using compatible index notation, simplify the surd and fractional factors, then combine the powers of x and present a fully simplified result.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2015-january-WMA01-q13c", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the discriminant to derive the condition for a line to meet a curve at only one point.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=32", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=24", "qpPages": [ 32, 32 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says part 13(c) was more demanding than the preceding intersection question. It was often omitted or left incomplete; many candidates were defeated by the algebra when eliminating a variable and few completed the required discriminant argument.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the line equation into the curve to obtain a quadratic, then set its discriminant to zero and simplify to the required equation in k.", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-legacy-2015-january-WMA01-q15c", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 15, "questionPart": "15(c)", "marks": 7, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the area between a curve, the y-axis and a horizontal line through its minimum point.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=40", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=26", "qpPages": [ 40, 40 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says part 15(c) distinguished between candidates. Many could integrate the curve but failed to identify the required bounded region, find the rectangle area, or combine the signed areas correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve over 0 ≤ x ≤ 4 and combine that signed area with the rectangle formed by the horizontal line y = -168. Equivalently, integrate y + 168 over the interval; the required area is 240 square units.", "sourceEvidencePages": { "examinerReport": [ 11 ] } }, { "id": "ial-legacy-2015-january-WMA01-q5a", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof", "sequences" ], "primaryTopicId": "proof", "summary": "Prove the formula for the sum of the first n terms of an arithmetic series.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Only 16% earned full marks on Question 5 and 26% earned none. Pearson notes that many proofs were incomplete or poorly structured, with errors in the last term, insufficient terms, or reasoning that was valid only for even n.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the sum forwards and backwards, align corresponding terms and add to obtain n times the sum of the first and last terms. Use the last term a + (n - 1)d to reach the stated formula without assuming the result.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WMA01-q11a", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Give the exact coordinates where a shifted sine curve meets both axes over the stated interval.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=26", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=20", "qpPages": [ 26, 26 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson says candidates were generally more familiar with part 11(b) than part 11(a). Many omitted the y-intercept or gave only two of the four x-intercepts, sometimes assuming an incorrect symmetry about the y-axis.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Set x = 0 for the y-intercept, then set sin(x - 60°) = 0 and list every x-intercept in -360° ≤ x ≤ 360°. Give exact coordinates, including the negative y-value at the origin.", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-legacy-2015-january-WMA01-q12b", "paperId": "ial-legacy-2015-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2015, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(b)", "marks": 4, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Find the first calendar year when a geometrically growing annual profit exceeds one million pounds.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_que_20150113.pdf#page=28", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA01_01_msc_20150305.pdf#page=22", "qpPages": [ 28, 28 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson reports that candidates often stopped at n = 15 or n > 14.5 without giving the year, used n instead of n - 1 as the exponent, or used a sum formula instead of the term formula.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Use 275000 × 1.1^(n-1) > 1000000, solve the inequality for the first integer term, and convert that term number into the corresponding calendar year.", "sourceEvidencePages": { "examinerReport": [ 9 ] } }, { "id": "ial-legacy-2015-january-WMA02-q13d", "paperId": "ial-legacy-2015-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2015, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(d)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Determine the time taken by the Ferris wheel to complete two revolutions.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_que_20150126.pdf#page=44", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_msc_20150305.pdf#page=27", "qpPages": [ 44, 44 ], "msPages": [ 27, 28 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says very few candidates were successful on part 13(d). Most attempts incorrectly multiplied a time from an earlier part by four.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "The model angle increases at 30 degrees per minute. Two revolutions correspond to 720 degrees, so divide 720 by 30 and include the required time unit.", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-legacy-2015-january-WMA02-q5b", "paperId": "ial-legacy-2015-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2015, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "binomial", "proof" ], "primaryTopicId": "binomial", "summary": "Use a binomial approximation at a specified x-value to obtain a fractional estimate of √10.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_que_20150126.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_msc_20150305.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes part 5(b) as very demanding. Many candidates substituted into only one side of the expression or assumed that √10 equalled the right-hand side without rearranging the given relation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute x = 1/20 into the expansion from part (a), equate the resulting approximation with the stated expression and rearrange to isolate √10. Simplify the resulting fraction.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WMA02-q8c", "paperId": "ial-legacy-2015-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2015, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an equation involving two exponential terms for the exact age of a car.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_que_20150126.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_msc_20150305.pdf#page=18", "qpPages": [ 24, 24 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson calls part 8(c) discriminating and says relatively few attempts were correct. The common error was taking logarithms of the separate terms instead of recognizing and solving a quadratic in e^(-0.1t).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = e^(-0.1t), so e^(-0.2t) = u². Solve the resulting quadratic for its positive root, then take logarithms to obtain the exact value t = 10 ln(9/7).", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2015-january-WMA02-q13b", "paperId": "ial-legacy-2015-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2015, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the maximum height of a Ferris-wheel passenger and the first time it occurs.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_que_20150126.pdf#page=44", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_msc_20150305.pdf#page=27", "qpPages": [ 44, 44 ], "msPages": [ 27, 28 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says only the more able candidates succeeded on part 13(b). Common errors were choosing the wrong cosine extreme for the maximum and losing the phase sign when finding the time.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the phase-shifted cosine model from part (a). The height is maximized when the cosine term is -1; solve the phase equation for its first valid time and round both answers to two decimal places.", "sourceEvidencePages": { "examinerReport": [ 9 ] } }, { "id": "ial-legacy-2015-january-WMA02-q4", "paperId": "ial-legacy-2015-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2015, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an integral exactly using the substitution x = 2 sin θ.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_que_20150126.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_msc_20150305.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson says Question 4 was not answered particularly well and only a relatively small proportion of solutions were fully correct. Most candidates stopped after substituting because they did not simplify 4 - (2 sin θ)² to 4 cos² θ.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute x = 2 sin θ and dx = 2 cos θ dθ. Simplify 4 - x² to 4 cos² θ so the integrand becomes a multiple of sec² θ, then integrate and evaluate the transformed limits.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WMA02-q6i", "paperId": "ial-legacy-2015-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2015, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx from the relation x = tan²(4y), expressing the result in x.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_que_20150126.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WMA02_01_msc_20150305.pdf#page=14", "qpPages": [ 16, 16 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson reports that candidates were split between the two parts of Question 6 and that those completing both were a minority. In part 6(i), many found dx/dy but stopped before converting the reciprocal derivative into a function of x.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x with respect to y, invert to obtain dy/dx, and replace tan²(4y) with x and tan(4y) with √x using the stated interval. Simplify to the required form.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WME01-q6a", "paperId": "ial-legacy-2015-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2015, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest distance a particle can be placed from one end of a rod while it remains in equilibrium on two supports.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_que_20150120.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_msc_20150305.pdf#page=12", "qpPages": [ 20, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies Question 6(a) as a challenging part. Many candidates did not realize that the greatest x occurs when the reaction at A is zero, so their equations retained too many unknowns and no valid progress was possible.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "At the greatest possible distance, the reaction at support A is zero and the rod is about to tip about B. Set R_A = 0, take moments about B, and solve for x.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WME01-q8c", "paperId": "ial-legacy-2015-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2015, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Decide whether a particle on a table reaches a pulley after a hanging particle hits the floor.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_que_20150120.pdf#page=28", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_msc_20150305.pdf#page=14", "qpPages": [ 28, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says part 8(c) was more challenging and involved several steps. Many found the impact speed but then incorrectly reused the earlier acceleration or used g as the later deceleration; others did not justify the comparison with the remaining distance.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the speed of A when B hits the floor. Then use the friction force alone to find A's new deceleration and stopping distance, and compare that distance with the remaining 0.3 m before the pulley.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2015-january-WME01-q3c", "paperId": "ial-legacy-2015-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2015, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when a moving particle is north-west of the origin from its position vector.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_que_20150120.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_msc_20150305.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson names Question 3(c) among the challenging parts. It required equating the negative i-component to the j-component; a number of candidates had no valid method or omitted the part, while equating same-sign components gave the wrong quadrant.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the position vector as (21 - 6t)i + (5 + 2t)j. For a north-west position, equate the negative i-component with the positive j-component and solve for t.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2015-january-WME01-q4", "paperId": "ial-legacy-2015-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2015, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the common height of two stones released half a second apart that hit the ground together.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_que_20150120.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WME01_01_msc_20150305.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson lists Question 4 among the paper's challenging questions. Candidates often used t and t + 0.5 for the two stones, or used inconsistent time variables within the equations, which prevented them from finding the common height.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Use one time variable consistently for the first stone and a time 0.5 seconds shorter for the second. Equate their expressions for the common height and solve for the shared impact time, then calculate the height.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2015-january-WST01-q7b", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a value k that divides a normal distribution into tails in a 1-to-3 probability ratio.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says the normal distribution remains challenging and nearly a quarter of candidates could not gain more than a couple of marks on Question 7. In part 7(b), many missed that P(W < k) = 0.75 or used the wrong sign when standardizing.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Use P(W < k) = 0.75, standardize with mean 99 and standard deviation 3.6, then use the closest appropriate z-value and round k to one decimal place.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WST01-q6e", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find E(Y²) when Y is a linear transformation of a discrete random variable.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=10", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes part 6(e) as quite challenging. A common error was confusing E(Y²) with [E(Y)]²; candidates who succeeded either formed the distribution of Y or used E(Y²) = Var(Y) + [E(Y)]².", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Use E(Y²) = E((3X - 1)²) = 9E(X²) - 6E(X) + 1, or calculate Var(Y) + [E(Y)]². Keep E(Y²) distinct from [E(Y)]².", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WST01-q5g", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(g)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the endpoints containing the middle 95% of a normal distribution.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=9", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says part 5(g) offered a challenge for the more able. Many candidates became lost in equations involving a, b and 0.95; some standardized with 1.64 instead of 1.96 or did not correctly solve for both endpoints.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the central 95% interval, so the standard normal cut-offs are ±1.96. Apply these to mean 73 and standard deviation 8 to obtain a and b.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2015-january-WST01-q3e", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Predict how adding an outlying data point changes the product moment correlation coefficient.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says only the best candidates fully answered part 3(e). Many asserted that coding left r unchanged or said r decreased without a convincing reason, showing weak understanding of what the correlation coefficient measures.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "The added point is well above the previous positive trend at a high milk price, so it weakens the linear association and r decreases. Explain the effect in relation to the trend or regression line.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2015-january-WST01-q1e", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for the sum of two independent observations, given both are odd.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson says parts 1(d) and 1(e) were answered correctly only by stronger candidates. In part 1(e), many did not recognize that conditional probability was required, and some omitted one of the (1,5), (3,3) or (5,1) cases from the numerator.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "List the ordered odd pairs that sum to 6: (1,5), (3,3) and (5,1). Divide their total probability by P(both observations are odd), and give the answer to three significant figures.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2015-january-WST01-q7d", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation when a stated percentile of a normal distribution is known.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson notes that Question 7 was challenging and describes errors in part 7(d), including using 0.84 instead of 0.8416, choosing a sign that produced a negative standard deviation, or manipulating the standardized equation incorrectly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardize the 20th percentile using z = -0.8416, set (116 - 120)/σ equal to that z-value, and solve for the positive standard deviation.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2015-january-WST01-q1d", "paperId": "ial-legacy-2015-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2015, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that two independent observations are both odd.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_que_20150115.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2015 Jan/WST01_01_msc_20150305.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson says parts 1(d) and 1(e) were answered correctly only by stronger candidates. In part 1(d), candidates commonly omitted ordered outcomes such as (1,3) and (3,1) or failed to include doubles when listing cases.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20150231.pdf", "tagConfidence": "reviewed", "answerGuide": "Add the probabilities of the odd outcomes for one observation, then square that probability because the observations are independent. Equivalently, add the probabilities of all ordered odd pairs.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2016-january-WMA01-q2ii", "paperId": "ial-legacy-2016-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2016, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Show an exact surd identity by rationalising the denominator and simplifying every step.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_que_20160112.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_msc_20160216.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson notes that many candidates lost the final mark by skipping the intermediate simplification √18 = 3√2 or failing to show all stages, despite reaching the correct value.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Multiply by the conjugate, simplify the denominator to a rational value, rewrite √18 as 3√2, and show the intermediate equality required by the question.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2016-january-WMA01-q8b", "paperId": "ial-legacy-2016-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2016, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a shifted trigonometric equation for every angle in the stated 0° to 360° interval.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_que_20160112.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_msc_20160216.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Question 8 as one of the paper’s most challenging questions. Candidates often found the reference angle but omitted one or two solutions when converting 2θ + 30° back to θ.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Use tan(2θ + 30°) = 3/7, find the reference angle, generate all four solutions in range, then halve the shifted angles and round to one decimal place.", "sourceEvidencePages": { "examinerReport": [ 4, 7 ] } }, { "id": "ial-legacy-2016-january-WMA01-q10b", "paperId": "ial-legacy-2016-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2016, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates of a stationary point after differentiating a fractional-power curve.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_que_20160112.pdf#page=26", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_msc_20160216.pdf#page=15", "qpPages": [ 26, 26 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson names Question 10 among the most challenging. The index manipulation after setting the derivative to zero caused difficulty; many mishandled fractional powers or gave x without substituting back to find the required coordinates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the derivative equal to zero, collect the powers of x on opposite sides, solve the resulting fractional-power equation, then substitute the positive x-value into the curve to obtain y.", "sourceEvidencePages": { "examinerReport": [ 4, 8 ] } }, { "id": "ial-legacy-2016-january-WMA01-q13a", "paperId": "ial-legacy-2016-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2016, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(a)", "marks": 4, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Use the no-real-roots condition to derive an inequality in a parameter.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_que_20160112.pdf#page=38", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_msc_20160216.pdf#page=18", "qpPages": [ 38, 38 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson calls Question 13 one of the paper’s most challenging questions and says many found part (a) difficult, with sign and algebra errors when applying the discriminant condition.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange the equation into a quadratic in x, identify its coefficients, use a negative discriminant, expand and simplify to the required quadratic inequality in k.", "sourceEvidencePages": { "examinerReport": [ 4, 9 ] } }, { "id": "ial-legacy-2016-january-WMA01-q16c", "paperId": "ial-legacy-2016-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2016, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 16, "questionPart": "16(c)", "marks": 8, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find exactly the area of a region bounded by two line segments and two arcs of a quadratic curve.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_que_20160112.pdf#page=48", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA01_01_msc_20160216.pdf#page=21", "qpPages": [ 48, 48 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson identifies Question 16 as one of the paper’s most challenging. In part (c), candidates often chose incomplete limits, confused the required curve areas with the trapezium, combined areas incorrectly, or gave a rounded answer instead of an exact value.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the intersection and x-intercept values to split the region into the appropriate curve areas and trapezium. Integrate the quadratic over each required interval, combine signed areas, and keep the result exact.", "sourceEvidencePages": { "examinerReport": [ 4, 11 ] } }, { "id": "ial-legacy-2016-january-WMA02-q4b", "paperId": "ial-legacy-2016-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2016, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Scale the volume of a solid of revolution to find the volume of a geometrically similar candle.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_que_20160119.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_msc_20160216.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that fewer than 5% of candidates answered part (b) successfully. Some integrated the curve again instead of using similarity, while others found the linear scale factor but did not cube it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the linear scale factor from the axial lengths, cube it to obtain the volume scale factor, and multiply the volume from part (a) by that factor.", "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-legacy-2016-january-WMA02-q6c", "paperId": "ial-legacy-2016-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2016, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Prove a stated expression for the cooling rate in terms of the current temperature.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_que_20160119.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_msc_20160216.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says very few candidates completed the proof. Many stopped after differentiating or struggled to rearrange the original model and substitute for the exponential term.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the exponential temperature model with respect to time, then use the original model to eliminate the exponential term and express the derivative using T.", "sourceEvidencePages": { "examinerReport": [ 4, 6 ] } }, { "id": "ial-legacy-2016-january-WMA02-q8b", "paperId": "ial-legacy-2016-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2016, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact integral involving θ² and a trigonometric function using integration by parts.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_que_20160119.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_msc_20160216.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that completely correct solutions were rare. Candidates often ignored the θ² factor, omitted the second integration by parts, lost a sign, or failed to substitute the lower limit.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the trigonometric expression in the form a + b cos 2θ, integrate by parts twice for the θ² cos 2θ term, then apply both limits and simplify the exact result.", "sourceEvidencePages": { "examinerReport": [ 4, 6, 7 ] } }, { "id": "ial-legacy-2016-january-WMA02-q10c", "paperId": "ial-legacy-2016-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2016, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the greatest and least values of a squared trigonometric expression after rewriting its inner function in amplitude-phase form.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_que_20160119.pdf#page=30", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_msc_20160216.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson notes that many candidates solved either the maximum or minimum correctly but rarely both. Common errors included forgetting the square on R or using the wrong extreme for the least value.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the amplitude R from part (a) to determine the maximum and minimum values of the inner expression, then square those values and apply the outer scale and constant separately for the greatest and least values.", "sourceEvidencePages": { "examinerReport": [ 4, 8 ] } }, { "id": "ial-legacy-2016-january-WMA02-q11e", "paperId": "ial-legacy-2016-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2016, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(e)", "marks": 3, "topicIds": [ "functions", "exponentials", "algebra" ], "primaryTopicId": "functions", "summary": "Compose two functions and express the result as a polynomial.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_que_20160119.pdf#page=34", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_msc_20160216.pdf#page=22", "qpPages": [ 34, 34 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson says candidates often reached an equivalent exponential-power expression but did not recognize that the question required a polynomial, leaving an unexpanded expression instead.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute g(x) into f, use the power law for logarithms and the inverse relationship between exponentials and logarithms, then expand the resulting cubic as a polynomial.", "sourceEvidencePages": { "examinerReport": [ 4, 8, 9 ] } }, { "id": "ial-legacy-2016-january-WMA02-q12d", "paperId": "ial-legacy-2016-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2016, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(d)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the shortest distance from a point on one line to a second line in three dimensions.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_que_20160119.pdf#page=38", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2016 Jan/WMA02_01_msc_20160216.pdf#page=24", "qpPages": [ 38, 38 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson identifies part (d) as the most problematic part of Question 12. Many candidates stopped after finding a distance with Pythagoras or used tangent rather than sine, suggesting the perpendicular-distance geometry was not understood.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the shortest-distance geometry: either find the perpendicular component of the displacement using the angle between the lines, or project onto a point on the second line and minimize the distance.", "sourceEvidencePages": { "examinerReport": [ 4, 9 ] } }, { "id": "ial-legacy-2016-january-WME01-q1a", "paperId": "ial-legacy-2016-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2016, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the driving force needed to accelerate a truck towing a trailer against resistance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20160125.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20160302.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says Question 1 had the lowest success on the paper and that understanding of connected bodies was weak. For part (a), candidates often omitted the tow-bar tension or one of the resistance/driving forces.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw or identify the forces on the truck, then apply Newton’s second law along the road, including the tow-bar tension and the truck’s resistance.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2016-january-WME01-q2a", "paperId": "ial-legacy-2016-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2016, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed and direction of a particle after a direct collision using impulse and momentum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20160125.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20160302.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports many sign errors in impulse-momentum calculations, often because candidates did not state a direction of motion. Vague descriptions of the final direction also lost marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, apply impulse as change in momentum with signed velocities, solve for the final velocity, then report its speed and direction explicitly.", "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-legacy-2016-january-WME01-q4c", "paperId": "ial-legacy-2016-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2016, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch a velocity-time graph for a stone moving upward and then falling below its starting point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20160125.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20160302.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says relatively few candidates produced a fully correct graph. A common error was drawing a V-shape for speed rather than a straight velocity line, and many missed the required time-axis intercept.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a single straight line with constant negative gradient for acceleration due to gravity, starting at the given upward velocity and marking the times when it crosses zero velocity and returns to the origin’s height.", "sourceEvidencePages": { "examinerReport": [ 4, 6 ] } }, { "id": "ial-legacy-2016-january-WME01-q5b", "paperId": "ial-legacy-2016-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2016, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the positions where either supporting string would exceed its maximum tension.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20160125.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20160302.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson notes that candidates did not always compare both tensions with the maximum, often reversed an inequality, or lost the endpoint by using strict inequalities instead of allowing equality.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the allowed tension limit to each of the two tension expressions from part (a), solve both inequalities, and combine them into the valid interval for x, including equality where the question permits it.", "sourceEvidencePages": { "examinerReport": [ 4, 7 ] } }, { "id": "ial-legacy-2016-january-WME01-q7d", "paperId": "ial-legacy-2016-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2016, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 7, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the initial distance when one particle stops at the pulley after the string breaks.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20160125.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20160302.pdf#page=14", "qpPages": [ 20, 20 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson says the final part of Question 7 discriminated at the top end. Many candidates found the speed at the break but did not calculate the new deceleration, while others forgot to add the first 0.5 m to the stopping distance.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the speed when the string breaks, calculate the new deceleration of the table particle under friction alone, use constant-acceleration equations to find its remaining stopping distance, then add the 0.5 m already travelled.", "sourceEvidencePages": { "examinerReport": [ 4, 8 ] } }, { "id": "ial-legacy-2016-january-WST01-q2f", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(f)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that three selected candidates all pass and exactly two achieve a merit grade.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says only the most able candidates obtained the correct answer. Common mistakes were using the wrong probability for passing without merit or forgetting the factor for which of the three candidates is the non-merit passer.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat the three selections with the required pass and merit outcomes, multiply the relevant probabilities, account for the choice of which candidate is the non-merit passer, and state the final probability.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2016-january-WST01-q3f", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Choose whether increased advertising should be spent on textbook or novel sales using regression gradients.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson calls part (f) quite challenging. Many candidates did not make the requested textbook-versus-novel comparison and instead suggested advertising more generally.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the estimated change in sales per pound for the textbook and novel regression lines, then state which type of book would gain greater overall sales and justify the choice.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2016-january-WST01-q4c", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability from a repeated-test tree diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=9", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that some candidates missed the conditional-probability requirement and gave 0.85, the unconditional probability, instead of conditioning on the certificate event.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Restrict the outcomes to students who were awarded a certificate, identify the paths for passing on the first or second attempt, and divide their combined probability by the probability of receiving a certificate.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2016-january-WST01-q5b", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of exceeding a medal threshold given that a hive has already exceeded a certificate threshold.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=10", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says the conditional probability caused difficulty. Some candidates used the wrong denominator or multiplied the two tail probabilities rather than taking their ratio.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the upper-tail probabilities beyond 50 kg and 39 kg, then divide the medal-tail probability by the certificate-tail probability to form the conditional probability.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2016-january-WST01-q5c", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the mean and standard deviation of a normal distribution from two cumulative probabilities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=10", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson notes that candidates who used the wrong sign for the lower-tail z-value obtained an impossible negative mean. Both parameters depended on correctly setting up and solving the standardized equations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert each cumulative probability to its z-score, form the two standardization equations for the stated values, solve simultaneously for μ and σ, and round both to two decimal places.", "sourceEvidencePages": { "examinerReport": [ 5, 6 ] } }, { "id": "ial-legacy-2016-january-WST01-q6a", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate the mean and standard deviation from the summary of ten rabbit weights.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson reports that almost all candidates found the mean but the standard deviation caught many out. Some used the supplied sum of squared deviations directly as the standard deviation or tried to reconstruct the raw sum incorrectly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide the total weight by 10 for the mean, then use the supplied sum of squared deviations divided by n to find the variance and take its square root for the standard deviation.", "sourceEvidencePages": { "examinerReport": [ 4, 6 ] } }, { "id": "ial-legacy-2016-january-WST01-q6d", "paperId": "ial-legacy-2016-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2016, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Determine how adding two weights affects the standard deviation, using the change in the sum of squared deviations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20160127.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20160302.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson says many candidates left this part blank or stated an increase/decrease without a reason. Correct explanations based on variance or a recalculated squared-deviation total were rare.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20160303.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the new sum of squared deviations about the updated mean with the original value, then state whether the standard deviation increases or decreases and explain why.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "cie-2011-w-11-q09ii", "paperId": "cie-2011-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two values of k for which the line is tangent to the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this part challenging or did not attempt it; it required simplifying to a three-term quadratic and applying the equal-roots condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the line and curve, form a quadratic in x, and set its discriminant to zero before solving for k." }, { "id": "cie-2011-w-11-q11iv", "paperId": "cie-2011-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(iv)", "marks": 4, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch f, g and the inverse of f on one diagram, showing their relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates sketched the half-parabola correctly or reflected it correctly in y = g to obtain the inverse.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the restricted half-parabola for f, draw g, and reflect f in the line y = g(x) to obtain the inverse; mark the matching domains." }, { "id": "cie-2011-w-11-q10iii", "paperId": "cie-2011-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 5, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the volume when the shaded region is rotated through 360 degrees about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates rotated the region about the x-axis instead of the stated y-axis.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express x as a function of y and integrate the circular cross-sectional area over the region's y-limits." }, { "id": "cie-2011-w-11-q08ii", "paperId": "cie-2011-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that no real value of p makes OA and OB perpendicular.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says relatively few candidates were successful with this part; errors in the scalar product also blocked progress.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the dot product from part (i), set it to zero, and show the resulting quadratic in p has no real roots." }, { "id": "cie-2011-w-12-q03ii", "paperId": "cie-2011-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector of the required magnitude in the direction of the given vector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few attempts were correct; finding a unit vector and then multiplying it by 28 challenged many candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the unit vector in the stated direction, then multiply it by 28, keeping the direction signs consistent." }, { "id": "cie-2011-w-12-q01ii", "paperId": "cie-2011-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the expression from part (i) to find the coefficient after substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says fewer than half recognised the connection to part (i); many expanded a different polynomial and missed the contribution of two terms.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute y = 2x − x² into the expansion from part (i), then collect both contributions to the required power." }, { "id": "cie-2011-w-12-q08ii", "paperId": "cie-2011-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume of the shaded region when it is rotated about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many integrated y or πy instead of πy², and few identified the correct integration limits.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate πy² with respect to x and use the x-values at which 8x − x² is zero as the limits." }, { "id": "cie-2011-w-12-q05ii", "paperId": "cie-2011-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the other value of θ satisfying the equation in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says very few candidates recognised the second solution by symmetry and many did not use the relationship between x and θ + 70 correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the symmetry of the trigonometric graph to find the second root, then apply the stated shift between x and θ." }, { "id": "cie-2011-w-12-q05iii", "paperId": "cie-2011-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "State the range of x for which the trigonometric inequality is satisfied.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says few candidates used the sketch graphs to obtain the correct range.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the two sketches from part (i) to identify where the required curve lies on the correct side." }, { "id": "cie-2011-w-13-q07ii", "paperId": "cie-2011-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the points of contact of the two tangents to the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the question was very testing and fully correct solutions were rare; even candidates with both m-values often did not find the contact points.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the discriminant condition to find both values of m, then substitute each into the line and curve to find the contact points." }, { "id": "cie-2011-w-13-q09iii", "paperId": "cie-2011-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(iii)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the compound-function inequality using the stated domains.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says a correct final solution was rare because candidates often missed a term in g(f(x)) or ignored the stated domains.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form g(f(x)), solve the resulting quadratic inequality, and intersect the solution set with the domains of both functions." }, { "id": "cie-2011-w-13-q10ii", "paperId": "cie-2011-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 7, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume generated by rotating the bounded region about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates misunderstood the axis and rotated the region about the x-axis.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use horizontal strips or an equivalent shell method with the y-axis as the axis of rotation and the matching limits." }, { "id": "cie-2011-w-21-q06ii", "paperId": "cie-2011-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "Rewrite the cubic equation in the required fixed-point form and find its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part challenged many candidates, who often could not rewrite the cubic into the required form.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange the cubic into the required iterative form, collecting the constant and reciprocal-power terms exactly." }, { "id": "cie-2011-w-21-q03", "paperId": "cie-2011-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find the x-coordinates where the curve y = ½ tan 2x has gradient 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says solving cos²2x = 1/4 was a challenge; many found only two values of 2x or only one value of x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate to obtain dy/dx = sec²2x, solve sec²2x = 4, and retain the x-values in the printed interval." }, { "id": "cie-2011-w-21-q07i", "paperId": "cie-2011-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for the parametric curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says differentiating the parametric expressions challenged many candidates, particularly the derivatives with respect to t.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t using the chain rule, then divide dy/dt by dx/dt." }, { "id": "cie-2011-w-21-q08ii", "paperId": "cie-2011-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the trigonometric identity proved earlier.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates who did not use the earlier identity often attempted an incorrect direct integration of cos 3x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the result from part (i) to rewrite the integrand as a sum of simple cosines, then integrate and apply the limits." }, { "id": "cie-2011-w-22-q04", "paperId": "cie-2011-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "exponentials", "integration", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find k from the equality involving the two exponential integrals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates answered correctly; common errors were substituting limits before integrating and failing to recognise a quadratic in e^(2k).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate both sides, apply the limits, reduce the equation to a quadratic in e^(2k), and solve for k." }, { "id": "cie-2011-w-22-q03ii", "paperId": "cie-2011-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use the trapezium rule with four ordinates to estimate the integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates ignored the trapezium rule, used five ordinates or selected the wrong number of intervals.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the specified four ordinates and common interval width in the trapezium rule, then round as requested." }, { "id": "cie-2011-w-22-q06i", "paperId": "cie-2011-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for the curve given in terms of θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates were challenged by differentiating both parametric expressions and commonly used an incorrect derivative for x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y separately with respect to θ, then form (dy/dθ)/(dx/dθ) and simplify." }, { "id": "cie-2011-w-22-q08iii", "paperId": "cie-2011-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 1, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the least value of 15 cos θ − 9 sin θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates noted that the least value is −3R.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the expression as R cos(θ − α), where R = √(15² + 9²) = √306, then use the least possible cosine value." }, { "id": "cie-2011-w-23-q04ii", "paperId": "cie-2011-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the identity from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates struggled to obtain a suitable integrand for the second part after proving the double-angle identity.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite cos²x using (1 + cos 2x)/2, integrate the resulting trigonometric terms, and apply the limits 0 to π/6." }, { "id": "cie-2011-w-23-q05", "paperId": "cie-2011-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find all solutions of the trigonometric equation in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often found only some of the four solutions and some used an incorrect identity.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use sec²2θ = 1 + tan²2θ to solve for tan 2θ, then list every valid angle in the full interval." }, { "id": "cie-2011-w-23-q07i", "paperId": "cie-2011-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate the product involving x and an exponential term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates were challenged by this part and often omitted the product rule or lost a factor of one-half.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the product and chain rules to both factors, retaining the one-half factor from differentiating the exponential." }, { "id": "cie-2011-w-23-q03", "paperId": "cie-2011-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation, checking for extraneous solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates omitted logarithm laws and some retained the false solution x = −1.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms using their laws, solve the resulting quadratic, and reject values outside the logarithm domains." }, { "id": "cie-2011-w-31-q04", "paperId": "cie-2011-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Solve the differential equation for x in terms of θ using the given condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates were challenged after separating variables; integrating the trigonometric fraction and removing logarithms were common difficulties.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate cos 2θ/sin²2θ as a logarithmic derivative, apply the condition, and solve for x." }, { "id": "cie-2011-w-31-q07ii", "paperId": "cie-2011-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find λ by equating the two cosines involving point P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often made sign or copying errors in these expressions; the question required equating the two cosines.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the dot-product expressions for cos AOP and cos BOP, equate them, and solve for λ." }, { "id": "cie-2011-w-31-q10b", "paperId": "cie-2011-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Find the greatest argument of a point in the shaded circular region on the Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few candidates found a correct method for the maximum argument; many relied on inexact measurements from a sketch.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the circle centred at 3i with radius 2, then use the tangent from the origin to determine the greatest argument." }, { "id": "cie-2011-w-31-q10a", "paperId": "cie-2011-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the two square roots of 1 − 2√6 i in exact x + iy form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says sign errors and miscopying were common, and some candidates lost the final mark when stating the roots.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set (x + iy)² = 1 − 2√6 i, equate real and imaginary parts, solve for x and y, and state both roots." }, { "id": "cie-2011-w-31-q05i", "paperId": "cie-2011-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Use the graph to establish that the equation has a root in the required interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls the graphical work more challenging; adequate parabola sketches were uncommon and many did not identify the intersection as a root.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch y = sec x and the given parabola accurately enough to show their intersection inside the specified interval." }, { "id": "cie-2011-w-31-q06ii", "paperId": "cie-2011-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find both solutions for θ after replacing x by 2θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says most candidates who found one solution missed the second solution in the interval.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the transformed cosine equation in the stated θ interval, including the solution arising from the negative angle for inverse cosine." }, { "id": "cie-2011-w-32-q04", "paperId": "cie-2011-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Solve the differential equation for x in terms of θ using the given condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates were challenged after separating variables; integrating the trigonometric fraction and removing logarithms were common difficulties.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate cos 2θ/sin²2θ as a logarithmic derivative, apply the condition, and solve for x." }, { "id": "cie-2011-w-32-q07ii", "paperId": "cie-2011-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find λ by equating the two cosines involving point P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often made sign or copying errors in these expressions; the question required equating the two cosines.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the dot-product expressions for cos AOP and cos BOP, equate them, and solve for λ." }, { "id": "cie-2011-w-32-q10b", "paperId": "cie-2011-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Find the greatest argument of a point in the shaded circular region on the Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few candidates found a correct method for the maximum argument; many relied on inexact measurements from a sketch.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch the circle centred at 3i with radius 2, then use the tangent from the origin to determine the greatest argument." }, { "id": "cie-2011-w-32-q10a", "paperId": "cie-2011-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the two square roots of 1 − 2√6 i in exact x + iy form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says sign errors and miscopying were common, and some candidates lost the final mark when stating the roots.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set (x + iy)² = 1 − 2√6 i, equate real and imaginary parts, solve for x and y, and state both roots." }, { "id": "cie-2011-w-32-q05i", "paperId": "cie-2011-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Use the graph to establish that the equation has a root in the required interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls the graphical work more challenging; adequate parabola sketches were uncommon and many did not identify the intersection as a root.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch y = sec x and the given parabola accurately enough to show their intersection inside the specified interval." }, { "id": "cie-2011-w-32-q06ii", "paperId": "cie-2011-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find both solutions for θ after replacing x by 2θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says most candidates who found one solution missed the second solution in the interval.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the transformed cosine equation in the stated θ interval, including the solution arising from the negative angle for inverse cosine." }, { "id": "cie-2011-w-33-q04i", "paperId": "cie-2011-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find N(t) from the differential equation and initial condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says relatively few candidates could turn the correct expression for N² into the required expression for N.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate N⁻¹/² dN = 1.2e⁻⁰·⁰²ᵗ dt, apply N(0) = 100, then square the resulting expression for √N." }, { "id": "cie-2011-w-33-q06ii", "paperId": "cie-2011-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the equation of the circle whose diameter endpoints represent w and w².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a minority identified the correct circle and many missed that the two complex points were the ends of a diameter.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use w = −1 + i and its square as the diameter endpoints; find the midpoint and radius, then write the circle equation." }, { "id": "cie-2011-w-33-q09iii", "paperId": "cie-2011-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(iii)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the values of a for which the distance between the parallel line and plane is 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates did not complete the distance argument after showing the line was parallel to the plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the point-to-plane distance formula for a point on the line, account for the absolute value, and solve both resulting cases for a." }, { "id": "cie-2011-w-33-q10i", "paperId": "cie-2011-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(i)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the integral after making the stated trigonometric substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates had difficulty simplifying the substituted integral and did not see the necessary factorisation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the sec²x derivative, factor the substituted numerator with the tangent identity, and integrate the simplified form." }, { "id": "cie-2011-w-41-q02", "paperId": "cie-2011-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension and the magnitude of the resultant force in the system.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates made progress with this question.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the force diagram, resolve forces in the relevant directions, and combine the components to find the requested resultant." }, { "id": "cie-2011-w-41-q07iv", "paperId": "cie-2011-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find when the particle returns to O after moving to A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says most candidates were challenged to develop a strategy for this final part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity to obtain displacement from O, then solve for the positive time after the particle has turned around when the displacement is zero." }, { "id": "cie-2011-w-41-q06ii", "paperId": "cie-2011-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Verify the car's speed at C for the motion from B to C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes parts (ii) and (iii) as more difficult problem-solving exercises.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a work–energy balance with the driving work, resistance, change in potential energy and the initial kinetic energy to obtain the speed at C." }, { "id": "cie-2011-w-41-q03i", "paperId": "cie-2011-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the resultant-force components parallel and perpendicular to AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says fully correct answers to the component parts were uncommon.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve all three forces parallel and perpendicular to AB, keeping component signs consistent with their directions, then sum each set of components." }, { "id": "cie-2011-w-42-q07iib", "paperId": "cie-2011-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)(b)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Express the approximate velocity v₁ in terms of t and show the stated expression for v₁ − v.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says parts (ii)(b) and (ii)(c) were more difficult than the first three parts.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the straight-line formula for v₁ on 0 ≤ t ≤ 400 and subtract the given quadratic velocity, preserving the completed-square form." }, { "id": "cie-2011-w-42-q06iii", "paperId": "cie-2011-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance BC using the speed at D and the different resistances on BC and CD.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates scored all four marks and often assumed a constant resistance equal to the sum of two values.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply a work–energy balance from B to D, split the resistance work across BC and CD, then use the known total distance BD to solve for BC." }, { "id": "cie-2011-w-42-q05ii", "paperId": "cie-2011-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the initial height of the particles above the floor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates ignored the motion of B while the string was slack or used the wrong acceleration.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the motion before A hits the floor to find the speed and fall time, then use B's continued upward motion for 0.3 seconds to recover the starting height." }, { "id": "cie-2011-w-42-q06i", "paperId": "cie-2011-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance AB from the lorry's work and energy changes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many omitted potential-energy change and used an approximation instead of the given value of sin θ.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use work-energy with the lorry's driving work, resistance, kinetic-energy change and potential-energy gain; use sin θ = 1/20." }, { "id": "cie-2011-w-43-q03", "paperId": "cie-2011-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total distance Q travels from X until it returns to X after P hits the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this one of the most challenging questions and describes confusion about the required distance and acceleration after P hit the ground.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the speed of Q at X, then use its upward motion under gravity to calculate the maximum rise and double that height." }, { "id": "cie-2011-w-43-q06", "paperId": "cie-2011-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find both tensions for which the ring is in limiting equilibrium on the rough vertical rod.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was challenging; candidates often assumed the normal reaction equalled the weight and considered only one friction direction.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the tension and weight horizontally and vertically; consider friction acting in both possible directions at limiting equilibrium." }, { "id": "cie-2011-w-43-q07iii", "paperId": "cie-2011-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the distance travelled while engine power is constant and acceleration varies.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates were expected to use work-energy and commonly assumed constant acceleration or misused power divided by speed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a work-energy balance with engine work, change in kinetic energy and resistance; do not assume constant acceleration or fixed force." }, { "id": "cie-2011-w-51-q06i", "paperId": "cie-2011-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the angular speed and kinetic energy in the first conical-pendulum configuration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question caused problems and few gained full marks; many used 0.9 m instead of the actual 0.7 m length.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct string length and geometry, resolve vertically for tension, then use radial acceleration to find angular speed and kinetic energy." }, { "id": "cie-2011-w-51-q04iii", "paperId": "cie-2011-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the least possible h from the moments of the composite solid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this question as challenging and gives a multi-body moment equation needed for this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the centre of mass of the combined cone and cylinder and solve the moment equation for the limiting value of h." }, { "id": "cie-2011-w-51-q06ii", "paperId": "cie-2011-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find tension and angular speed in the second conical-pendulum configuration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part caused problems and required first recovering the string length from a right-angled triangle.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the geometry to find the string length and circular radius, resolve vertically for tension, then apply the radial equation." }, { "id": "cie-2011-w-52-q06i", "paperId": "cie-2011-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the angular speed and kinetic energy in the first conical-pendulum configuration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question caused problems and few gained full marks; many used 0.9 m instead of the actual 0.7 m length.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct string length and geometry, resolve vertically for tension, then use radial acceleration to find angular speed and kinetic energy." }, { "id": "cie-2011-w-52-q04iii", "paperId": "cie-2011-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the least possible h from the moments of the composite solid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this question as challenging and gives a multi-body moment equation needed for this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the centre of mass of the combined cone and cylinder and solve the moment equation for the limiting value of h." }, { "id": "cie-2011-w-52-q06ii", "paperId": "cie-2011-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find tension and angular speed in the second conical-pendulum configuration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part caused problems and required first recovering the string length from a right-angled triangle.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the geometry to find the string length and circular radius, resolve vertically for tension, then apply the radial equation." }, { "id": "cie-2011-w-53-q06ii", "paperId": "cie-2011-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the area A from the moment equation for the composite object with a hole.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates could set up a correct moment equation for this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about O, include the hemisphere, cylinder and removed material with correct signs, and solve for A." }, { "id": "cie-2011-w-53-q06i", "paperId": "cie-2011-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of the composite hemisphere and cylinder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Question 6 the most challenging and gives the three-term moment equation required in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about O for the hemisphere, cylinder and combined body, then solve for the distance of the combined centre of mass." }, { "id": "cie-2011-w-53-q06iii", "paperId": "cie-2011-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Find how much further the cylindrical hole can be extended while keeping the centre of mass at O.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part required extending the hole into the hemisphere, a geometric step candidates needed to identify.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the symmetry and the marked 0.48 m hole length to calculate the permitted increase, 2(0.6 − 0.48) = 0.24 m." }, { "id": "cie-2011-w-53-q05iii", "paperId": "cie-2011-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the tension in the part of the string attached to O, given that P and R have equal kinetic energies.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates completed this part; the solution links the kinetic-energy condition to the final radial-force equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the equal kinetic energies and known radii to find P's mass, then apply the radial equation to P using the known angular speed and the tension between P and Q." }, { "id": "cie-2011-w-61-q07ii", "paperId": "cie-2011-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Given that exactly two selected balls have the same number, find the probability that both are numbered 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the question required reversing the conditional-probability statement and applying the conditional-probability formula.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the probability of both balls being numbered 2, then divide by the probability that exactly two of the three selected balls share a number." }, { "id": "cie-2011-w-61-q06bii", "paperId": "cie-2011-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Arrange the four groups of three personal possessions, keeping each owner’s group together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates could identify the separate groupings but struggled to combine them into the required count.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat each owner’s three distinguishable possessions as one block, arrange the four blocks, then count the 3! internal arrangements in each block." }, { "id": "cie-2011-w-61-q07iii", "paperId": "cie-2011-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether the two events are independent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the independence definition required all relevant probabilities to be stated and substituted; some candidates confused it with other event relationships.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State the relevant probabilities and test P(X ∩ Y) = P(X)P(Y), or use the equivalent conditional-probability definition." }, { "id": "cie-2011-w-62-q06ii", "paperId": "cie-2011-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability where the condition is implicit in the wording.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was taxing because candidates had to recognise a conditional-probability question without being told explicitly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the restricted sample space, calculate the joint and conditioning probabilities, and divide." }, { "id": "cie-2011-w-62-q07ii", "paperId": "cie-2011-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the proportion of winter days on which the minimum temperature is below zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says standardising and cancelling a fraction proved too difficult for many candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardise the temperature using the given mean and standard deviation, then use the stated probability to solve for the proportion." }, { "id": "cie-2011-w-62-q07iii", "paperId": "cie-2011-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected number of days with a minimum temperature above three times the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates found the relevant probability but did not understand how to use the sample size of 70.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardise 3μ under the stated normal model to obtain P(Z > 1), then multiply that probability by 70 days." }, { "id": "cie-2011-w-62-q04iv", "paperId": "cie-2011-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iv)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a histogram from the grouped data after finding the class frequencies.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes widespread confusion between cumulative frequency and frequency density, with many incorrect histogram heights.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert cumulative frequencies to class frequencies, divide by class widths, then plot frequency density against the intervals." }, { "id": "cie-2011-w-63-q02i", "paperId": "cie-2011-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a randomly chosen teenager is female or watches ‘Kops are Kids’, or both.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates were challenged by confusion between P(F and W) and P(F or W), often using conditional probabilities instead of the required probabilities.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the two-way table to calculate P(F) + P(W) − P(F and W), with all probabilities taken from the full group of 30 teenagers." }, { "id": "cie-2011-w-63-q06i", "paperId": "cie-2011-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate the probability using the conditional proportion and the binomial distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says attempts at this conditional-probability and binomial part were generally less successful; many used 0.37 instead of the conditional proportion 37/83.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Restrict the sample to Rhesus-positive people to find the success probability, then sum the required binomial probabilities." }, { "id": "cie-2011-w-71-q05ii", "paperId": "cie-2011-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the hypothesis test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates generally found Type II error challenging, despite some successful solutions.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the alternative mean, calculate the probability of not rejecting the null hypothesis using the rejection region from part (i)." }, { "id": "cie-2011-w-71-q07ii", "paperId": "cie-2011-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use Figure 2 to find P(U < 0.5).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates correctly identified the line in the second figure and few used the more successful trapezium-area method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the area under the density curve from 0 to 0.5; in Figure 2 this is the area of a trapezium under the linear segment." }, { "id": "cie-2011-w-71-q07i", "paperId": "cie-2011-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Identify the variables with the largest median and largest standard deviation from the density graphs, explaining the latter choice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates gave a convincing argument for the variables and their relative likelihoods.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the cumulative area for medians, then compare the spread of each density and state that higher and lower values are more likely for the widest one." }, { "id": "cie-2011-w-71-q02", "paperId": "cie-2011-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the engineering student’s claim is justified using a normal approximation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many omitted the hypotheses or gave an unqualified conclusion without the comparison needed to justify it.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State both hypotheses, approximate B(100, 0.2) by N(20, 16), apply the continuity correction at 26.5, and compare z = 1.625 with the 5% one-tailed critical value 1.645." }, { "id": "cie-2011-w-72-q05ii", "paperId": "cie-2011-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the hypothesis test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates generally found Type II error challenging, despite some successful solutions.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the alternative mean, calculate the probability of not rejecting the null hypothesis using the rejection region from part (i)." }, { "id": "cie-2011-w-72-q07ii", "paperId": "cie-2011-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use Figure 2 to find P(U < 0.5).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates correctly identified the line in the second figure and few used the more successful trapezium-area method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the area under the density curve from 0 to 0.5; in Figure 2 this is the area of a trapezium under the linear segment." }, { "id": "cie-2011-w-72-q07i", "paperId": "cie-2011-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Identify the variables with the largest median and largest standard deviation from the density graphs, explaining the latter choice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates gave a convincing argument for the variables and their relative likelihoods.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the cumulative area for medians, then compare the spread of each density and state that higher and lower values are more likely for the widest one." }, { "id": "cie-2011-w-72-q02", "paperId": "cie-2011-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the engineering student’s claim is justified using a normal approximation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many omitted the hypotheses or gave an unqualified conclusion without the comparison needed to justify it.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State both hypotheses, approximate B(100, 0.2) by N(20, 16), apply the continuity correction at 26.5, and compare z = 1.625 with the 5% one-tailed critical value 1.645." }, { "id": "cie-2011-w-73-q03i", "paperId": "cie-2011-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe how to select a random sample using random-number tables.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates needed to explain numbering, forming three-digit values from random digits, and discarding invalid or repeated numbers.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Number the population, read consecutive three-digit groups from a random-number table, and discard out-of-range or repeated numbers." }, { "id": "cie-2011-w-73-q03iii", "paperId": "cie-2011-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(iii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the population to which the sample conclusion refers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many mistook the population for the club members or the sample instead of all cafe-spending amounts.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the measured variable and its values for all club members as the population." }, { "id": "cie-2011-w-73-q05ii", "paperId": "cie-2011-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Describe a Type I error in the context of the driving-distance test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates needed a clear contextual statement and wording that identifies the incorrect conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the test concludes the mean driving distance decreased when, in fact, it has not decreased." }, { "id": "cie-2011-w-73-q02", "paperId": "cie-2011-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the sample size from the stated confidence-interval width for a proportion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w11_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes errors in the sampling-distribution variance, square root, and confidence-interval width when solving for n.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the standard error √(0.35 × 0.65/n), the correct critical z-value, and solve the interval-width equation for n." }, { "id": "cie-2011-s-11-q02", "paperId": "cie-2011-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Related rates for the volume of a spherical balloon", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The June 2011 examiner report says candidates had the least success with Q2; many missed differentiation and the chain rule or confused the variables v, r and t.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate V = 4πr³/3 with respect to time, then substitute the fixed volume rate and r = 10 to find dr/dt." }, { "id": "cie-2011-s-11-q07ii", "paperId": "cie-2011-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Solve a gradient inequality using the critical values", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q7(ii), the report says relatively few candidates applied the correct inequalities to the critical values after finding them.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find where the gradient equals 1/3, then test the intervals split by those critical values and write the strict inequality solution." }, { "id": "cie-2011-s-11-q08ii", "paperId": "cie-2011-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model repeated annual donations with a percentage increase", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q8(ii), the report identifies a significant error: using r = 0.1 instead of the 1.1 multiplier.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the geometric progression multiplier 1.05 for a 5% increase, then apply the relevant term or sum formula for Model 2." }, { "id": "cie-2011-s-11-q10iii", "paperId": "cie-2011-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a line through a point and the midpoint of two coordinates", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q10(iii), the report says inaccurate midpoint work often cost candidates the final three marks.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the midpoint of AP accurately, then use it with Q to calculate the gradient and line equation." }, { "id": "cie-2011-s-12-q03i", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the roots of a quadratic to identify its coefficients", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes Q3 as poorly answered and says most candidates did not use the given roots directly to form the factors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the quadratic as a product of factors from roots −3 and 5, expand, and compare coefficients." }, { "id": "cie-2011-s-12-q03ii", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Choose a parameter so a quadratic has equal roots", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says Q3 was poorly answered and notes that many candidates confused the roots condition with the discriminant.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the known p and q, then set the discriminant to zero and solve for r." }, { "id": "cie-2011-s-12-q08ii", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Find a position vector in a parallelogram", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q8(ii), only a small number of solutions were correct; candidates often missed OD = OA + AD and AD = BC.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the parallelogram relation AD = BC, then calculate OD = OA + AD. A quick sketch helps keep the vector directions clear." }, { "id": "cie-2011-s-12-q09ia", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)(a)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Find the range of a function involving cos²x", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q9(i)(a), correct solutions were rare because candidates often used the range of cos x for cos²x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use 0 ≤ cos²x ≤ 1, substitute the endpoint values into the function, and state the full range." }, { "id": "cie-2011-s-12-q09ib", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation exactly over the stated interval", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "For Q9(i)(b), the report notes that many missed the exact-answer requirement or omitted the ± solutions when solving for cos x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange to an equation for cos x, retain both positive and negative cosine solutions, and give exact angles in the interval." }, { "id": "cie-2011-s-12-q10a", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the perimeter of the smallest sector in an arithmetic sequence of angles", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Q10 was poorly answered. For the sector part, only a small minority used the 360° total; correct perimeters were rare, often because degrees were used in s = rθ.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic progression and the 360° total to find the smallest angle, convert it to radians for arc length, then add two radii." }, { "id": "cie-2011-s-12-q10bi", "paperId": "cie-2011-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(i)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the first term parameter of a geometric progression", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "For the geometric-progression part of Q10, the report says many did not use equal successive ratios and that failing to cancel a common factor often produced an unusable cubic.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the square of the middle term equal to the product of the first and third terms, solve for k, and retain the value consistent with positive terms." }, { "id": "cie-2011-s-13-q08i", "paperId": "cie-2011-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity with squared functions", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q8(i), the report highlights algebra and factorisation errors, especially sign errors, across the methods candidates tried.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose a side to transform, use tanθ = sinθ/cosθ and sin²θ = 1 − cos²θ, then factor carefully and keep signs consistent." }, { "id": "cie-2011-s-13-q09iv", "paperId": "cie-2011-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2011, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(iv)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Relate the normal angle to the tangent gradient", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q9(iv), only a small number of candidates recognised how the normal gradient determines k.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the tangent gradient at P, take the negative reciprocal for the normal, and match it to tan(tan⁻¹k)." }, { "id": "cie-2011-s-21-q02", "paperId": "cie-2011-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find an exact gradient from parametric equations", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q2, the report says most candidates had the right approach but were thwarted by incorrect t-derivatives.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t, form dy/dx = (dy/dt)/(dx/dt), and substitute t = π/6 exactly." }, { "id": "cie-2011-s-21-q03", "paperId": "cie-2011-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Recover K and m from a straight-line log graph", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q3, a majority found the model problem intractable; the report identifies logarithmic linearisation and simultaneous equations as the productive approach.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain ln y = ln K + m ln x; use the two plotted points to find the gradient m and intercept ln K." }, { "id": "cie-2011-s-21-q06a", "paperId": "cie-2011-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Integrate an exponential expression exactly", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q6(a), the report says only strong candidates coped well with the integration; weaker candidates often did not attempt it or could not expand the integrand.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand or simplify the integrand before integrating, then apply the limits with consistent algebra." }, { "id": "cie-2011-s-21-q08iii", "paperId": "cie-2011-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iii)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find extreme values of a squared trigonometric expression", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q8(iii), the report calls the last part difficult and says candidates usually failed to connect it to the amplitude form from the earlier parts.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the earlier amplitude form A sin(θ + α) to identify the range of the bracket, then square that range and add the constant." }, { "id": "cie-2011-s-22-q05", "paperId": "cie-2011-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find a tangent to an implicitly defined curve", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q5, many candidates did no differentiation or missed the chain rule; others found the gradient at the point incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly using the chain rule on y², evaluate dy/dx at (2, −1), then use the point-gradient equation." }, { "id": "cie-2011-s-22-q08i", "paperId": "cie-2011-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove an identity involving cosecant and secant", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q8(i), many candidates reached an intermediate identity but did not use sin²2θ = 4sin²θcos²θ.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use reciprocal and Pythagorean identities, then apply sin²2θ = 4sin²θcos²θ to reach the required expression." }, { "id": "cie-2011-s-23-q05", "paperId": "cie-2011-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find a tangent to an implicitly defined curve", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q5, many candidates did no differentiation or missed the chain rule; others found the gradient at the point incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly using the chain rule on y², evaluate dy/dx at (2, −1), then use the point-gradient equation." }, { "id": "cie-2011-s-23-q08i", "paperId": "cie-2011-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2011, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove an identity involving cosecant and secant", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q8(i), many candidates reached an intermediate identity but did not use sin²2θ = 4sin²θcos²θ.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use reciprocal and Pythagorean identities, then apply sin²2θ = 4sin²θcos²θ to reach the required expression." }, { "id": "cie-2011-s-31-q03i", "paperId": "cie-2011-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find an equation of a plane from a normal vector", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q3(i), the report says many candidates failed to use AB as the plane normal and B as a point in the plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use AB as the normal to the plane and point B as a known point, then substitute into the plane equation." }, { "id": "cie-2011-s-31-q03ii", "paperId": "cie-2011-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the acute angle between a plane and the y-axis", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q3(ii), few candidates gave a correct vector for the y-axis and many made scalar-product errors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Represent the y-axis with its direction vector, then use the angle between that line and the plane normal, applying the complementary angle as needed." }, { "id": "cie-2011-s-31-q04ii", "paperId": "cie-2011-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "Use a transformed cubic equation to find a numerical root", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q4(ii), few candidates spotted the connection to the roots of f(x); consequently, the required positive root was rarely chosen.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the transformed equation to the roots of f(x), identify the positive root required for 3ʸ, then solve for y at the requested accuracy." }, { "id": "cie-2011-s-31-q07ii", "paperId": "cie-2011-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate an exact integral after substitution", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q7(ii), candidates often knew integration by parts was required but struggled to identify which terms to differentiate and integrate, with sign errors common.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stated substitution and transformed limits, then integrate by parts with the correct choices and simplify signs carefully." }, { "id": "cie-2011-s-31-q08ii", "paperId": "cie-2011-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the minimum distance from the origin to a locus line", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "For Q8(ii), few candidates produced the sketch needed to see that the closest point on the locus gives the required distance.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot the line specified by the argument condition from the point found earlier, then use the perpendicular distance from the origin to the line." }, { "id": "cie-2011-s-32-q05i", "paperId": "cie-2011-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate a parametric curve to find dy/dx", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Q5(i) discriminated well and that applying the chain rule in both derivatives left scope for mistakes in the division and simplification.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule to both x(t) and y(t), then divide the derivatives and simplify." }, { "id": "cie-2011-s-32-q07aii", "paperId": "cie-2011-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a conjugate argument to determine a parameter", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q7(a)(ii), forming the equation for a from the conjugate proved challenging; candidates often equated the ratio to the wrong angle expression.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the conjugate from part (i), express its argument with the correct quadrant, and equate its tangent to the given value to solve for a." }, { "id": "cie-2011-s-32-q07b", "paperId": "cie-2011-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Interpret a region on an Argand diagram", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q7(b), the report warns that unequal axis scales caused serious errors when drawing the required Argand diagram.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the loci with equal scales on both axes, mark their intersection or overlap, and select the region satisfying all conditions." }, { "id": "cie-2011-s-32-q10iii", "paperId": "cie-2011-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 2, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find where a tangent to y = x²e⁻ˣ passes through the origin", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q10(iii), only a minority had a sound method such as equating the gradient with y/x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate y = x²e⁻ˣ, evaluate the gradient at P, and equate it to the slope y/x of OP before solving for x." }, { "id": "cie-2011-s-33-q04ii", "paperId": "cie-2011-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a transformed trigonometric equation over an interval", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q4(ii), many candidates struggled to obtain and solve the resulting simple equation in tan²θ.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the result from part (i) to solve for tan²θ, then find every θ in the stated interval and check the endpoints and quadrants." }, { "id": "cie-2011-s-33-q07ii", "paperId": "cie-2011-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State the moduli and arguments of complex roots", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q7(ii), many candidates did not state both moduli clearly or selected arguments without checking the roots’ positions on the Argand diagram.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert each root to modulus-argument form and choose arguments from the actual quadrants on the Argand diagram." }, { "id": "cie-2011-s-33-q07iii", "paperId": "cie-2011-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Verify a power identity using the roots from part (ii)", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q7(iii), very few candidates used the information from part (ii); most made little progress by multiplying out the factors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the modulus and argument information from part (ii) to raise each root to the sixth power, showing the angle and sign." }, { "id": "cie-2011-s-33-q08i", "paperId": "cie-2011-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Locate the maximum of a product of trigonometric functions", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q8(i), candidates often missed the chain rule in differentiating both trigonometric factors and then struggled to find the correct solutions.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product and chain rules, set the derivative to zero, solve the resulting trigonometric equation, and select the maximum in the interval." }, { "id": "cie-2011-s-33-q08ii", "paperId": "cie-2011-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2011, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find an area using u = cos x", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "For Q8(ii), candidates often became confused when substituting dx and expressing cos²x in terms of u; many omitted essential limit-substitution work.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = cos x, replace dx and the remaining trigonometric terms consistently, transform the limits, and evaluate the integral." }, { "id": "cie-2011-s-41-q03", "paperId": "cie-2011-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on a ring in equilibrium with two strings", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Q3 proved difficult for most candidates; many knew which directions to resolve but could not form the equations correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw a clear force diagram and resolve each tension and weight into horizontal and vertical components before solving the equilibrium equations." }, { "id": "cie-2011-s-41-q04i", "paperId": "cie-2011-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces and friction on an inclined plane", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q4(i), the report says the major difficulty was obtaining the correct equations for forces parallel to the inclined plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the force diagram for this case, resolve parallel and perpendicular to the plane, and use limiting friction with the normal reaction." }, { "id": "cie-2011-s-41-q06i", "paperId": "cie-2011-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Integrate a velocity model and solve for the travel time", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q6(i), the report says the quartic equation defeated the majority, although those who recognized it as quadratic in t² could solve it.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate v(t) from the initial to final time, set the distance equal to 64 m, solve the resulting equation in t², and keep the physically valid time." }, { "id": "cie-2011-s-41-q07iii", "paperId": "cie-2011-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when a slack string becomes taut again", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q7(iii), successful candidates changed to acceleration due to gravity after slackening and allowed for A to rise and fall; 0.4 s was a common wrong answer.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "After B hits the floor, model A under gravity alone; account for its upward motion and return downward before finding when the string becomes taut." }, { "id": "cie-2011-s-42-q04ii", "paperId": "cie-2011-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction of the resultant after removing the 12 N force", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q4(ii), very few candidates obtained both components of the resultant, despite the values being available from the previous part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the resultant of the remaining forces into horizontal and vertical components, use R² = X² + Y² for its magnitude and tanφ = Y/X for its direction, retaining the correct quadrant." }, { "id": "cie-2011-s-42-q05ii", "paperId": "cie-2011-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Relate the velocities of two particles at equal height conditions", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Q5(ii) the most difficult part of the paper and says a common error was using one velocity for both particles.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use separate velocity symbols for P and Q, apply the given height relation to each particle’s motion, and solve the resulting equations." }, { "id": "cie-2011-s-43-q03i", "paperId": "cie-2011-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find speeds for connected particles moving on different slopes", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q3(i), the report says a fully correct energy solution for P was rare; some candidates wrongly gave both particles acceleration g.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct acceleration for P along its slope and distinguish P’s and Q’s speeds when forming any energy equations." }, { "id": "cie-2011-s-43-q07i", "paperId": "cie-2011-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2011, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "integration", "kinematics" ], "primaryTopicId": "integration", "summary": "Use a velocity model and an endpoint condition to show initial speed is zero", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q7(i), many solutions omitted the constant of integration instead of using v = 0 at t = 20.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the acceleration or velocity relation, use v = 0 at t = 20 to determine the constant, then evaluate the speed at t = 0." }, { "id": "cie-2011-s-51-q02ii", "paperId": "cie-2011-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle of a suspended lamina from its centre-of-mass geometry", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q2(ii), the report says a solution was not often seen and points to using AG with the sine rule.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the centre of mass from part (i), find AG in triangle AOG, and apply the sine rule to obtain the required angle." }, { "id": "cie-2011-s-51-q07", "paperId": "cie-2011-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 12, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Model two particles moving in a rotating disc groove", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Q7 as the most difficult question on the paper and requires Newton’s second law for both particles.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law radially to both particles; first solve for tension and angular velocity, then relate their radii and speeds." }, { "id": "cie-2011-s-52-q02ii", "paperId": "cie-2011-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Locate the suspension point for a composite lamina", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q2(ii), a correct answer was very rarely seen; the report outlines using tan(ADO) and cos(ADO) to determine XD.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the centre of mass from part (i) and the geometry of triangle ADO to relate h and XD, then solve for XD." }, { "id": "cie-2011-s-52-q04i", "paperId": "cie-2011-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find speed when a particle on an elastic string has zero acceleration", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Q4 the paper’s most difficult question and says the five-term energy equation often had missing or incorrect terms.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the equilibrium extension first, then write the complete energy balance including kinetic, elastic and gravitational potential energy at both states." }, { "id": "cie-2011-s-52-q04ii", "paperId": "cie-2011-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use elastic and kinetic energy to establish the upward displacement", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Q4 is identified as the paper’s most difficult question; the report specifically says part (ii) uses initial elastic plus kinetic energy equalling final potential energy.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the initial elastic and kinetic energy equal to the final gravitational potential energy, noting that final kinetic energy is zero." }, { "id": "cie-2011-s-52-q06ii", "paperId": "cie-2011-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find horizontal range for a projectile launched below the horizontal", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For Q6(ii), sign errors were common and some candidates used the same-level flight-time formula despite the different launch and landing heights.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a signed vertical displacement with the launch angle below horizontal, solve the vertical equation for flight time, then calculate horizontal distance." }, { "id": "cie-2011-s-53-q06ii", "paperId": "cie-2011-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 7, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Find where a particle stops and determine whether friction keeps it there", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q6(ii), the report says many candidates could not explain why the particle did not subsequently move and needed to compare the two forces.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables and integrate to find the stopping distance; then compare the force toward O with the available friction force." }, { "id": "cie-2011-s-53-q07i", "paperId": "cie-2011-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2011, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a prism with a quadrant removed", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Q7 as the most difficult question; many candidates made errors when taking moments after finding the quadrant’s centre of mass.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a composite-body moment equation about O, subtracting the removed quadrant’s contribution from the full prism." }, { "id": "cie-2011-s-61-q04iii", "paperId": "cie-2011-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements when one item is omitted", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q4(iii), few candidates realised that three different options had to be summed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the distinct possible omitted-item choices, count arrangements for each case, and add the totals." }, { "id": "cie-2011-s-61-q07aii", "paperId": "cie-2011-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the smallest number of dice needed to exceed a probability threshold", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q7(a)(ii), successful solutions were rare even though candidates who solved the index inequality generally selected the correct integer.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write P(at least one 3) = 1 − (5/6)^n, solve the strict inequality, and choose the smallest integer that satisfies it." }, { "id": "cie-2011-s-61-q07b", "paperId": "cie-2011-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the winner probability in a no-replacement turn-taking game", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q7(b), the report says candidates often missed the three successful outcomes and the no-replacement condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a tree or a sum of mutually exclusive paths, tracking alternating turns and the changing bag contents after each draw." }, { "id": "cie-2011-s-62-q03", "paperId": "cie-2011-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 7, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Update a coded mean and standard deviation after adding one value", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Q3 the most difficult question on the paper; candidates often added 29 without converting it to coded form, losing the new standard deviation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the coded sums to recover the original totals, add the value 29 consistently, and calculate the new standard deviation for 37 values." }, { "id": "cie-2011-s-62-q06ii", "paperId": "cie-2011-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a binomial probability from a normal-model straw length", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q6(ii), the report says this part required thought and discriminated between candidates who could reason through a problem and those who relied on routine examples.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use z = 1.282 to find the probability that one straw lies between 21.5 cm and 22.5 cm, then apply a binomial model with n = 4 for fewer than two." }, { "id": "cie-2011-s-63-q04ii", "paperId": "cie-2011-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Test pairs of events for independence", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q4(ii), most candidates struggled to apply the independence condition; only a small proportion calculated a joint event and used the required equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For each pair, calculate the intersection probability and compare it with the product of the individual probabilities." }, { "id": "cie-2011-s-63-q06", "paperId": "cie-2011-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2011, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 11, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Solve a multi-part binomial problem about repeated Sudoku results", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Q6 the most difficult question for many candidates and notes that the final part required using the binomial distribution twice.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use binomial probabilities for the number correct in one month, then use the resulting monthly probability in a second binomial model over five months." }, { "id": "cie-2011-s-71-q02b", "paperId": "cie-2011-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Recover a confidence level from a one-sided interval event", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q2(b), the report says the part was poorly answered and many candidates did not understand what was required.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use symmetry of the sampling distribution to translate the probability that the whole interval lies below the population mean into the corresponding normal-tail probability." }, { "id": "cie-2011-s-71-q06iv", "paperId": "cie-2011-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iv)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Estimate a 12-month injury count probability with a normal approximation", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q6(iv), the report says this question was not well attempted and highlights missing or incorrect continuity corrections when standardising.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "With mean 2.1 per month, use a 12-month Poisson mean and variance of 25.2; apply the correct continuity correction before standardising." }, { "id": "cie-2011-s-72-q03iii", "paperId": "cie-2011-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(iii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance of a goal bonus over ten future matches", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q3(iii), the report says a correct answer required two steps and proved challenging for many candidates.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the probability of at least one goal in one match, then raise that probability to the tenth power for success in every next match." }, { "id": "cie-2011-s-72-q06iii", "paperId": "cie-2011-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Explain a Type II error in a die-bias test", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For Q6(iii), the report says Type II errors caused problems and this contextual explanation was not well answered.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Describe failing to reject the fair-die assumption when the die is in fact biased, using the question’s context and outcome." }, { "id": "cie-2011-s-72-q07iv", "paperId": "cie-2011-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find a quartile from a continuous probability density", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For Q7(iv), the report says this part was more challenging; candidates often used wrong limits or failed to reject an invalid root.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the density using the correct interval limits, solve the resulting equation, and reject any root outside the support." }, { "id": "cie-2011-s-73-q02ii", "paperId": "cie-2011-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Explain why a confidence interval is approximate", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For Q2(ii), a significant number of candidates did not justify why the confidence interval was approximate.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Refer to the estimated variance or the normal approximation to a binomial distribution, as appropriate to the method used." }, { "id": "cie-2011-s-73-q06iii", "paperId": "cie-2011-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2011, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the chance that at least one of four brake pads needs early replacement", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s11_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls Q6(iii) more demanding and says a significant number did not recognise how to use the density to find the initial probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s11_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the density from 2 to 2.2 to find the replacement probability for one pad, then use 1 − P(none of four)." }, { "id": "ial-legacy-2017-january-WME01-q7e", "paperId": "ial-legacy-2017-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2017, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 2, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph for P as the string first pulls it upward, then goes slack, and later becomes taut again.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_que_20170123.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_rms_20170301.pdf#page=17", "qpPages": [ 24, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report says the final velocity-time sketch rarely earned both available marks. Common errors included drawing a speed-time graph, changing the gradient at the wrong point, and omitting a key velocity-axis label.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Show the initial increase in velocity while the string is taut, then a straight-line decrease under gravity after Q reaches the ground. Mark the instant P stops, continue into negative velocity, and label the key times and speeds requested.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2017-january-WME01-q3", "paperId": "ial-legacy-2017-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2017, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction of the resultant of two forces whose lines of action form an obtuse angle.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_que_20170123.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_rms_20170301.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Question 3 as more challenging. Candidates often used the wrong angle in the cosine rule or combined force components incorrectly, and some did not obtain the correct direction of the resultant.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve both forces along perpendicular directions, then combine the components and find the angle to P. Alternatively, use the cosine and sine rules with the correct included triangle angle; distinguish it from the 120-degree angle shown between the force lines.", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-legacy-2017-january-WME01-q2c", "paperId": "ial-legacy-2017-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2017, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the time when a moving particle's velocity is parallel to a specified vector.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_que_20170123.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_rms_20170301.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The examiner report says most candidates could not form the starting equation for this part. Common approaches confused displacement with velocity or substituted the force where acceleration was required.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the force and mass to obtain the acceleration, write the velocity components as functions of time, and set their ratio equal to the corresponding ratio in the given direction vector.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2017-january-WME01-q5b", "paperId": "ial-legacy-2017-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2017, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use momentum conservation to find both possible mass ratios after a direct collision.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_que_20170123.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_rms_20170301.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes this part as challenging because there are two possible post-collision directions for P. Most candidates considered only one case; others reached the second value without justifying it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up signed momentum equations for both possible directions of P after impact. Solve each case separately and retain both physically valid values of k.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2017-january-WME01-q4c", "paperId": "ial-legacy-2017-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2017, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Explain how the particle model is used for a package placed on a plank.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_que_20170123.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WME01_01_rms_20170301.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The examiner report says very few candidates gave a correct response. Many said that the weight acts at a point but did not identify the point; the mark scheme accepts the package's position C or E.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the package's weight is treated as acting at one point, its centre of mass, at the package's position on the plank.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2017-january-WST01-q6c", "paperId": "ial-legacy-2017-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2017, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance that an inspection shuts down the rice-filling machine after checking one or two bags.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WST01_01_que_20170120.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_MS%20Checked.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report says this part was beyond most candidates. Many found the probability in one tail but failed to include both tails, and others doubled the tail probability instead of squaring it for two bags.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the probability that one bag lies outside the allowed interval by adding the two equal tail probabilities, then square that probability for two independent bag selections.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2017-january-WST01-q7c", "paperId": "ial-legacy-2017-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2017, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Construct the cumulative distribution function of the square of a discrete random variable.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WST01_01_que_20170120.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_MS%20Checked.pdf#page=12", "qpPages": [ 26, 26 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls the final part more challenging and says only the strongest candidates handled it well. Few recognized that the cumulative probability for Y = 9 is obtained from the cumulative probability for X = 3; some incorrectly squared probabilities.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the possible squared values, then calculate each cumulative probability by summing the original probabilities up to the corresponding value of X. Label the cumulative probabilities clearly at each step.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2017-january-WST01-q4f", "paperId": "ial-legacy-2017-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2017, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(f)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that a player's first-round score is greater than the score in round two.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WST01_01_que_20170120.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_MS%20Checked.pdf#page=9", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The examiner report says this part defeated all but the most able candidates and had few correct solutions. A case-by-case enumeration of the valid score pairs was a successful approach.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "List the mutually exclusive score pairs with X greater than Y, multiply each pair's independent probabilities, and add the results.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2017-january-WST01-q1f", "paperId": "ial-legacy-2017-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2017, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(f)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Estimate the probability that two tomatoes selected from the sample are both within 0.75 g of the mean.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WST01_01_que_20170120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_MS%20Checked.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says very few candidates accounted for selection without replacement. Common incorrect approaches stopped after finding the one-tomato proportion or squared it as if the selections were independent.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the histogram to estimate how many weights fall between 5.5 g and 7 g. Since two different tomatoes are selected, use the conditional probability for the second selection after the first has qualified.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2017-january-WST01-q3c", "paperId": "ial-legacy-2017-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2017, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the least-squares regression equation for freezing temperature from water salinity.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WST01_01_que_20170120.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_MS%20Checked.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The examiner report says only a minority obtained both regression coefficients correctly to the requested precision. Premature rounding and sign errors were common after many candidates had made a sound start.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the gradient from Sxy/Sxx, use the sample means to find the intercept, and retain sufficient precision until both coefficients are rounded to three significant figures.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2017-january-WST01-q1d", "paperId": "ial-legacy-2017-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2017, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the estimated proportion above a weight threshold to compare the median with that threshold.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2017 Jan/WST01_01_que_20170120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_MS%20Checked.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this part was challenging and only stronger candidates gave a clear explanation. A common error was to misread the proportion above 6.25 g and infer the median was below the threshold.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20170301.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the proportion above 6.25 g with one half. If more than half the observations are above the threshold, the median is above it.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "cie-2010-w-11-q11iii", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(iii)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the values of k for which the line intersects the curve twice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists Question 11 among the more challenging questions and says this part required considerable care; only a minority completed it without error.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the line and curve, form a quadratic in x, and require a positive discriminant for two distinct intersections." }, { "id": "cie-2010-w-11-q07iii", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch the graph of the function over its stated domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates found this part particularly difficult and often failed to use the information from the first two parts when sketching.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the range and exact value from the earlier parts to locate the graph, then show the correct shape and endpoint behaviour on the given interval." }, { "id": "cie-2010-w-11-q04i", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the trigonometric identity by transforming one side into the other.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists part (i) among the more challenging questions and says relatively few candidates earned full marks; many treated the identity as an equation to solve.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Start with the left-hand side, replace tan x by sin x/cos x, use sin^2 x = 1 - cos^2 x, then factor 1 - cos^2 x." }, { "id": "cie-2010-w-11-q09iii", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(iii)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the area of the shaded region between the two touching circles and their common tangent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The general comments identify Question 9(iii) as one of the more challenging parts. The specific comments note errors in the tangent geometry and area calculation. The examiner report later refers to a Question 9(iv) accuracy hint, although the QP has only parts (i)-(iii); this is treated as a report typo, and no (iv) item is indexed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the circle radii and the tangent geometry to find the required sector and triangle areas, then combine them with the correct signs." }, { "id": "cie-2010-w-11-q01", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the expression after expanding the bracketed function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report lists Question 1 among the more challenging questions and says many candidates tried to integrate the expression as written or lost the product term when expanding.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand the bracket first, retain every product term, then integrate each resulting power of x." }, { "id": "cie-2010-w-11-q11ii", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(ii)", "marks": 4, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the bounded region about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report lists Question 11 among the more challenging questions and says candidates often forgot to square the curve or mishandled integration of a function of a linear function.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the disc method, integrate pi times the square of the curve, and account for the linear inner function when integrating." }, { "id": "cie-2010-w-11-q11i", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(i)", "marks": 3, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Differentiate the rational function and decide whether it has stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report lists Question 11 among the more challenging questions and says some candidates omitted the derivative of the inner linear function or failed to establish that no stationary point exists.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule to the inner linear function and check whether the derivative can equal zero on the curve's domain." }, { "id": "cie-2010-w-11-q07i", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "State the range of the tangent-defined function on its stated domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says part (i) was difficult and candidates needed to use the restricted input interval to see that the tangent term stays positive.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the interval for x to determine the range of the tangent input, then state the resulting strict bound for f(x)." }, { "id": "cie-2010-w-11-q07ii", "paperId": "cie-2010-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Evaluate the function exactly at the specified angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "The report says many candidates missed the instruction to give an exact value and answered with a decimal.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the exact angle into the function and use an exact trigonometric value; do not convert the result to a decimal." }, { "id": "cie-2010-w-12-q07iii", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find h(x) from the composition f = h g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The general comments single out Question 7(iii) as a part that caused most candidates difficulty; only a few candidates used the direct relationship from the earlier parts.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use h(g(x)) = f(x), substitute g(x) = x - 2, then express the result in terms of the input to h." }, { "id": "cie-2010-w-12-q09ii", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Express vector BG in the given unit-vector basis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The general comments identify Question 9(ii) as especially difficult; fewer than half of the solutions gave vector BG correctly, with the j and k components causing most difficulty.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate G using the square's scale factor and height, then subtract the position vector of B component by component." }, { "id": "cie-2010-w-12-q09i", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 2, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the height of the horizontal square is 4 cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The general comments identify Question 9(i) as especially difficult. The report says many candidates wrote down a value without a valid explanation; successful approaches used similar triangles or the isosceles geometry.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the similar cross-sections of the square pyramid to relate the side-length change to the vertical height." }, { "id": "cie-2010-w-12-q10i", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Express the tank height in terms of x and derive the total surface area formula.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The general comments identify Question 10(i) as one of the parts causing most candidates difficulty; many counted seven or eight faces instead of the six specified surfaces.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use volume = base area times height, then add the external base, four sides, and lid areas using their stated dimensions." }, { "id": "cie-2010-w-12-q07i", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Complete the square and state the range of the restricted quadratic function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says at least half of candidates failed to recognise the range from the completed-square form and restricted domain.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the quadratic in vertex form, use the domain x > 2 to identify its minimum, and state the range with the correct strictness." }, { "id": "cie-2010-w-12-q05b", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Show the stated inequality for a geometric progression whose partial sum is below 90% of its infinite sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this algebraic part caused more difficulty and many candidates omitted it; others struggled to simplify the resulting inequality.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the finite and infinite geometric-sum formulas, divide out their common factors, and rearrange to isolate r^n." }, { "id": "cie-2010-w-12-q06i", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the values of k for which the line and parabola have no common points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says only a minority solved the resulting inequality correctly after finding the discriminant boundary values.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the equations, form the quadratic intersection condition, and solve the discriminant inequality with the correct interval direction." }, { "id": "cie-2010-w-12-q03ii", "paperId": "cie-2010-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the snake's instantaneous growth rate at age five.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says about a third of candidates did not interpret the rate of growth as dx/dt at t = 5, and often substituted the age into the wrong quantity.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x with respect to t, then substitute t = 5 into dx/dt rather than into x." }, { "id": "cie-2010-w-13-q09bii", "paperId": "cie-2010-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find m when consecutive partial sums of the arithmetic progression are equal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls the algebra in this part quite difficult and notes that hardly any candidates used the consequence that equal consecutive partial sums make the next term zero.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate S_m and S_(m+1), simplify the resulting algebra, or observe that the next term must be zero and solve for m." }, { "id": "cie-2010-w-21-q08iii", "paperId": "cie-2010-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of x sin x using the result from part (ii).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most attempts used false integrals such as -x cos x; correct solutions depended on the derivative from part (ii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative found in part (ii) as an antiderivative identity and substitute the upper and lower limits." }, { "id": "cie-2010-w-21-q03", "paperId": "cie-2010-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Show the exact value of the definite integral involving (e^x + 1)^-2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the majority incorrectly treated integration of the expanded expression as if it remained a single squared exponential term.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as a sum of exponential terms, integrate each term, and apply the limits exactly." }, { "id": "cie-2010-w-21-q02", "paperId": "cie-2010-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation using logarithms to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says isolating x after taking logarithms proved beyond many candidates, who made serious algebraic errors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides, collect the terms involving x, and solve the resulting linear equation in x." }, { "id": "cie-2010-w-21-q05", "paperId": "cie-2010-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the cotangent and cosecant equation for every angle in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates used incorrect identities and failed to form a quadratic in cotangent or tangent.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use cosec^2(theta) = 1 + cot^2(theta) to form a quadratic in cot(theta), then find all angles in 0 to 360 degrees." }, { "id": "cie-2010-w-21-q08ii", "paperId": "cie-2010-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate sin x - x cos x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many candidates differentiated x cos x as a single term instead of applying the product rule.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate both terms, using the product rule on x cos x, then simplify the cancellation." }, { "id": "cie-2010-w-21-q08i", "paperId": "cie-2010-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show that the normal at Q passes through the specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates omitted a term from the product-rule derivative or failed to form the normal equation after finding its gradient.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x sin x with the product rule, evaluate the tangent gradient at Q, take its negative reciprocal, and verify the point lies on the normal." }, { "id": "cie-2010-w-22-q08iii", "paperId": "cie-2010-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of x sin x using the result from part (ii).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most attempts used false integrals such as -x cos x; correct solutions depended on the derivative from part (ii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative found in part (ii) as an antiderivative identity and substitute the upper and lower limits." }, { "id": "cie-2010-w-22-q03", "paperId": "cie-2010-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Show the exact value of the definite integral involving (e^x + 1)^-2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the majority incorrectly treated integration of the expanded expression as if it remained a single squared exponential term.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as a sum of exponential terms, integrate each term, and apply the limits exactly." }, { "id": "cie-2010-w-22-q02", "paperId": "cie-2010-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation using logarithms to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says isolating x after taking logarithms proved beyond many candidates, who made serious algebraic errors.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides, collect the terms involving x, and solve the resulting linear equation in x." }, { "id": "cie-2010-w-22-q05", "paperId": "cie-2010-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the cotangent and cosecant equation for every angle in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates used incorrect identities and failed to form a quadratic in cotangent or tangent.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use cosec^2(theta) = 1 + cot^2(theta) to form a quadratic in cot(theta), then find all angles in 0 to 360 degrees." }, { "id": "cie-2010-w-22-q08ii", "paperId": "cie-2010-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate sin x - x cos x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many candidates differentiated x cos x as a single term instead of applying the product rule.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate both terms, using the product rule on x cos x, then simplify the cancellation." }, { "id": "cie-2010-w-22-q08i", "paperId": "cie-2010-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show that the normal at Q passes through the specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates omitted a term from the product-rule derivative or failed to form the normal equation after finding its gradient.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x sin x with the product rule, evaluate the tangent gradient at Q, take its negative reciprocal, and verify the point lies on the normal." }, { "id": "cie-2010-w-31-q07iii", "paperId": "cie-2010-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the plane containing AB that is perpendicular to plane OAB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates identified the required normal direction and most gave the equation of plane OAB instead of the requested plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a normal direction perpendicular to plane OAB and containing AB, then form the plane equation through A and B." }, { "id": "cie-2010-w-31-q04i", "paperId": "cie-2010-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "functions", "numerical-methods" ], "primaryTopicId": "functions", "summary": "Use graph sketches to show the equation has exactly one root in the given interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says correct sketches were rare and many candidates drew a straight line for the quadratic or gave cot x the wrong concavity and endpoint behaviour.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch y = 4x^2 - 1 and y = cot x on the interval, using the correct curvature, intercepts, and asymptotic behaviour to establish one intersection." }, { "id": "cie-2010-w-31-q07ii", "paperId": "cie-2010-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the position vector of the point where OP is perpendicular to AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes frequent damaging algebraic slips when solving for the parameter in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write P as a point on AB, then set OP dot AB equal to zero and solve for the parameter." }, { "id": "cie-2010-w-31-q10iv", "paperId": "cie-2010-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iv)", "marks": 1, "topicIds": [ "functions", "exponentials" ], "primaryTopicId": "functions", "summary": "State the limiting mass as time becomes very large.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_31.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a few candidates answered this part correctly and often used equality wording instead of saying the mass tends to its limiting value.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the solved expression to take the limit as t tends to infinity and state the limiting value, not an equality at a finite time." }, { "id": "cie-2010-w-31-q10ii", "paperId": "cie-2010-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the differential equation for the mass formed as a function of time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_31.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Question 10 was among those done least well; the report identifies errors in separating and integrating, especially sign and logarithm/exponential manipulation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the variables, integrate both sides, apply x = 0 when t = 0, and rearrange for x." }, { "id": "cie-2010-w-32-q07iii", "paperId": "cie-2010-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the plane containing AB that is perpendicular to plane OAB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates identified the required normal direction and most gave the equation of plane OAB instead of the requested plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a normal direction perpendicular to plane OAB and containing AB, then form the plane equation through A and B." }, { "id": "cie-2010-w-32-q04i", "paperId": "cie-2010-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "functions", "numerical-methods" ], "primaryTopicId": "functions", "summary": "Use graph sketches to show the equation has exactly one root in the given interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says correct sketches were rare and many candidates drew a straight line for the quadratic or gave cot x the wrong concavity and endpoint behaviour.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch y = 4x^2 - 1 and y = cot x on the interval, using the correct curvature, intercepts, and asymptotic behaviour to establish one intersection." }, { "id": "cie-2010-w-32-q07ii", "paperId": "cie-2010-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the position vector of the point where OP is perpendicular to AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes frequent damaging algebraic slips when solving for the parameter in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write P as a point on AB, then set OP dot AB equal to zero and solve for the parameter." }, { "id": "cie-2010-w-32-q10iv", "paperId": "cie-2010-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iv)", "marks": 1, "topicIds": [ "functions", "exponentials" ], "primaryTopicId": "functions", "summary": "State the limiting mass as time becomes very large.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a few candidates answered this part correctly and often used equality wording instead of saying the mass tends to its limiting value.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the solved expression to take the limit as t tends to infinity and state the limiting value, not an equality at a finite time." }, { "id": "cie-2010-w-32-q10ii", "paperId": "cie-2010-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the differential equation for the mass formed as a function of time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Question 10 was among those done least well; the report identifies errors in separating and integrating, especially sign and logarithm/exponential manipulation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the variables, integrate both sides, apply x = 0 when t = 0, and rearrange for x." }, { "id": "cie-2010-w-33-q10iib", "paperId": "cie-2010-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)(b)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find all six roots of p(z^2) = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The general comments say the end of Question 10 challenged even the strongest candidates; the report says many misunderstood the substitution and only a minority completed the complex square-root work.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take the square roots of each root from part (a), including complex roots, and express the resulting six values fully." }, { "id": "cie-2010-w-33-q08iib", "paperId": "cie-2010-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)(b)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the smallest positive angle satisfying the equation with half-angle arguments.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says a correct answer was unusual; many candidates ignored the negative cosine value or failed to consider the alternative angle when solving for theta.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the R cos(theta - alpha) form, solve for the half-angle with the sign of the cosine taken into account, then choose the smallest positive theta in the valid range." }, { "id": "cie-2010-w-33-q10iia", "paperId": "cie-2010-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find all three roots of the cubic equation p(z) = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes confusion between factors and roots, and candidates often omitted the known root after solving the quadratic.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the known factor z + 2, divide the cubic, solve the remaining quadratic, and distinguish the roots of the equation from the factors." }, { "id": "cie-2010-w-41-q03", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use equilibrium of three forces at X to find an angle and an unknown weight.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the worst-attempted question: most candidates scored no marks, introduced acceleration despite equilibrium, or assumed unsupported 45-degree angles.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat the three string tensions meeting at X as concurrent forces in equilibrium and resolve horizontally and vertically or use a force triangle." }, { "id": "cie-2010-w-41-q07ii", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and tension after the external force is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates applied Newton's second law to Q correctly; common errors retained the removed force or reused the old friction force.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recalculate the normal reaction and friction after removing the 3.2 N force, then apply Newton's second law to each particle with consistent signs." }, { "id": "cie-2010-w-41-q07i", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction when the connected-particle system is in limiting equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates wrongly introduced acceleration despite the explicit equilibrium condition and often equated the normal reaction directly to the weight.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces on P and Q, use the equilibrium condition and limiting friction, and calculate the normal reaction before finding the coefficient." }, { "id": "cie-2010-w-41-q06ii", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "energy-momentum", "algebra" ], "primaryTopicId": "energy-momentum", "summary": "Find the release height H from the stated ratio of speeds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates made progress in part (ii), where the speed values were less accessible than in part (i)(c).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy to express the speeds at B and the ground in terms of H and h, form the given ratio, and solve for H." }, { "id": "cie-2010-w-41-q06ic", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)(c)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the ratio of impact speed to speed at B when the particle is released from A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says only a few candidates made progress in part (i)(c), although its values were more accessible than those in part (ii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of mechanical energy from A to B and from A to ground to find both speeds, then simplify their ratio." }, { "id": "cie-2010-w-41-q05iii", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the work done by friction from A to X, where the speed is the average speed for A to B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says scoring on this later part was usually limited to a single mark for finding distance AX.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the time and distance to X from the constant-acceleration motion, then use the work-energy relation for the section AX." }, { "id": "cie-2010-w-41-q05ii", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the stated work against friction to find the plane's angle of inclination.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says after the straightforward first part, later scoring was usually limited to single marks for finding the energy gain or friction force in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the loss in gravitational potential energy and the change in kinetic energy to the work done against friction, using the distance along the plane." }, { "id": "cie-2010-w-41-q04ii", "paperId": "cie-2010-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Show that particle Q has the same displacement as P after ten seconds, given that their velocities are then equal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says many candidates failed to recognise that Q’s acceleration depended on time and did not use calculus to connect its velocity and displacement with P’s.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate Q’s acceleration to obtain its velocity using the initial condition, then integrate velocity to find its displacement and compare it with P’s constant-acceleration displacement." }, { "id": "cie-2010-w-42-q03", "paperId": "cie-2010-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Resolve four horizontal forces to find the direction angle and repeated force magnitude.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The general comments identify Question 3 as the most difficult of the early questions; the report says incorrect complementary-angle identities caused many candidates to stall.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the two known forces and the two equal unknown forces along perpendicular axes, then solve the simultaneous equations for alpha and F." }, { "id": "cie-2010-w-42-q05ii", "paperId": "cie-2010-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the velocities of two particles at the instant they reach the same height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The general comments identify Question 5 as the most difficult later question, and the report says this part was poorly attempted with muddled time handling.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate their height expressions using the launch delay, solve for the meeting time, then use each particle's own elapsed time to find its velocity." }, { "id": "cie-2010-w-42-q07i", "paperId": "cie-2010-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 7, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the two turning times of the velocity function and the distance travelled by the second time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says use of limits in the integration was not well attempted and some candidates used an average-acceleration shortcut instead of differentiating.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity to find the intervals where it increases or decreases, solve for T1 and T2, then integrate velocity from 0 to T2." }, { "id": "cie-2010-w-43-q03ii", "paperId": "cie-2010-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the mass of Q in limiting equilibrium on the inclined prism.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies common errors in omitting the weight component, using an incorrect friction expression, or assuming acceleration in an equilibrium situation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the plane, include the tension and friction directions correctly, and use limiting friction mu R." }, { "id": "cie-2010-w-43-q07iii", "paperId": "cie-2010-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show that the car cannot reach 30 m/s while its speed is increasing.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates were unsure what was required in the middle parts of Question 7; it specifically distinguishes the increasing-speed argument expected here from setting acceleration to zero.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the power-speed relation and constant resistance to determine the acceleration as speed approaches 30 m/s, then show the limiting condition prevents a higher increasing speed." }, { "id": "cie-2010-w-43-q07iv", "paperId": "cie-2010-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Explain why the car's speed is approximately constant between B and C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates were unsure what was required in parts (iii), (iv), and (v), and warns against treating a zero-acceleration equation as a constant-speed argument in part (iii).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the engine's power to the resistive force at the limiting speed and explain why the resulting acceleration is close to zero." }, { "id": "cie-2010-w-43-q07v", "paperId": "cie-2010-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(v)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "State the approximate constant speed and its maximum possible error.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates were unsure what was required in the middle parts of Question 7, including parts (iii)-(v).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the limiting speed implied by the constant power and resistance, then state the bound on the difference from that value over BC." }, { "id": "cie-2010-w-43-q07vi", "paperId": "cie-2010-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(vi)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Assuming constant speed from B to C, estimate the travel time and distance from the engine's work.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report contrasts this final part's required constant-speed assumption with candidates who incorrectly used constant-acceleration formulae.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use work = power times time to find the duration, then multiply by the assumed constant speed to obtain BC." }, { "id": "cie-2010-w-43-q03i", "paperId": "cie-2010-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Show that the string tension is 3 N from the resultant force on the pulley.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report calls Question 3 one of the more challenging questions and notes candidates sometimes searched for arbitrary calculations that rounded to the given tension.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the angle between the two string segments and resolve their equal tensions along perpendicular axes, or apply the cosine rule to the force triangle." }, { "id": "cie-2010-w-51-q07iii", "paperId": "cie-2010-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the trajectory's angle to the horizontal when the projectile reaches A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates differentiated the trajectory and used its gradient; most attempted a longer time-based method incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the trajectory to obtain dy/dx at A, then use the local gradient to determine the angle to the horizontal." }, { "id": "cie-2010-w-51-q06ii", "paperId": "cie-2010-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the cyclist's differential equation for v(t).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes sign errors in the logarithmic integral, omitted constants, and missing limits prevented further progress.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate with the correct negative sign for ln(15-v), apply the initial speed, and rearrange for v." }, { "id": "cie-2010-w-51-q06iii", "paperId": "cie-2010-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find the distance travelled during the first nine seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often made errors integrating the exponential term and omitted the integration constant or limits.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity function from t = 0 to t = 9 and evaluate the definite integral accurately." }, { "id": "cie-2010-w-51-q05ii", "paperId": "cie-2010-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the maximum speed of the particle attached to the elastic string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates omitted the elastic energy stored at the start when forming the energy equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy from release to the lowest point, including the initial and final elastic energies and gravitational potential-energy change." }, { "id": "cie-2010-w-52-q07iii", "paperId": "cie-2010-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the trajectory's angle to the horizontal when the projectile reaches A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates differentiated the trajectory and used its gradient; most attempted a longer time-based method incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the trajectory to obtain dy/dx at A, then use the local gradient to determine the angle to the horizontal." }, { "id": "cie-2010-w-52-q06ii", "paperId": "cie-2010-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the cyclist's differential equation for v(t).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes sign errors in the logarithmic integral, omitted constants, and missing limits prevented further progress.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate with the correct negative sign for ln(15-v), apply the initial speed, and rearrange for v." }, { "id": "cie-2010-w-52-q06iii", "paperId": "cie-2010-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find the distance travelled during the first nine seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often made errors integrating the exponential term and omitted the integration constant or limits.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity function from t = 0 to t = 9 and evaluate the definite integral accurately." }, { "id": "cie-2010-w-52-q05ii", "paperId": "cie-2010-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the maximum speed of the particle attached to the elastic string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates omitted the elastic energy stored at the start when forming the energy equation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy from release to the lowest point, including the initial and final elastic energies and gravitational potential-energy change." }, { "id": "cie-2010-w-53-q06iii", "paperId": "cie-2010-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "differentiation", "integration", "mechanics" ], "primaryTopicId": "differentiation", "summary": "Find the speed at time 2T after the particle has reversed direction down the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates answered this part correctly and often reused the upward-motion differential equation after the particle began moving down the plane.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the downward-motion differential equation with the friction direction reversed, solve it from the instantaneous-rest condition at T, and evaluate the speed at 2T." }, { "id": "cie-2010-w-53-q03ii", "paperId": "cie-2010-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the tension in AP and the angle it makes with the vertical.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often omitted the tension in PQ when resolving vertically, producing an incorrect equation for the tension in AP.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the forces on P vertically and horizontally, include the tension in PQ, and use the circular-motion condition for the horizontal component." }, { "id": "cie-2010-w-61-q06v", "paperId": "cie-2010-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(v)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count all arrangements possible using Beryl's twelve coloured pegs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says part (v) extended the earlier cases and candidates had to recognise and add the possibilities using four, three, or two different colours.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Partition the arrangements by how many distinct colours appear, count the two-, three-, and four-colour cases, then add them." }, { "id": "cie-2010-w-61-q06iv", "paperId": "cie-2010-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iv)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements that use exactly three different colours.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the last two parts of Question 6 were more difficult and that part (iv) required recognising a generalisation of part (ii) and multiplying by the colour choices.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose the three colours, count the colour multiplicities that fill four holes, and account for the positions of identical pegs without overcounting." }, { "id": "cie-2010-w-62-q06iii", "paperId": "cie-2010-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the probability of more than 35 or fewer than 27 late trains in 90 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the same failure to add the two disjoint tail probabilities also caused losses in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Approximate the binomial distribution with its correct mean and variance, use continuity corrections for both tails, and add the two probabilities." }, { "id": "cie-2010-w-62-q02i", "paperId": "cie-2010-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Calculate the two coded-data sums using x - 50.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says coded-data methods caused problems and candidates needed to handle the shift consistently rather than mixing coded and uncoded sums.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given sum and standard deviation to evaluate sum(x-50) and sum(x-50)^2, preserving the shift term for all 16 scores." }, { "id": "cie-2010-w-62-q02ii", "paperId": "cie-2010-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Find the new mean and standard deviation after one additional score is included.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates mixed coded and uncoded totals while finding the new mean and variance.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Update the total and sum of squares for all 17 marks, then calculate the mean and variance from the updated totals." }, { "id": "cie-2010-w-62-q06ii", "paperId": "cie-2010-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of more than seven or fewer than two late trains in nine days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates calculated both tail probabilities separately but did not add them, losing marks.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the binomial distribution with n = 9 and p = 0.3, calculate both disjoint tails, and add their probabilities." }, { "id": "cie-2010-w-62-q01", "paperId": "cie-2010-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find p from the probability distribution of the discrete random variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many candidates had difficulty solving the quadratic and many who solved it kept the invalid negative probability value.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the listed probabilities to sum to one, solve the quadratic, and reject any root that would make a probability negative." }, { "id": "cie-2010-w-62-q07iii", "paperId": "cie-2010-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count committees containing exactly one of Albert and Tracey.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates forgot to reduce the number of available men after fixing Albert or Tracey in this part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the cases with Albert but not Tracey and Tracey but not Albert, reducing the available pool after fixing each person, then add the counts." }, { "id": "cie-2010-w-63-q06ii", "paperId": "cie-2010-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count valid seating arrangements under the business, student, and couple seating restrictions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates had difficulty multiplying the independent arrangement counts and often used the wrong number of choices for students or seat pairs.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Arrange the three business people in the front row, place students in window seats, then place each couple on a permitted side and row without overlap." }, { "id": "cie-2010-w-63-q06iii", "paperId": "cie-2010-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that Mrs Lin sits behind a student while Mrs Brown is in the front row.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often overlooked that only ten seats, not eleven, could place Mrs Lin directly behind a student when Mrs Brown was already in the front row.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the arrangements satisfying both seat conditions, respecting the front row and the reduced number of seats available behind students, then divide by all arrangements." }, { "id": "cie-2010-w-63-q04iii", "paperId": "cie-2010-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Calculate the sum of squared deviations from 50 using the given standard deviation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The general comments identify Question 4 as one of the more challenging questions; the report notes power errors and use of the wrong shifted mean in the sum-of-squares calculation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the standard-deviation identity with the mean shift from 50, then multiply by the sample size and simplify carefully." }, { "id": "cie-2010-w-71-q06i", "paperId": "cie-2010-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the die is biased toward sixes using the ten observed throws.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question was particularly challenging, with many candidates stating the wrong null probability, using the wrong tail or distribution, or omitting a clear comparison.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State hypotheses for P(6), use the binomial model under the fair-die null, and compare the relevant tail probability with the 10% significance level." }, { "id": "cie-2010-w-71-q06ii", "paperId": "cie-2010-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the repeated die test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates were unable to find the Type I error probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the rejection region from the stated significance test under the null hypothesis that the die is fair." }, { "id": "cie-2010-w-71-q06iii", "paperId": "cie-2010-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Explain a Type II error in the context of the die-bias test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was not answered well and required an explanation in the context of the die test.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the test fails to detect the bias and concludes the die is fair when it is actually biased toward sixes." }, { "id": "cie-2010-w-71-q05ii", "paperId": "cie-2010-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the sample indicates a lower mean final mark at Grinford.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says explanations in context remain difficult and stresses using the population mean in the hypotheses and making a non-definite conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set hypotheses about the population mean, standardise the sample mean using the distribution of F, compare with the one-tailed critical value, and give a contextual conclusion." }, { "id": "cie-2010-w-72-q06i", "paperId": "cie-2010-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the die is biased toward sixes using the ten observed throws.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this question was particularly challenging, with many candidates stating the wrong null probability, using the wrong tail or distribution, or omitting a clear comparison.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State hypotheses for P(6), use the binomial model under the fair-die null, and compare the relevant tail probability with the 10% significance level." }, { "id": "cie-2010-w-72-q06ii", "paperId": "cie-2010-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the repeated die test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates were unable to find the Type I error probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the rejection region from the stated significance test under the null hypothesis that the die is fair." }, { "id": "cie-2010-w-72-q06iii", "paperId": "cie-2010-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Explain a Type II error in the context of the die-bias test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was not answered well and required an explanation in the context of the die test.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the test fails to detect the bias and concludes the die is fair when it is actually biased toward sixes." }, { "id": "cie-2010-w-72-q05ii", "paperId": "cie-2010-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the sample indicates a lower mean final mark at Grinford.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says explanations in context remain difficult and stresses using the population mean in the hypotheses and making a non-definite conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set hypotheses about the population mean, standardise the sample mean using the distribution of F, compare with the one-tailed critical value, and give a contextual conclusion." }, { "id": "cie-2010-w-73-q06i", "paperId": "cie-2010-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 8, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at the 1% level whether the mean chemical amount differs between drugs A and B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Question 6 was among the most demanding and details errors in hypotheses, unbiased variance, tail probability, and the final comparison.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State two-tailed hypotheses, calculate the unbiased sample variance, standardise the difference using the large-sample standard error, compare with the critical value, and conclude in context." }, { "id": "cie-2010-w-73-q07i", "paperId": "cie-2010-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the publicity campaign increased the mean weekly house sales.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The general comments identify Question 7 as more demanding; the report says many candidates used only the probability of exactly five sales or compared the wrong Poisson tail with the significance level.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Model the three-week total with a Poisson distribution, set hypotheses for the increased mean, calculate the upper-tail probability for five or more sales, compare with 5%, and conclude in context." }, { "id": "cie-2010-w-73-q06ii", "paperId": "cie-2010-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Decide whether a normal population assumption is needed for the test and justify the answer.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The general comments identify part 6(ii) as more demanding; the report says many candidates incorrectly assumed the population itself had to be normal.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the large sample size and Central Limit Theorem to explain why the sample mean is approximately normal without assuming the population itself is normal." }, { "id": "cie-2010-w-73-q03iii", "paperId": "cie-2010-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(iii)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Explain how a second confidence interval can lead to a different conclusion when the population mean is unchanged.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The general comments identify part 3(iii) as one of the more demanding questions; the report says many candidates incorrectly claimed the variance changed instead of recognising sample-to-sample variation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Explain that a new random sample can have a different sample mean and hence a different confidence interval, even though the population mean has not changed." }, { "id": "cie-2010-w-73-q02ii", "paperId": "cie-2010-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Use the random-number table to select the first three readers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The general comments identify Question 2 as more demanding; the report says many attempts incorrectly divided the supplied blocks instead of extracting valid four-digit numbers consistently.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read consecutive four-digit groups consistently, reject any number above 7302, and keep the first three valid unique reader numbers." }, { "id": "cie-2010-w-73-q02i", "paperId": "cie-2010-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Give two reasons the returned questionnaire would not produce a random sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w10_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The general comments identify Question 2 as more demanding; the report says two distinct sources of selection bias were required and many attempts did not identify both.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Explain both voluntary-response bias and the fact that only buyers of that magazine issue could respond." }, { "id": "ial-legacy-2018-january-WME01-q8c", "paperId": "ial-legacy-2018-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2018, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance block A travels before it momentarily stops after block B reaches the ground.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_que_20180126.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_rms_20180308.pdf#page=18", "qpPages": [ 24, 24 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report says this was the most difficult part of the final question: candidates often found only one of the two required distances, and premature rounding commonly spoiled the final total.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the first-stage distance with the original acceleration, then use the new acceleration after the string goes slack to find the additional distance. Add both distances, keeping sufficient precision until the final value.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2018-january-WME01-q2", "paperId": "ial-legacy-2018-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2018, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Derive a condition on the friction coefficient for a particle to remain at rest under an angled force.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_que_20180126.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_rms_20180308.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The examiner report says this question was poorly answered. Many candidates assumed friction was already at its limiting value, losing the final marks; others skipped the horizontal force equation and lost the opening marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the applied force horizontally and vertically, use the resulting reaction and the friction inequality, then rearrange for the stated condition. The force need not be at limiting friction.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2018-january-WME01-q3b", "paperId": "ial-legacy-2018-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2018, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use momentum conservation and the post-collision direction to prove an upper bound on the second particle’s mass parameter.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_que_20180126.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_rms_20180308.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The examiner report says many candidates missed the significance of the second particle’s stated direction after impact. Correct solutions were uncommon; sign mistakes often led to an unsupported numerical value instead of the required bound.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Write signed momentum before and after impact, then use the stated post-impact direction to impose the sign condition needed for the inequality.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2018-january-WME01-q5b", "paperId": "ial-legacy-2018-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2018, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Use the cyclist’s speed-time profile and total distance to determine the unknown duration T.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_que_20180126.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_rms_20180308.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The examiner report says question 5 was unusually difficult. Many candidates used an incorrect trapezium for the distance calculation; a correct solution needed separate areas for the changing-speed and constant-speed phases.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the constant speed, add the areas for each phase of the correct speed-time graph, form an equation in T, and solve for the value consistent with the stated condition.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2018-january-WME01-q5a", "paperId": "ial-legacy-2018-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2018, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Sketch the cyclist’s speed-time graph across the acceleration, steady-speed, and deceleration stages.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_que_20180126.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WME01_01_rms_20180308.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says a significant number misread the timing and drew a trapezium that reached zero speed at the wrong time. Even candidates with the intended shape sometimes labelled it incorrectly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the initial rising segment, the constant-speed plateau, and the final falling segment with the phase durations in their stated order.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2018-january-WST01-q6d", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for how many draws are needed when the second selected sock is blue.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=11", "qpPages": [ 20, 20 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report identifies this as the most difficult part of question 6. Only a few candidates recognized the need for a conditional probability and then formed the correct ratio.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Restrict outcomes to those with a blue second draw, then form the ratio of outcomes in which both blue socks have appeared by draw three to the eligible outcomes.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2018-january-WST01-q7c", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Compare two normal distributions to find a common threshold and the matching tail probability.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=12", "qpPages": [ 24, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The examiner report calls this a strong challenge for the best candidates. Many formed an incorrect equation while standardizing, often through a sign error; only the strongest responses reached the requested probability.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the standardized scores at the shared threshold to find it, then standardize that threshold for either distribution and evaluate the matching tail.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2018-january-WST01-q6e", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the last possible draw is needed to obtain both blue socks.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=11", "qpPages": [ 20, 20 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says it was rare to see all four required probability products and the correct combined answer. Several candidates gave only one valid sequence.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Add the probabilities of the four possible positions of the first blue sock among the first four draws, using conditional probabilities without replacement.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2018-january-WST01-q7b", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Solve for a lower threshold from a specified probability under a normal distribution.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=12", "qpPages": [ 24, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this later part of question 7 challenged candidates: many reached a standard-score magnitude but chose the wrong sign, so only stronger candidates obtained the threshold.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the interval probability into a cumulative probability, identify the correct signed standard score, and back-transform to the original scale.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2018-january-WST01-q6c", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the second blue sock is drawn on the third selection.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=11", "qpPages": [ 20, 20 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that candidates often missed one of the valid sequences in this calculation or included an impossible ordering.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the mutually exclusive orders for the first three draws that contain exactly one blue in the first two positions and the second blue in position three.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2018-january-WST01-q4bii", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the specified expectation and probability total to solve for two unknown probabilities.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=9", "qpPages": [ 12, 12 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says candidates often formed a second equation incorrectly, including setting both equations to zero. Correct probability values were needed before later variance calculations could proceed.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the zero expected value as one equation and the condition that the probabilities sum to one as the other, then solve the simultaneous equations.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2018-january-WST01-q1eiii", "paperId": "ial-legacy-2018-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2018, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(e)(iii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Reason about how raising every score in only one class changes the variance of the combined group.", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_que_20180118.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/Maths/Old/2018 Jan/WST01_01_rms_20180308.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report identifies this as the most challenging part of question 1. Candidates often made a one-sided claim about adding scores without comparing how the two group means changed relative to each other.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20180308.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the distance between the two class means before and after the change; the within-class variance of class A stays fixed, but the class means move farther apart.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "cie-2009-w-11-q05i", "paperId": "cie-2009-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the arc length in a semicircle-and-sector diagram where the sector radius is not given directly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The matching report calls Question 5 difficult: most candidates could not find the sector radius, and few used trigonometry or Pythagoras (report p.2).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the diagram to determine the right triangle, radius, and angle before applying the arc-length formula in radians." }, { "id": "cie-2009-w-11-q05ii", "paperId": "cie-2009-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the shaded area formed by subtracting a triangle from a circular sector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often failed to identify the 45-degree sector angle and used the sector formula with degrees instead of radians (report p.2).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the sector angle and radius first, then subtract the triangle area from the sector area." }, { "id": "cie-2009-w-11-q06i", "paperId": "cie-2009-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use perpendicular tangent gradients at two x-values to determine a constant in dy/dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 6 was described as difficult; only a minority used the gradients k−4 and k−6 with the perpendicular-gradient condition (report p.2).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Evaluate both tangent gradients and use the condition that their product is -1." }, { "id": "cie-2009-w-11-q08i", "paperId": "cie-2009-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(i)", "marks": 6, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Form and solve equations for the first term and common ratio of an arithmetic and geometric progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes that only a minority solved for r or d after forming the AP/GP relations in part (i) (report p.2).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate each stated term into an equation, then solve the simultaneous equations without assuming a ratio prematurely." }, { "id": "cie-2009-w-11-q09ii", "paperId": "cie-2009-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector of a specified magnitude in the direction of AC.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "This vector part was reported as poorly answered; only a small minority found the unit vector in AC's direction and scaled it to magnitude 30 (report p.3).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find AC, divide by its magnitude to form a unit vector, then scale to the required length." }, { "id": "cie-2009-w-11-q10iii", "paperId": "cie-2009-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Show that an equation involving an inverse function has no real solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says only about half of candidates recognized that no real solutions required a negative discriminant (report p.3).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange to a quadratic and use its discriminant to establish that it has no real roots." }, { "id": "cie-2009-w-12-q04i", "paperId": "cie-2009-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Find the range of a sine function over a specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The range was poorly understood, and candidates struggled to find the endpoint values 2 and 8 (report p.5).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the maximum and minimum sine values attained on the interval, including both endpoints." }, { "id": "cie-2009-w-12-q04ii", "paperId": "cie-2009-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch one complete cycle of a transformed sine graph on the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the sketch caused difficulty because candidates missed the cycle's endpoint values and its concave-down then concave-up shape (report p.5).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Mark the endpoint values, midline, turning points, and the correct concavity through the cycle." }, { "id": "cie-2009-w-12-q09iv", "paperId": "cie-2009-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(iv)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Calculate the area of a rectangle from a coordinate-geometry construction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report explicitly describes part (iv) as very difficult and says the usual side-length-product approach was rarely successful (report p.6).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the previously found coordinates or lengths to obtain perpendicular side lengths, then multiply them." }, { "id": "cie-2009-w-12-q10ic", "paperId": "cie-2009-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2009, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the acute angle between a tangent and a given straight line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies this as the least well-answered part of Question 10; most candidates could not find the angle between the two gradients (report p.7).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the tangent gradient, compare it with the line gradient, and use the tangent formula for the angle between lines." }, { "id": "cie-2009-w-21-q02", "paperId": "cie-2009-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2009, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation whose argument contains a difference of terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a small number could cope with this logarithm question and many incorrectly split the logarithm of a difference (report p.8).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use log laws only on products or quotients; equate arguments after rewriting 2 ln x as ln(x²), then check the domain." }, { "id": "cie-2009-w-21-q05i", "paperId": "cie-2009-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2009, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express cos²(2x) in terms of cos(4x) for use in a later integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says almost everyone struggled with this identity part; many gave incorrect forms for cos²(2x) (report p.9).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the double-angle identity with the angle replaced by 2x and rearrange for the squared cosine." }, { "id": "cie-2009-w-21-q08ii", "paperId": "cie-2009-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2009, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 7, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation to show a tangent condition and find another tangent equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates could not differentiate 2xy correctly; sign errors were common and only about half formed the implicit derivative correctly (report p.10).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate every product term, collect dy/dx terms, and substitute each point before forming the line equation." }, { "id": "cie-2009-w-22-q03i", "paperId": "cie-2009-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2009, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate an integral using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report notes that many candidates ignored the instruction to use the trapezium rule and that endpoint secant values were often evaluated incorrectly (report p.11).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stated interval width and evaluate the integrand at both endpoints and the single interior point." }, { "id": "cie-2009-w-22-q04ii", "paperId": "cie-2009-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2009, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find an exact parameter value at which a parametric curve has a specified gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says simplifying the exponential equation and giving the corresponding exact t-value proved difficult for many candidates (report p.12).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use dy/dx=(dy/dt)/(dx/dt), substitute the target gradient, then solve the resulting exponential equation exactly." }, { "id": "cie-2009-w-22-q06ii", "paperId": "cie-2009-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2009, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a transformed sine equation for all angles in a full-turn interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Although many found one angle, the report says relatively few understood that the shifted-cosine equation gives two values (report p.12).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the sine-cosine combination as one shifted cosine and include both solutions in the interval." }, { "id": "cie-2009-w-31-q01", "paperId": "cie-2009-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a linear-modulus inequality without introducing extraneous critical values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Completely sound solutions were rare; the report highlights invalid reliance on a squared inequality and failure to apply the modulus case conditions (report pp.14–15).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the modulus by its sign intervals and retain each interval restriction when solving." }, { "id": "cie-2009-w-31-q06i", "paperId": "cie-2009-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector equation for the line through a midpoint and a trisection point in 3D.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says most candidates found the midpoint vector incorrectly and many also lacked a sound method for the trisection point (report p.16).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate each point's position vector from the given vectors, then use one point and the direction vector between them." }, { "id": "cie-2009-w-31-q10i", "paperId": "cie-2009-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Derive a differential equation for a sphere's radius from its changing volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only the strongest candidates progressed from the volume growth relation to the required differential equation (report p.16).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate V with respect to r, relate dV/dt to dr/dt, and substitute the sphere's volume and surface-area formulae." }, { "id": "cie-2009-w-31-q10iii", "paperId": "cie-2009-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 1, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Determine the allowed time interval implied by the radius-growth model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report states that there were very few correct answers to this final model-interpretation part (report p.16).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the expression for r(t) and the physical condition that the modeled radius remains valid." }, { "id": "cie-2009-w-32-q03i", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate an equation involving both x and y implicitly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies product-rule errors and sign mistakes in the implicit differentiation as common difficulties (report p.17).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the product rule to x²y and the chain rule to y³, keeping all dy/dx terms together before rearranging." }, { "id": "cie-2009-w-32-q06i", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use a trigonometric substitution to transform a definite integral and its limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This substitution part was particularly difficult; common errors included the wrong x² substitution, replacing dx incorrectly, and failing to change limits (report p.18).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the substitution, simplify with 1+tan²θ=sec²θ, and convert both integration limits." }, { "id": "cie-2009-w-32-q06ii", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact integral obtained after the trigonometric substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not recognize how to integrate cos²θ and produced incorrect forms, despite the preceding part's setup (report p.18).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity for cos²θ, integrate exactly, and retain the transformed limits." }, { "id": "cie-2009-w-32-q07iii", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Prove an exact angle relation using two complex numbers on an Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only a few candidates recognized the argument-angle link, and decimal approximations were not accepted as proof of the exact result (report p.19).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Relate the angle to the difference of the arguments and justify the exact value rather than relying on decimal approximations." }, { "id": "cie-2009-w-32-q08ii", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand a rational expression in ascending powers up to the x² term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that many candidates using partial fractions made errors in the geometric expansions or multiplying the series by the remaining factor (report p.19).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the partial fractions from part (i), expand each factor to the required order, then multiply and collect terms." }, { "id": "cie-2009-w-32-q09iii", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(iii)", "marks": 3, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Solve a differential-equation model for θ at a specified time in terms of A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report describes many errors in this part, especially false logarithm identities and failure to remove logarithms fully (report p.20).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the time into the integrated logarithmic relationship and isolate θ without misusing log laws." }, { "id": "cie-2009-w-32-q10ii", "paperId": "cie-2009-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2009, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the perpendicular distance between two parallel planes in vector form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Many candidates made little progress because they did not know where to start or found the distance from the origin instead of the plane separation (report p.20).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a common normal and project the displacement between the planes onto that normal; avoid measuring from the origin to a foot point." }, { "id": "cie-2009-w-41-q04i", "paperId": "cie-2009-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension in a longer string from a three-force equilibrium diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes repeated incorrect angle assumptions, equal-tension assumptions, and force-resolution errors in this part (report p.22).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces using the stated 3-4-5 geometry and distinguish the tensions in the two strings." }, { "id": "cie-2009-w-41-q06iii", "paperId": "cie-2009-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show that a second particle reaches the ground a specified time after the first.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Although most handled earlier stages, very few candidates found the time taken by Q in this part (report p.23).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Track the two particles' motion separately and solve for each particle's time to the ground." }, { "id": "cie-2009-w-41-q07ii", "paperId": "cie-2009-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the acceleration in the second velocity stage equals a specified negative value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates used a correct method; the report points to confusion about the given acceleration parameter and setting accelerations equal incorrectly (report p.23).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the second-stage velocity and solve the stated acceleration equation within that stage's time interval." }, { "id": "cie-2009-w-41-q07iii", "paperId": "cie-2009-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find displacement at the end of a two-stage velocity model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates obtained the correct displacement; many used the wrong limits or omitted the first-stage displacement (report p.23).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate velocity over the second stage and include the displacement accumulated in the first stage through the integration constant or separate areas." }, { "id": "cie-2009-w-42-q02ii", "paperId": "cie-2009-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find driving work for a lorry descending a hill with resistance and braking forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 2 was one of the least well answered; candidates frequently combined potential-energy and resistive-work terms with incorrect signs (report p.24).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply energy conservation with a consistent sign convention for gravitational potential-energy loss, kinetic-energy change, and resistive work." }, { "id": "cie-2009-w-42-q02iii", "paperId": "cie-2009-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(iii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find driving work over a horizontal journey from changes in speed and constant resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number ignored the instruction to use energy and instead applied Newton's law, losing marks even when the final value was correct (report p.24).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the work-energy principle and include both the kinetic-energy change and work against resistance." }, { "id": "cie-2009-w-42-q06ii", "paperId": "cie-2009-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find a particle's displacement after a connected-particle system changes stage.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says solutions to parts (ii) and (iii) were often incomplete because candidates treated the motion as only one stage (report p.25).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the motion while the string is taut, then continue with the correct motion after the other particle reaches the ground." }, { "id": "cie-2009-w-42-q07ii", "paperId": "cie-2009-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2009, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time when a motorcyclist's average speed reaches a stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates rarely completed this part; the report says many equated the target average speed to instantaneous speed from the first stage (report p.25).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form average speed as total distance divided by total elapsed time, using the displacement from the first stage as well as the second." }, { "id": "cie-2009-w-51-q01", "paperId": "cie-2009-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find spring compression and plate separation from equal opposing forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates found the correct plate distance; many used 40 N instead of the 20 N force from each plate or treated compression as extension (report p.26).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Hooke's law to the spring's total force and add the compression to the natural-length geometry." }, { "id": "cie-2009-w-51-q04i", "paperId": "cie-2009-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve a velocity vector into its horizontal component using an inverse-tangent angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes that the correct horizontal component was often given without valid working and warns against taking the inverse-tangent angle from the wrong ratio (report p.27).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the direction ratios to determine sin and cos of the angle before resolving the velocity." }, { "id": "cie-2009-w-51-q05i", "paperId": "cie-2009-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use moments to find the force in equilibrium with a lamina's centre of mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates used an incorrect centre-of-mass distance or moment arm in this part (report p.27).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the lamina's centre of mass, calculate its perpendicular moment arm, then balance moments about the pivot." }, { "id": "cie-2009-w-51-q06iii", "paperId": "cie-2009-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the maximum angular speed before a particle loses contact with a table.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes part (iii) as not particularly well done and notes that some candidates reused an earlier tension value (report p.27).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the limiting contact condition to find the critical tension, then relate that tension to the angular speed." }, { "id": "cie-2009-w-52-q01", "paperId": "cie-2009-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Decide which face a released triangular prism falls onto using its centre of mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a few candidates scored full marks; the report says many treated the prism as separate rods rather than using its cross-section and centre of mass (report p.28).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of the triangular cross-section and compare the relevant tipping angle with 90 degrees." }, { "id": "cie-2009-w-52-q06i", "paperId": "cie-2009-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find string tension by taking moments about a point for a uniform cone.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The most common error was using two-thirds rather than four-thirds of the cone length for the weight's moment arm (report p.29).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the cone's centre-of-mass position and the correct perpendicular distance from the pivot to the weight's line of action." }, { "id": "cie-2009-w-52-q07ii", "paperId": "cie-2009-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2009, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the resisting force from a particle's speed-dependent motion along a line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes that Newton's second law was often used with omitted force terms or sign errors in this part (report p.29).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a=dv/dt or a=v(dv/dx), then apply Newton's second law with all force terms and signs included." }, { "id": "cie-2009-w-61-q03ii", "paperId": "cie-2009-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the shortest and longest times within the middle 67% of a normal distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates struggled to reverse the normal tables for the central 67% and often misread which groups were the extremes (report p.30).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the normal table in reverse to find the two symmetric z-values, then convert them to journey times." }, { "id": "cie-2009-w-61-q05ai", "paperId": "cie-2009-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count four-digit numbers from a restricted digit set that lie between 5000 and 6000 without repetition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Part (a) was a stumbling block; a common error was treating the thousands digit as two choices instead of fixing it at 5 (report p.31).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Fix the first digit as 5, then arrange distinct remaining digits in the available positions." }, { "id": "cie-2009-w-61-q05aii", "paperId": "cie-2009-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Repeat the four-digit count when repeated digits are allowed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says allowing repeated digits confused candidates, while those who understood part (a)(i) generally succeeded here (report p.31).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Keep the leading digit fixed at 5 and count the remaining places independently when repetition is allowed." }, { "id": "cie-2009-w-61-q05bii", "paperId": "cie-2009-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count teams under a condition that a set of three cousins is either all selected or all excluded.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only the best candidates recognized the efficient all-in/all-out combination count; attempts based on listing options often omitted cases (report p.31).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the all-in and all-out cases and count the remaining team selections in each case." }, { "id": "cie-2009-w-62-q01ii", "paperId": "cie-2009-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the standard deviation from a box-and-whisker plot and a normal-model assumption.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 1 was identified as particularly challenging; many candidates did not connect the box plot with the normal distribution or the 75th percentile (report p.32).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the quartiles as the corresponding normal percentiles and solve for the standard deviation." }, { "id": "cie-2009-w-62-q04aii", "paperId": "cie-2009-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count odd numbers above 500 using selected digits without repetition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as the paper's most difficult part; candidates commonly missed cases or confused the leading and final digit restrictions (report p.33).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split by number of digits and possible leading digits, while enforcing the odd final digit and no-repetition rule." }, { "id": "cie-2009-w-62-q07ii", "paperId": "cie-2009-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2009, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to find a threshold matching a specified upper-tail probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says most candidates found this inverse normal-table calculation very demanding and often confused probabilities with z-values (report p.34).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the given tail probability with the known upper tail beyond 128, find the corresponding z-value, and solve for k." }, { "id": "cie-2009-w-71-q04i", "paperId": "cie-2009-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Verify the Poisson critical region for a one-tailed test at the 5% level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This question was not well attempted; candidates often compared individual Poisson probabilities with 0.05 instead of summing the tail and checking the boundary (report p.36).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate and accumulate the relevant Poisson probabilities, compare the tail with 0.05, and verify the boundary value." }, { "id": "cie-2009-w-71-q04ii", "paperId": "cie-2009-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the Type II error probability under the alternative Poisson mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was only slightly better attempted and candidates commonly omitted P(4) from the acceptance-region probability (report p.36).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the acceptance region from part (i) and sum the alternative-model probabilities over that region." }, { "id": "cie-2009-w-71-q05iv", "paperId": "cie-2009-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iv)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a binomial probability for exactly three observations exceeding the upper quartile.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Not many candidates used the correct binomial parameters; some wrongly used the quartile value as a probability (report p.36).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use p=0.25 for exceeding the upper quartile and calculate the exactly-three binomial term for five observations." }, { "id": "cie-2009-w-71-q06ii", "paperId": "cie-2009-w-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses and interpret a Type I error in the photographer's context.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says part (ii) was not well attempted and candidates often calculated a Type I error instead of explaining its contextual meaning (report p.36).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State the null and one-sided alternative, give the test's significance probability, and describe the false rejection in context." }, { "id": "cie-2009-w-72-q03i", "paperId": "cie-2009-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a suitable approximation to assess whether overbooking exceeds the number of seats.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this question very demanding; candidates often modeled the wrong group, chose a normal model, or summed the wrong Poisson outcomes (report p.38).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Model the number of non-arrivals with a Poisson approximation and evaluate the small counts that leave too many passengers arriving." }, { "id": "cie-2009-w-72-q04i", "paperId": "cie-2009-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the two-tailed binomial acceptance region for a test with a maximum Type I error probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was not well answered; candidates frequently failed to sum the binomial probabilities or used a one-tailed or normal method (report p.38).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Accumulate binomial tail probabilities symmetrically and include values only while the total Type I error stays within the limit." }, { "id": "cie-2009-w-72-q05i", "paperId": "cie-2009-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test whether sample evidence supports a population mean below a stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates omitted the required probability or critical-value comparison, and often failed to compare the calculated critical mean correctly (report p.38).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up one-sided hypotheses, standardize the sample mean, compare consistently with a critical value or p-value, and conclude in context." }, { "id": "cie-2009-w-72-q07a", "paperId": "cie-2009-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2009, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Recover a linear transformation's constants from its means and standard deviations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w09_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates correctly used Var(Y)=b²Var(X) to determine the scale factor, making this transformation question particularly difficult (report p.39).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w09_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the standard-deviation relationship to find the positive scale factor, then use the mean relationship to solve for the shift." }, { "id": "cie-2010-s-11-q01", "paperId": "cie-2010-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express tangent and sine values in terms of k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates gained little or no credit: many mishandled the tangent angle identities, and only very few made progress on the sine part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Represent tan x = k with a right triangle, then use the triangle ratios and angle identities for π − x and π/2 − x. Keep the acute-angle and positive-k conditions in view." }, { "id": "cie-2010-s-11-q05", "paperId": "cie-2010-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": null, "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the range and solve an equation for a trigonometric function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates mishandled the sign and reversed the greatest and least values; very few found both cosine solutions or gave accurate answers in radians.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use sin²x + cos²x = 1 to rewrite f as a constant plus a cosine term. Determine its extrema from the cosine range, then solve the equation over 0 ≤ x ≤ π, checking both cosine signs and radian answers." }, { "id": "cie-2010-s-11-q08", "paperId": "cie-2010-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": null, "marks": 10, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find triangle vertices and a perpendicular-bisector intersection.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says errors in forming the line equations for C were very common; midpoint and perpendicular-bisector errors were also common.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "First find the gradient of AB and use the given gradient relationship to determine m. Form AC and BC, solve for C, then find the midpoint of AB and intersect its perpendicular bisector with BC." }, { "id": "cie-2010-s-11-q10", "paperId": "cie-2010-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": null, "marks": 11, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use position vectors to find a unit vector, diagonal angle and perimeter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says very few candidates attempted the unit vector, and many found the angle between OA and OC instead of the required angle between OB and AC.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use OB = OA + OC, divide by its magnitude for the unit vector, and use a scalar product for the angle between OB and AC. Find both side lengths and add them twice for the perimeter." }, { "id": "cie-2010-s-12-q03", "paperId": "cie-2010-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the range of a quadratic function and a parameter for equal roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the range part was poorly answered: only a small minority evaluated f at the stationary point, and many gave the inequality in the wrong direction.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the quadratic’s stationary value to state its range. Substitute f into g, set the resulting quadratic equal to k, then set its discriminant to zero for equal roots." }, { "id": "cie-2010-s-12-q07", "paperId": "cie-2010-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 8, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Sum an arithmetic sequence and evaluate a geometric progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says finding the 41 terms in the inclusive arithmetic sum proved too difficult for most candidates; it notes that the progression part was generally more successful.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "List the multiples of 5 inclusively to identify the first term, last term and number of terms, then use the arithmetic-series formula. For the geometric progression, use its first-three-term sum to find the first term and then the convergent infinite-sum formula." }, { "id": "cie-2010-s-12-q08", "paperId": "cie-2010-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": null, "marks": 8, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Maximise the volume of a rectangular block with fixed surface area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates struggled to eliminate h; around half found x = 4 but did not calculate the stationary volume.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the surface-area equation to eliminate h, substitute into V, differentiate with respect to x, and use the sign of the second derivative (or a change in derivative sign) to classify the stationary value." }, { "id": "cie-2010-s-12-q10", "paperId": "cie-2010-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": null, "marks": 9, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find a tangent and solve an increasing-function inequality.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates missed the y-intercept or gave the normal instead of the tangent; the increasing-function inequality was badly answered and often ignored.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the composite expression using the chain rule, find the curve’s y-intercept and gradient there, and form the tangent. For the final part solve dy/dx > 0, tracking the quadratic inequality’s sign." }, { "id": "cie-2010-s-12-q11", "paperId": "cie-2010-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": null, "marks": 10, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Solve and analyse a sine function, including its inverse interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report describes the question as poorly answered overall: many gave degree answers instead of radians, missed the second solution, or could not connect the no-solution part to the sketch.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the sine equation in the stated domain, sketch the full cycle, use the range to identify k-values with no solution, then choose the largest one-to-one interval and invert the function at the requested value." }, { "id": "cie-2010-s-13-q06", "paperId": "cie-2010-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": null, "marks": 8, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find an angle and perimeter from three position vectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only about half of candidates selected the required vector pair for the angle; others used unrelated vectors or attempted an inappropriate triple scalar product.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Subtract position vectors to form the two sides meeting at B, use their scalar product to find angle ABC, then calculate all three side lengths and add them." }, { "id": "cie-2010-s-13-q07", "paperId": "cie-2010-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 8, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Calculate the perimeter and area of a circular plate with two cut-out segments.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates could not assemble a coherent perimeter or area method: common errors included using the wrong arc angle, omitting straight edges, and adding sectors to get the full circle.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given subtended angle and the remaining arc angle to calculate arc lengths; add the straight edges for perimeter. For area, subtract the two removed circular segments from the full circle, or combine the matching sector and triangle areas." }, { "id": "cie-2010-s-13-q08", "paperId": "cie-2010-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": null, "marks": 9, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a rhombus’s perpendicular bisector, vertices and area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates did not understand the perpendicular-bisector condition; in the area part, candidates confused diagonal halving and some treated the area as a product of sides.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the midpoint and gradient of AC, form its perpendicular bisector, and use the y-axis condition and rhombus symmetry to locate B and D. Find the area from the diagonals or base and height." }, { "id": "cie-2010-s-13-q09", "paperId": "cie-2010-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2010, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": null, "marks": 11, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Find curve-line intersections, a stationary point and a volume of revolution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says some candidates failed to find the stationary point, integrated y rather than y², omitted the line boundary, or used incorrect limits in the volume calculation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Solve the curve equation with y = 5 for A and B, differentiate for the minimum point M, then integrate the difference between the outer and inner squared radii over the shaded interval." }, { "id": "cie-2010-s-21-q07", "paperId": "cie-2010-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 8, "topicIds": [ "functions", "numerical-methods" ], "primaryTopicId": "functions", "summary": "Use graphs and fixed-point iteration to locate a unique root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says a significant number failed the graph part, the iteration justification was often omitted, and some candidates used a numerical approach in place of the explanation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch y = e^(2x) and y = 2 − x to establish the unique intersection, check the root interval by substitution, show the iteration’s fixed point satisfies the original equation, and iterate to the required accuracy." }, { "id": "cie-2010-s-21-q08", "paperId": "cie-2010-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": null, "marks": 10, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Prove cotangent derivatives and evaluate trigonometric integrals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says there were few successful attempts at the later parts; many omitted the required cosec²x integral or could not integrate cosec²x correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate cot x using the quotient rule, rewrite cot²x with cosec²x, and use the resulting derivative when integrating. Apply the identities and exact limits carefully in the final integral." }, { "id": "cie-2010-s-22-q02", "paperId": "cie-2010-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Estimate a curved area using the trapezium rule and judge the error direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates attempted exact integration instead, used the wrong number of ordinates or formula, and few compared the trapezia with the exact area.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use ordinates at x = 2, 2.5 and 3 with width 0.5. For the error direction, examine the curve’s concavity on [2,3] and compare the chords with the curve." }, { "id": "cie-2010-s-22-q04", "paperId": "cie-2010-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate a trigonometric integral exactly using identities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates failed to use the required identity or exact tangent values for the second integral, though a sizeable minority completed it correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity to establish the first integral, then rewrite tan²x as sec²x − 1 and integrate using tan x − x. Substitute the exact endpoint values rather than decimals." }, { "id": "cie-2010-s-22-q08", "paperId": "cie-2010-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": null, "marks": 9, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and solve the resulting equation over a full turn.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says more candidates than expected could not reduce the equation to the required double-angle form, often misusing tan x or sec x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the shifted sine and cosine terms into √3 sin x, multiply the equation by cos x, reduce to an equation for sin 2x, and list every solution in 0° < x < 360°." }, { "id": "cie-2010-s-23-q02", "paperId": "cie-2010-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Estimate a curved area using the trapezium rule and judge the error direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates attempted exact integration instead, used the wrong number of ordinates or formula, and few compared the trapezia with the exact area.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use ordinates at x = 2, 2.5 and 3 with width 0.5. For the error direction, examine the curve’s concavity on [2,3] and compare the chords with the curve." }, { "id": "cie-2010-s-23-q04", "paperId": "cie-2010-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate a trigonometric integral exactly using identities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates failed to use the required identity or exact tangent values for the second integral, though a sizeable minority completed it correctly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity to establish the first integral, then rewrite tan²x as sec²x − 1 and integrate using tan x − x. Substitute the exact endpoint values rather than decimals." }, { "id": "cie-2010-s-23-q08", "paperId": "cie-2010-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2010, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": null, "marks": 9, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and solve the resulting equation over a full turn.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says more candidates than expected could not reduce the equation to the required double-angle form, often misusing tan x or sec x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the shifted sine and cosine terms into √3 sin x, multiply the equation by cos x, reduce to an equation for sin 2x, and list every solution in 0° < x < 360°." }, { "id": "cie-2010-s-31-q01", "paperId": "cie-2010-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a parameterised inequality with absolute values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Q1 among the questions done least well; many candidates lost the parameter, squared incorrectly, or failed to combine the two valid boundary values.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split at the critical points where each expression inside an absolute value changes sign, solve the resulting linear inequalities on each interval, and combine the valid intervals while retaining a > 0." }, { "id": "cie-2010-s-31-q03", "paperId": "cie-2010-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use logarithms to solve for constants in a power relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Q3 among the least-well-done questions and says many candidates multiplied logarithms instead of adding them or applied the index to the wrong variables.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms of xⁿy = C to obtain a linear relation between ln y and ln x. Substitute both data pairs to solve the simultaneous equations for n and C." }, { "id": "cie-2010-s-31-q07", "paperId": "cie-2010-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 9, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Use an Argand diagram to find a constrained complex-number point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the Argand question among the least-well-done questions; many candidates misplaced the perpendicular bisector or circle and misunderstood the least-argument requirement.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot 1, i and 2 + 2i. Draw the perpendicular-bisector locus from the equal-distance condition and the unit circle about u, shade their intersection, then locate the least-argument point and calculate its modulus." }, { "id": "cie-2010-s-32-q03", "paperId": "cie-2010-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Evaluate exact trigonometric values without a calculator.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says a considerable number could not obtain the exact sine value without a calculator, and later errors included incorrect tangent triple-angle formulae.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the 3–4–5 triangle to find the exact sine and tangent values, apply angle-difference formulae for a − 30°, and use double- and triple-angle formulae for tan 2a and tan 3a." }, { "id": "cie-2010-s-32-q07", "paperId": "cie-2010-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 8, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Solve and interpret a differential equation involving exponential and trigonometric terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls the question poorly answered: few completed the integrations and constant correctly, and many gave incorrect long-term behaviour or claimed x was proportional to t.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate the exponential and sec²x terms, use x = 0 when t = 0 to find the constant, then state the limiting value and use dx/dt’s sign to explain monotonic increase." }, { "id": "cie-2010-s-33-q02", "paperId": "cie-2010-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the gradient and intercept of a logarithmic graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates often did not take logarithms of the full exponential term or forgot the coefficient of ln y when finding the gradient and A.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Take logarithms of y³ = Ae^(2x) to get a linear equation for ln y in terms of x, read off its gradient and use the intercept condition to solve for A." }, { "id": "cie-2010-s-33-q04", "paperId": "cie-2010-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 7, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve a differential equation for x² using the initial condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often mishandled the differential equation’s fraction, guessed the integral, and made errors when exponentiating and applying the initial condition.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set u = x² so that du/dt = 2x(dx/dt), separate the resulting first-order equation, integrate, and use x = 1 at t = 0 to determine the constant." }, { "id": "cie-2010-s-33-q07", "paperId": "cie-2010-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 8, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Prove a triple-angle identity and use it for an exact definite integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many ignored the identity’s role in the integral and tried integration by parts; it also stresses exact limit substitution rather than calculator evaluation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Derive cos 3θ from the addition formula, rearrange to express cos³θ as a combination of cos 3θ and cos θ, then integrate and evaluate the exact sine values at both limits." }, { "id": "cie-2010-s-33-q08", "paperId": "cie-2010-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2010, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": null, "marks": 9, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Verify complex roots and sketch a constrained region on an Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the Argand diagram proved a real problem: candidates often placed the circle in the wrong quadrant, drew the argument ray from the wrong point, or shaded the wrong region.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute 1 + i√3 into the cubic, use conjugacy for the other root, then plot the point and construct the unit-circle and argument boundaries before shading their intersection." }, { "id": "cie-2010-s-41-q02", "paperId": "cie-2010-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Interpret a piecewise velocity-time graph for a cutting tool.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates used constant-acceleration equations in part (iii) even though they did not apply; some assumed a particular time for the maximum speed without explaining why it was time-independent.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read gradients for acceleration and signed areas for displacement from each straight segment. Find the greatest return speed from the downward portion of the graph." }, { "id": "cie-2010-s-41-q06", "paperId": "cie-2010-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": null, "marks": 11, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Model connected particles with friction and a change after one particle lands.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this connected-particle situation was more challenging than usual; significant numbers used A’s weight instead of friction or carried the wrong acceleration/distance into the second stage.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Before B reaches the floor, resolve forces on each particle with friction on A and use the string constraint. After B lands, use A’s remaining distance and the new acceleration to find its speed at the pulley." }, { "id": "cie-2010-s-41-q07", "paperId": "cie-2010-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 11, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find a piecewise velocity and distance after t = 15.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes repeated errors using s(15) = 225 to determine the integration constant and in equating the later distance to the total rather than the extra distance.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the distance integral over 0 to 15 to find A, apply continuity to obtain B, integrate the later velocity for total distance, and solve for the time and speed at 315 m." }, { "id": "cie-2010-s-42-q02", "paperId": "cie-2010-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Interpret a piecewise velocity-time graph for a cutting tool.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates used constant-acceleration equations in part (iii) even though they did not apply; some assumed a particular time for the maximum speed without explaining why it was time-independent.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Read gradients for acceleration and signed areas for displacement from each straight segment. Find the greatest return speed from the downward portion of the graph." }, { "id": "cie-2010-s-42-q06", "paperId": "cie-2010-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": null, "marks": 11, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Model connected particles with friction and a change after one particle lands.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this connected-particle situation was more challenging than usual; significant numbers used A’s weight instead of friction or carried the wrong acceleration/distance into the second stage.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Before B reaches the floor, resolve forces on each particle with friction on A and use the string constraint. After B lands, use A’s remaining distance and the new acceleration to find its speed at the pulley." }, { "id": "cie-2010-s-42-q07", "paperId": "cie-2010-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 11, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find a piecewise velocity and distance after t = 15.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes repeated errors using s(15) = 225 to determine the integration constant and in equating the later distance to the total rather than the extra distance.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the distance integral over 0 to 15 to find A, apply continuity to obtain B, integrate the later velocity for total distance, and solve for the time and speed at 315 m." }, { "id": "cie-2010-s-43-q03", "paperId": "cie-2010-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": null, "marks": 7, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to find a load’s speed after a force change.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates confused kinetic energy at B with its gain and some ignored the instruction to use an energy approach; premature rounding also produced common wrong speeds.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate work by the force over each half-distance, subtract the equal resistance work to find the kinetic-energy gain, then use the initial and final kinetic energies to solve for speed." }, { "id": "cie-2010-s-43-q05", "paperId": "cie-2010-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Model a billiard ball’s motion before and after rebounding.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates assumed the rebound speed stayed unchanged, giving a common incorrect time, and often drew a speed-time graph or omitted the timing offset.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use constant deceleration to find the first travel distance and deceleration, reverse the velocity sign at the wall while applying the rebound speed condition, then calculate time to rest and sketch velocity against time." }, { "id": "cie-2010-s-43-q06", "paperId": "cie-2010-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": null, "marks": 9, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find when two particles on an incline are a specified distance apart.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates omitted P’s initial O-to-A travel, used t − 2 instead of t + 2, or added the two displacements.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find acceleration from P’s first two seconds, then account for P having already travelled for two seconds when Q starts. Equate the two positions with the correct signed separation." }, { "id": "cie-2010-s-43-q07", "paperId": "cie-2010-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2010, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse friction when two stacked boxes move without slipping.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the unfamiliar two-interface friction setup caused difficulty; many failed to isolate box A in part (ii) or included extra/missing forces in the final part.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the limiting floor friction to find its coefficient, then consider box A alone to bound the acceleration from inter-box friction. Apply the same acceleration to both boxes to find the maximum force P." }, { "id": "cie-2010-s-51-q07", "paperId": "cie-2010-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 11, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Solve motion under a position-dependent resistive force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Q7 the most difficult question on the paper; many lost signs or constants in the differential equation, and very few completed the final algebra and inequality explanation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law, use a = v dv/dx to derive v(x), then use v = dx/dt and the initial condition to solve for x(t). Show x < 5 from the exponential form." }, { "id": "cie-2010-s-52-q07", "paperId": "cie-2010-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 11, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Solve motion under a position-dependent resistive force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Q7 the most difficult question on the paper; many lost signs or constants in the differential equation, and very few completed the final algebra and inequality explanation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law, use a = v dv/dx to derive v(x), then use v = dx/dt and the initial condition to solve for x(t). Show x < 5 from the exponential form." }, { "id": "cie-2010-s-53-q06", "paperId": "cie-2010-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": null, "marks": 9, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Solve the motion of a particle opposed by a force proportional to √v.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Q6 was difficult for many candidates; errors included losing a minus sign, omitting the integration constant, and failing to integrate the resulting expression in x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use m v dv/dx = −3√v to derive the given velocity-displacement relation, then separate variables in dx/dt = v and integrate with x = 0 at t = 0 to find x at 0.5 s." }, { "id": "cie-2010-s-53-q07", "paperId": "cie-2010-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2010, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 11, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the maximum speed and tension of a particle on an elastic string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Q6 and Q7 as discriminating questions; it notes sign errors in energy equations and difficulty finding the greatest speed and tension conditions.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy to establish v² as a function of extension, maximize that quadratic for the greatest speed, then find the maximum extension at a turning point and use Hooke’s law for the greatest tension." }, { "id": "cie-2010-s-61-q03", "paperId": "cie-2010-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find parameters of a normal distribution from two tail probabilities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes the response as disappointing: candidates often used the wrong z-values or signs, approximated too early, and then obtained inaccurate μ and σ.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert both tail probabilities to standard normal z-values and solve the two linear equations for μ and σ, preserving signs and working at sufficient precision." }, { "id": "cie-2010-s-61-q05", "paperId": "cie-2010-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "binomial", "distributions" ], "primaryTopicId": "binomial", "summary": "Use a binomial model and decide when a normal approximation is valid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few candidates showed familiarity with the binomial distribution; common errors included giving the wrong approximation condition, omitting the continuity correction, and using the wrong tail.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set X ~ Bin(n, 0.25) for each period. Check np and nq for approximation eligibility, and for the 40-day probability apply a continuity correction before using the normal distribution." }, { "id": "cie-2010-s-61-q06", "paperId": "cie-2010-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": null, "marks": 9, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count mug allocations and arrangements under parity and separation conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few correct solutions were submitted for any part; candidates often ignored the parity relation or failed to account for all adjacent-mug cases.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For odd allocations, sum the cases where both people receive odd counts. For the arrangements, use permutations with identical objects and subtract cases where the restricted mugs are adjacent." }, { "id": "cie-2010-s-61-q07", "paperId": "cie-2010-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 10, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find multi-question probabilities and a conditional probability in a quiz tree.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says very few candidates gave reasonable solutions in part (iii); many multiplied the first-question probability by the second without accounting for the friend/audience outcome branches.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the tree to combine Peter’s answer and help outcomes for two questions, then divide the probability of both correct with the audience on the first help by the probability both are correct." }, { "id": "cie-2010-s-62-q01", "paperId": "cie-2010-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Choose and justify an average for a small data set with an outlier.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates gave fictional or generic reasons and showed little understanding of when mean, median or mode is useful.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate mean and standard deviation, identify the outlier, select the most representative centre, and explain why the other two measures do not suit this data set." }, { "id": "cie-2010-s-62-q02", "paperId": "cie-2010-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Combine a normal probability with a binomial sample calculation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says a very large proportion found the wrong probability in part (i); premature rounding then caused further errors in the binomial calculation.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardize the single-pencil length to find P(X > 10.9), then take its complement for p = P(X < 10.9). Use p in a Bin(6, p) model and sum the cases with at least two short pencils." }, { "id": "cie-2010-s-62-q04", "paperId": "cie-2010-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Relate μ and σ from two normal-distribution conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates could not eliminate one variable; dividing by 3/5 and expressing μ in terms of σ caused further difficulty.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Standardize each event, use 5σ = 3μ to solve the first probability, then invert the second normal-table probability and rearrange the resulting z equation for μ in terms of σ." }, { "id": "cie-2010-s-62-q05", "paperId": "cie-2010-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Calculate event probabilities and test exclusivity and independence.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most candidates struggled to distinguish mutually exclusive from independent events and to apply each condition to these events.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the ordered outcomes satisfying Q and R, compute their intersection, then compare P(Q ∩ R) with 0 for exclusivity and with P(Q)P(R) for independence." }, { "id": "cie-2010-s-63-q01", "paperId": "cie-2010-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "probability", "binomial" ], "primaryTopicId": "probability", "summary": "Find the probability of selecting exactly three red sweets without replacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Q1 as one of the most challenging questions; many incorrectly treated red and non-red probabilities as constant or used 49 instead of 39 non-red sweets.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a hypergeometric count: choose 3 from the 13 red sweets and 4 from the 39 non-red sweets, divided by choosing any 7 of the 52 sweets." }, { "id": "cie-2010-s-63-q04", "paperId": "cie-2010-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2010, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 8, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count drink-can arrangements under endpoint and adjacency restrictions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Q4 as one of the most challenging questions; candidates often failed to account for identical cans and the restricted arrangements.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For the endpoint condition, split by which drink occupies both ends and permute the remaining multiset. For the second part, treat the three cola cans as a block and exclude arrangements where the green-tea cans are adjacent." }, { "id": "cie-2010-s-71-q01", "paperId": "cie-2010-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 2, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Interpret a probability-density graph for train waiting time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_71.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates got both parts correct; many mishandled the density calculation or described train lateness instead of the implication for arrival times.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use total area under the density to find k, then interpret the graph as a uniform distribution on waiting times from 0 to 12 minutes and describe what it implies about arrival intervals." }, { "id": "cie-2010-s-71-q05", "paperId": "cie-2010-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Normalize an exponential density and find a percentile wait time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says part (ii) was not well attempted and a common error in part (iii) was equating the lower-tail integral to 0.8 instead of using the specified upper-tail probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the density over 0 to 5 to determine k, sketch its support and shape, then set the upper-tail probability to 0.8 and solve for the corresponding time." }, { "id": "cie-2010-s-71-q07", "paperId": "cie-2010-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 10, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Conduct a Poisson test and calculate a Type II error probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_71.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_71.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often failed to interpret the Type I error in context, omitted essential comparisons, or used the wrong probability sums for the Type II error.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the null use the Poisson mean 5.2 to determine the lower-tail critical region and Type I probability, test the observed count 2, then find the acceptance-region probability under mean 4.1." }, { "id": "cie-2010-s-72-q01", "paperId": "cie-2010-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 2, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Interpret a probability-density graph for train waiting time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_72.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates got both parts correct; many mishandled the density calculation or described train lateness instead of the implication for arrival times.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use total area under the density to find k, then interpret the graph as a uniform distribution on waiting times from 0 to 12 minutes and describe what it implies about arrival intervals." }, { "id": "cie-2010-s-72-q05", "paperId": "cie-2010-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Normalize an exponential density and find a percentile wait time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says part (ii) was not well attempted and a common error in part (iii) was equating the lower-tail integral to 0.8 instead of using the specified upper-tail probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the density over 0 to 5 to determine k, sketch its support and shape, then set the upper-tail probability to 0.8 and solve for the corresponding time." }, { "id": "cie-2010-s-72-q07", "paperId": "cie-2010-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 10, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Conduct a Poisson test and calculate a Type II error probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_72.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_72.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates often failed to interpret the Type I error in context, omitted essential comparisons, or used the wrong probability sums for the Type II error.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the null use the Poisson mean 5.2 to determine the lower-tail critical region and Type I probability, test the observed count 2, then find the acceptance-region probability under mean 4.1." }, { "id": "cie-2010-s-73-q01", "paperId": "cie-2010-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether support for a party has decreased using a binomial model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates often calculated only P(X = 3), used an inappropriate normal approximation, or failed to compare the test statistic with 0.025.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State hypotheses for the support proportion, calculate the lower-tail binomial probability P(X ≤ 3) under p = 1/3, compare with 0.025, and state a contextual conclusion." }, { "id": "cie-2010-s-73-q05", "paperId": "cie-2010-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find moments of a density supported on x ≥ 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_73.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls the support x > 1 the major obstacle: many used incorrect limits such as 0 to 1 or 1 to 2, so they could earn only method marks.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the support limits 1 to infinity to normalize the density, then integrate x f(x) and x² f(x); subtract [E(X)]² to obtain the variance." }, { "id": "cie-2010-s-73-q07", "paperId": "cie-2010-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2010, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 11, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Apply a Poisson model, scale its rate and use a normal approximation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_qp_73.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s10_ms_73.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says contextual model assumptions caused difficulty and a large number missed the continuity correction or used the wrong tail when estimating vaccine shortages.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_s10_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State the contextual Poisson assumptions, rescale the mean from 15 to 10 minutes for part (ii), then for the 20-hour total use a normal approximation with continuity correction at 370.5." }, { "id": "cie-2008-w-1-q05i", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Determine the constants from the function's maximum and minimum values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes the question as poorly answered and says the positivity condition was commonly used with the signs reversed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up equations using the maximum and minimum of the cosine function, then solve for both positive constants." }, { "id": "cie-2008-w-1-q05ii", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over the full stated degree interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates gave only one solution after making cosine the subject.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the constants from part (i), isolate the cosine value, and include every solution in the interval." }, { "id": "cie-2008-w-1-q05iii", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": "5(iii)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch one complete cycle of the transformed cosine function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes that many sketches had sharp V-shaped sections instead of the required sinusoidal curve.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Mark the maximum, minimum, and key quarter-cycle points before drawing a smooth cosine curve over the full interval." }, { "id": "cie-2008-w-1-q07i", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Form the area expression from the wire split between a square and a circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates did not form the two simultaneous equations needed and that squaring the radius expression caused substantial algebra difficulty.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the circle radius in terms of the square side using the wire-length condition, substitute into the area, and simplify carefully." }, { "id": "cie-2008-w-1-q09i", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the shaded area between the curve, the y-axis, and the horizontal boundary.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says only a small proportion reached the correct area, commonly through an incorrect integration constant or omitted rectangle subtraction.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose an integration variable that matches the limits and subtract the integrated curved area from the enclosing rectangle if integrating with respect to x." }, { "id": "cie-2008-w-1-q09ii", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 9, "questionPart": "9(ii)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume generated by rotating the shaded region about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates used the wrong axis or failed to subtract the curve-generated volume from the cylinder.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up the disk or washer integral about the x-axis and account for the surrounding cylinder volume where needed." }, { "id": "cie-2008-w-1-q09iii", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 9, "questionPart": "9(iii)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the acute angle between the tangents at the two stated points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says very few correct solutions were seen and notes that many missed the link between tangent gradient and the angle to the x-axis.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate to obtain both tangent gradients, then use the tangent-angle relationship and select the acute angle." }, { "id": "cie-2008-w-1-q10ii", "paperId": "cie-2008-w-1", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2008, "session": "October/November", "paperCode": "9709/1", "variant": "Variant not numbered", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "functions", "algebra", "differentiation" ], "primaryTopicId": "functions", "summary": "Compose the functions and establish the maximum value of the resulting quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_1.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_1.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says candidates often failed to simplify the composition to the correct quadratic and commonly lost the sign while completing the square or differentiating.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the inner function into the outer function, simplify the quadratic without changing its sign, and find its stationary maximum." }, { "id": "cie-2008-w-2-q03", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Recover the constants in an exponential model from a straight-line logarithmic graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates did not attempt the question and stresses that ln y, rather than y, is linear in x.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the vertical intercept and gradient of the ln-transformed graph to find the model constants, keeping the logarithm applied to y rather than y itself." }, { "id": "cie-2008-w-2-q04ii", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find every solution of the trigonometric equation in the given interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says around half obtained an incorrect tangent equation and that the second solution was only rarely found.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the equivalent tangent equation, identify all quadrants allowed by the interval, and list both valid angles." }, { "id": "cie-2008-w-2-q05", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Verify the exact value of a definite integral involving a logarithmic term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates integrated the second term correctly and many used false logarithm rules or decimal approximations where an exact result was required.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate each term using the correct logarithm rule, keep exact logarithms throughout, and simplify the final expression exactly." }, { "id": "cie-2008-w-2-q06", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": null, "marks": 7, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact point where the second derivative of a product function is zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most candidates missed the product rule, and those with a correct first derivative often struggled to obtain the second derivative.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the product rule twice, simplify the second derivative, solve its zero condition, and substitute back to find the exact coordinates." }, { "id": "cie-2008-w-2-q08i", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the two linked identities involving secant and tangent functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says roundabout arguments were common in part (a) and identifies cancelling a sine factor across a sum as a serious error in part (b).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite secant and tangent in terms of sine and cosine, combine over common denominators, use a Pythagorean identity, and do not cancel terms across a sum." }, { "id": "cie-2008-w-2-q08ii", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 8, "questionPart": "8(ii)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate secant using the reciprocal cosine expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says differentiation in this part was often poor and states the required derivative rule.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate 1/cos x using the quotient or reciprocal rule and express the result as sec x tan x." }, { "id": "cie-2008-w-2-q08iii", "paperId": "cie-2008-w-2", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2008, "session": "October/November", "paperCode": "9709/2", "variant": "Variant not numbered", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral using the identities from the earlier parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_2.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_2.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says few integrations were correct and that many substituted the limits into the integrand instead of the antiderivative.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand using the identity already proved, integrate the resulting secant expression, and substitute the limits into the antiderivative." }, { "id": "cie-2008-w-3-q05ii", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Justify the solution set of a polynomial inequality after determining a parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates who found that the quadratic factor had no real roots usually omitted the additional sign argument needed to justify the inequality.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Factor using the known quadratic divisor, locate the real roots, and show the sign of the polynomial on the relevant intervals." }, { "id": "cie-2008-w-3-q06ii", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the double-angle trigonometric equation and retain all solutions in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as one of the least well answered parts; candidates often mishandled the 2θ argument and commonly missed the second solution.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the amplitude form from part (i) with argument 2θ, solve for every allowed angle, and convert each result back to θ." }, { "id": "cie-2008-w-3-q08i", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 8, "questionPart": "8(i)", "marks": 4, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Derive the differential equation describing the tank's changing liquid depth.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates failed to interpret the flow-rate relationship correctly and often treated dh/dt itself as the flow rate.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the volume-depth relation with respect to time and combine the inflow and depth-dependent leakage rates using the chain rule." }, { "id": "cie-2008-w-3-q08ii", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 8, "questionPart": "8(ii)", "marks": 1, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Verify the stated rational-function decomposition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates treated verification at one value as sufficient instead of proving the identity algebraically.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the right-hand terms over a common denominator and simplify the numerator to match the left-hand expression." }, { "id": "cie-2008-w-3-q08iii", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 8, "questionPart": "8(iii)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve the tank-depth differential equation and express time in terms of depth.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says separation attempts were often incorrect and even correct separations frequently failed in using the identity or integrating the resulting expression.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate variables, use the partial-fraction identity from part (ii), integrate both sides, and apply the initial condition for an initially empty tank." }, { "id": "cie-2008-w-3-q10iii", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 10, "questionPart": "10(iii)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Explain geometrically why the three represented complex numbers form an equilateral triangle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this part was rarely answered correctly and identifies confusion between modulus as distance from the origin and side length of the triangle.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the equal moduli and the arguments found earlier to establish equal distances and equal central angles between the vertices." }, { "id": "cie-2008-w-3-q10iv", "paperId": "cie-2008-w-3", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2008, "session": "October/November", "paperCode": "9709/3", "variant": "Variant not numbered", "questionNumber": 10, "questionPart": "10(iv)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the other two vertices of the equilateral triangle from one complex vertex.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_3.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_3.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says hardly any candidates identified the two required rotated numbers and that irrelevant square-root work was common.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rotate the given complex number by the two required angles about the origin and simplify the results into exact Cartesian form." }, { "id": "cie-2008-w-4-q01i", "paperId": "cie-2008-w-4", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2008, "session": "October/November", "paperCode": "9709/4", "variant": "Variant not numbered", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve both components of the resultant force relative to the 10 N force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_4.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_4.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says part (i) was poorly attempted and that candidates often gave the component of one force instead of the resultant, especially in part (a).", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Add the signed components of both forces along and perpendicular to the stated direction, distinguishing the resultant from either force on its own." }, { "id": "cie-2008-w-4-q04", "paperId": "cie-2008-w-4", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2008, "session": "October/November", "paperCode": "9709/4", "variant": "Variant not numbered", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the work done by the crane using the load's change in energy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_4.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_4.pdf", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this question was poorly attempted; many candidates found the displacement incorrectly or used work equals force times distance despite the instruction to use energy.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the vertical displacement from the constant-acceleration information, then equate the crane's work to the changes in gravitational potential and kinetic energy." }, { "id": "cie-2008-w-4-q06ii", "paperId": "cie-2008-w-4", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2008, "session": "October/November", "paperCode": "9709/4", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the train's maximum speed from total distance and journey time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_4.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_4.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was poorly attempted and lists several incorrect speed values caused by treating the accelerating and decelerating stages incorrectly.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the area under the three-stage velocity-time graph, including the triangular acceleration and deceleration sections and the constant-speed section." }, { "id": "cie-2008-w-4-q07ii", "paperId": "cie-2008-w-4", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2008, "session": "October/November", "paperCode": "9709/4", "variant": "Variant not numbered", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance fallen after the acceleration changes at point A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_4.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_4.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates often mishandled both integration constants and the final evaluation time, with many integrating over the wrong interval.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the acceleration to get velocity, apply the matching velocity condition at A, then integrate velocity over the correct post-A time interval and add the first-stage distance." }, { "id": "cie-2008-w-5-q05i", "paperId": "cie-2008-w-5", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2008, "session": "October/November", "paperCode": "9709/5", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the rope tension for the block held in equilibrium by the inclined rope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_5.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_5.pdf", "qpPages": [ 4, 4 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Question 5 the paper's most difficult question and describes a frequent incorrect moment equation that omitted one force's moment contribution.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw a clear force diagram, take moments about the correct corner using perpendicular distances, and resolve only if needed after finding the tension." }, { "id": "cie-2008-w-5-q05ii", "paperId": "cie-2008-w-5", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2008, "session": "October/November", "paperCode": "9709/5", "variant": "Variant not numbered", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction when the block is on the point of slipping.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_5.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_5.pdf", "qpPages": [ 4, 4 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates generally attempted this part more successfully but often used incorrectly calculated friction or normal forces.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use equilibrium to find the horizontal friction force and normal reaction, then take their ratio at limiting equilibrium." }, { "id": "cie-2008-w-5-q06i", "paperId": "cie-2008-w-5", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2008, "session": "October/November", "paperCode": "9709/5", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Derive the speed-extension relationship for the descending elastic-string particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_5.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_5.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates reached the expanded quadratic but skipped the required steps to show the stated completed-square form.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write an energy balance from release to extension x and complete the square in the resulting expression for v squared." }, { "id": "cie-2008-w-5-q06ii", "paperId": "cie-2008-w-5", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2008, "session": "October/November", "paperCode": "9709/5", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance from the fixed point at the lowest position.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_5.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_5.pdf", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates found the extension value but forgot to convert it to the distance from O.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use zero speed to identify the turning-point extension, then add the string's natural length to obtain the distance from O." }, { "id": "cie-2008-w-5-q07iii", "paperId": "cie-2008-w-5", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2008, "session": "October/November", "paperCode": "9709/5", "variant": "Variant not numbered", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the direction of motion as the projectile passes through the specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_5.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_5.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says only a handful differentiated the trajectory from part (i) to obtain the direction of motion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the trajectory equation to obtain its tangent gradient at the point, then convert that gradient to a direction angle and identify whether it is upward or downward." }, { "id": "cie-2008-w-6-q04i", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count distinct arrangements when several house styles repeat.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates treated the houses as distinct and gave 12 factorial instead of accounting for repeated styles.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a multinomial arrangement count, dividing by the factorials for repeated house styles." }, { "id": "cie-2008-w-6-q04ii", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements subject to the two fixed groups of house styles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates often could not find the group counts and, when they did, commonly added instead of multiplying them.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the permitted arrangements within each group separately and multiply the two counts." }, { "id": "cie-2008-w-6-q04iii", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count selections that contain exactly one house from each of two specified styles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies using the wrong remaining pool in the combination calculation as a frequent error.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose one house from each required style, then choose the other two from the remaining eligible houses without double-counting." }, { "id": "cie-2008-w-6-q06i", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Set up the three-stage independent traffic-light tree correctly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the three-stage tree caused difficulty for many candidates and that many received no marks on the question.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw each stage from both prior outcomes and label the stop and no-stop probabilities at every branch." }, { "id": "cie-2008-w-6-q06iv", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 6, "questionPart": "6(iv)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability given that exactly two traffic lights required a stop.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this conditional-probability part was universally poorly done; many candidates found only one numerator case or did not divide by the conditioning probability.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Count the qualifying outcomes in the numerator and divide by the probability of the conditioning event from part (iii)." }, { "id": "cie-2008-w-6-q07i", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability of at least seven odd scores in eight throws.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says weaker candidates often missed either the odd-score probability or the binomial model and frequently mishandled the at-least-seven tail.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the single-throw probability of an odd score, model the number of odd scores binomially, and sum the outcomes 7 and 8." }, { "id": "cie-2008-w-6-q07iv", "paperId": "cie-2008-w-6", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2008, "session": "October/November", "paperCode": "9709/6", "variant": "Variant not numbered", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the two-throw sum exceeds its expected value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_6.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_6.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says this part required more understanding than many candidates showed.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the distribution table and the expectation from part (iii), then sum the probabilities of all values above that expectation." }, { "id": "cie-2008-w-7-q01i", "paperId": "cie-2008-w-7", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2008, "session": "October/November", "paperCode": "9709/7", "variant": "Variant not numbered", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Explain why selecting a child by summing two dice does not give equal chances.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_7.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_7.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a minority attempted the explanation well and warns that simply asserting unequal chances without relating them to the dice outcomes was insufficient.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the number of dice outcomes that produce different child numbers and explain why their selection probabilities are unequal." }, { "id": "cie-2008-w-7-q01ii", "paperId": "cie-2008-w-7", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2008, "session": "October/November", "paperCode": "9709/7", "variant": "Variant not numbered", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Describe a valid method for selecting any one of the eleven children at random.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_7.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_7.pdf", "qpPages": [ 2, 2 ], "msPages": [ 3, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not give enough detail to describe a complete valid selection method.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Give a procedure that assigns an equal number of outcomes to each child, such as using a random integer from 1 to 11 and repeating any excluded result." }, { "id": "cie-2008-w-7-q02", "paperId": "cie-2008-w-7", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2008, "session": "October/November", "paperCode": "9709/7", "variant": "Variant not numbered", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out a two-sided normal hypothesis test for the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_7.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_7.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists common errors including using a one-tail test, using a single observation rather than the sample mean, and omitting the comparison needed to justify the conclusion.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State hypotheses using the population mean, calculate the sample mean and standardized test statistic, compare with the two-sided critical value, and state a justified conclusion." }, { "id": "cie-2008-w-7-q04iii", "paperId": "cie-2008-w-7", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2008, "session": "October/November", "paperCode": "9709/7", "variant": "Variant not numbered", "questionNumber": 4, "questionPart": "4(iii)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Interpret the expected number of confidence intervals that miss the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_qp_7.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w08_ms_7.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was poorly attempted and many candidates who calculated an interval could not interpret its meaning.", "difficultySourceUrl": "https://pastpapers.papacambridge.com/directories/CAIE/CAIE-pastpapers/upload/9709_w08_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the confidence level to find the complementary proportion and multiply it by the number of samples." }, { "id": "cie-2024-s-11-q01a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rewrite a surd expression and solve the related equation exactly. Part 1(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Keep the surd algebra exact, then substitute the transformed expression into the equation and solve." }, { "id": "cie-2024-s-11-q01b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rewrite a surd expression and solve the related equation exactly. Part 1(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Keep the surd algebra exact, then substitute the transformed expression into the equation and solve." }, { "id": "cie-2024-s-11-q02a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe graph transformations and find the transformed function. Part 2(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Name each transformation with its direction and scale; substitute x carefully to obtain the final function." }, { "id": "cie-2024-s-11-q02b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe graph transformations and find the transformed function. Part 2(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Name each transformation with its direction and scale; substitute x carefully to obtain the final function." }, { "id": "cie-2024-s-11-q03a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use binomial coefficients to determine an expansion coefficient. Part 3(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify which binomial terms can contribute to the requested power before multiplying coefficients." }, { "id": "cie-2024-s-11-q03b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use binomial coefficients to determine an expansion coefficient. Part 3(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify which binomial terms can contribute to the requested power before multiplying coefficients." }, { "id": "cie-2024-s-11-q04a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Estimate a derivative from function values and chord gradients. Part 4(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given nearby function values to form chord gradients, then infer the limiting gradient." }, { "id": "cie-2024-s-11-q04b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Estimate a derivative from function values and chord gradients. Part 4(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given nearby function values to form chord gradients, then infer the limiting gradient." }, { "id": "cie-2024-s-11-q04c", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Estimate a derivative from function values and chord gradients. Part 4(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given nearby function values to form chord gradients, then infer the limiting gradient." }, { "id": "cie-2024-s-11-q05a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and solve the resulting equation. Part 5(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Transform the equation using the proved identity and check every solution in the specified interval." }, { "id": "cie-2024-s-11-q05b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and solve the resulting equation. Part 5(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Transform the equation using the proved identity and check every solution in the specified interval." }, { "id": "cie-2024-s-11-q06a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Apply graph transformations, inverses and function equations. Part 6(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For 9709/11, the report notes that only a few candidates gained full marks on the inverse-graph sketch; the y = x mirror line was often missing.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Track domain and range through each transformation before finding inverse or composite-function solutions." }, { "id": "cie-2024-s-11-q06b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Apply graph transformations, inverses and function equations. Part 6(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Track domain and range through each transformation before finding inverse or composite-function solutions." }, { "id": "cie-2024-s-11-q06c", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Apply graph transformations, inverses and function equations. Part 6(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Track domain and range through each transformation before finding inverse or composite-function solutions." }, { "id": "cie-2024-s-11-q06d", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Apply graph transformations, inverses and function equations. Part 6(d).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Track domain and range through each transformation before finding inverse or composite-function solutions." }, { "id": "cie-2024-s-11-q07a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use radian measure and sector geometry to solve a linked geometry problem. Part 7(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use radians consistently and connect arc length, sector area and the given parallel-line geometry." }, { "id": "cie-2024-s-11-q07b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use radian measure and sector geometry to solve a linked geometry problem. Part 7(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use radians consistently and connect arc length, sector area and the given parallel-line geometry." }, { "id": "cie-2024-s-11-q08a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Solve arithmetic and geometric progression problems. Part 8(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the nth-term or sum formula first and check the progression condition before using infinity." }, { "id": "cie-2024-s-11-q08b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Solve arithmetic and geometric progression problems. Part 8(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=13", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the nth-term or sum formula first and check the progression condition before using infinity." }, { "id": "cie-2024-s-11-q09", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find a volume of revolution using integration. Part 9.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Square the radius function in the volume integral and apply the stated limits." }, { "id": "cie-2024-s-11-q10", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10", "marks": 8, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find tangent gradients and contact points for a circle. Part 10.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the circle and use tangency (a repeated root); recover each contact point." }, { "id": "cie-2024-s-11-q11a", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use differentiation to analyse a curve and find a triangle area from tangent and normal lines. Part 11(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate to determine the interval, then write the tangent and normal equations before calculating area." }, { "id": "cie-2024-s-11-q11b", "paperId": "cie-2024-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 8, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use differentiation to analyse a curve and find a triangle area from tangent and normal lines. Part 11(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_11.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_11.pdf#page=18", "qpPages": [ 17, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For 9709/11, the report says many candidates could not complete the tangent and normal equations and area method in this part.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate to determine the interval, then write the tangent and normal equations before calculating area." }, { "id": "cie-2024-s-21-q01", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of a trigonometric curve. Part 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For 9709/21, very few fully correct stationary-point solutions were seen after differentiation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate, set the derivative to zero and use the given interval and accuracy." }, { "id": "cie-2024-s-21-q02", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation to find a curve gradient. Part 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides with respect to x, retaining dy/dx terms, then substitute the point." }, { "id": "cie-2024-s-21-q03a", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch and solve a modulus inequality, then apply it to an exponential inequality. Part 3(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph intersections to obtain the critical values and check the direction of each inequality." }, { "id": "cie-2024-s-21-q03b", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch and solve a modulus inequality, then apply it to an exponential inequality. Part 3(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph intersections to obtain the critical values and check the direction of each inequality." }, { "id": "cie-2024-s-21-q03c", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch and solve a modulus inequality, then apply it to an exponential inequality. Part 3(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph intersections to obtain the critical values and check the direction of each inequality." }, { "id": "cie-2024-s-21-q04a", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove and use a tangent identity to solve a trigonometric equation. Part 4(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity to reduce the equation to a quadratic in tan θ, then check the interval." }, { "id": "cie-2024-s-21-q04b", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove and use a tangent identity to solve a trigonometric equation. Part 4(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity to reduce the equation to a quadratic in tan θ, then check the interval." }, { "id": "cie-2024-s-21-q05a", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Differentiate a curve and locate its stationary point numerically. Part 5(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with the quotient rule, verify a sign change and iterate with the specified formula." }, { "id": "cie-2024-s-21-q05b", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Differentiate a curve and locate its stationary point numerically. Part 5(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with the quotient rule, verify a sign change and iterate with the specified formula." }, { "id": "cie-2024-s-21-q05c", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Differentiate a curve and locate its stationary point numerically. Part 5(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with the quotient rule, verify a sign change and iterate with the specified formula." }, { "id": "cie-2024-s-21-q06a", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Approximate an area with the trapezium rule and find a volume of revolution. Part 6(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For 9709/21, only a small minority produced correct trapezium-rule solutions.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use equal-width ordinates for the trapezium rule; for volume integrate πy² with exact limits." }, { "id": "cie-2024-s-21-q06b", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Approximate an area with the trapezium rule and find a volume of revolution. Part 6(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For 9709/21, the report says very few correct volume solutions were seen; many stopped after finding an integrand and did not use the required double-angle identity.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use equal-width ordinates for the trapezium rule; for volume integrate πy² with exact limits." }, { "id": "cie-2024-s-21-q07a", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide a polynomial and determine constants from a remainder condition. Part 7(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show polynomial division and use the remainder condition to solve for the unknown constants." }, { "id": "cie-2024-s-21-q07b", "paperId": "cie-2024-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide a polynomial and determine constants from a remainder condition. Part 7(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_21.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show polynomial division and use the remainder condition to solve for the unknown constants." }, { "id": "cie-2024-s-31-q01", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a product using the binomial theorem. Part 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose binomial terms by their powers and simplify coefficients only after multiplying." }, { "id": "cie-2024-s-31-q02", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation to a stated accuracy. Part 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms only when domains allow it, then solve and apply the requested rounding." }, { "id": "cie-2024-s-31-q03", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Transform a relation into a straight-line form to determine constants. Part 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms of the relation and identify the linear variables before solving for constants." }, { "id": "cie-2024-s-31-q04a", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Represent complex numbers in polar form and combine them. Part 4(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find modulus and principal argument before applying polar multiplication or division." }, { "id": "cie-2024-s-31-q04b", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Represent complex numbers in polar form and combine them. Part 4(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "For 9709/31, only a small minority knew how to divide complex numbers in polar form.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Find modulus and principal argument before applying polar multiplication or division." }, { "id": "cie-2024-s-31-q05", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a trigonometric-exponential curve and find stationary points. Part 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the product or quotient rule and chain rule, then solve dy/dx = 0 on the given domain." }, { "id": "cie-2024-s-31-q06a", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use graphing, bracketing and iteration to locate a root. Part 6(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the root bracket and each iteration; use the stated fixed-point formula accurately." }, { "id": "cie-2024-s-31-q06b", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use graphing, bracketing and iteration to locate a root. Part 6(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the root bracket and each iteration; use the stated fixed-point formula accurately." }, { "id": "cie-2024-s-31-q06c", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use graphing, bracketing and iteration to locate a root. Part 6(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the root bracket and each iteration; use the stated fixed-point formula accurately." }, { "id": "cie-2024-s-31-q06d", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use graphing, bracketing and iteration to locate a root. Part 6(d).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the root bracket and each iteration; use the stated fixed-point formula accurately." }, { "id": "cie-2024-s-31-q06e", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(e)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use graphing, bracketing and iteration to locate a root. Part 6(e).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the root bracket and each iteration; use the stated fixed-point formula accurately." }, { "id": "cie-2024-s-31-q07a", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use complex loci to find the least distance between points. Part 7(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw each locus correctly, then minimise the distance between a point on each locus." }, { "id": "cie-2024-s-31-q07b", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use complex loci to find the least distance between points. Part 7(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For 9709/31, only a small minority understood that the shortest distance between the complex loci was required.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw each locus correctly, then minimise the distance between a point on each locus." }, { "id": "cie-2024-s-31-q08", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate a trigonometric integral using a substitution. Part 8.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For 9709/31, only a minority showed awareness of the substitution needed for this exact integral.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Make the substitution and change limits or back-substitute consistently; retain exact form." }, { "id": "cie-2024-s-31-q09a", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find line relationships and the image of a point under reflection. Part 9(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use direction vectors for intersections and perpendicularity, then apply the reflection formula." }, { "id": "cie-2024-s-31-q09b", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find line relationships and the image of a point under reflection. Part 9(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use direction vectors for intersections and perpendicularity, then apply the reflection formula." }, { "id": "cie-2024-s-31-q09c", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find line relationships and the image of a point under reflection. Part 9(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "For 9709/31, few responses correctly found the point where the two lines intersect.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use direction vectors for intersections and perpendicularity, then apply the reflection formula." }, { "id": "cie-2024-s-31-q09d", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find line relationships and the image of a point under reflection. Part 9(d).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For 9709/31, the report says very few candidates found the reflected point; most did not reach this final vector step.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use direction vectors for intersections and perpendicularity, then apply the reflection formula." }, { "id": "cie-2024-s-31-q10a", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation and integration by parts. Part 10(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly before integrating by parts; include constants and exact limits." }, { "id": "cie-2024-s-31-q10b", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 7, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation and integration by parts. Part 10(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For 9709/31, only a minority recognised integration by parts and completed the integral.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly before integrating by parts; include constants and exact limits." }, { "id": "cie-2024-s-31-q11a", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Form and solve a differential equation for a growth model. Part 11(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=14", "qpPages": [ 18, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up the rate equation from the context, separate variables and apply the initial condition." }, { "id": "cie-2024-s-31-q11b", "paperId": "cie-2024-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 9, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Form and solve a differential equation for a growth model. Part 11(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_31.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up the rate equation from the context, separate variables and apply the initial condition." }, { "id": "cie-2024-s-41-q01a", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Represent motion with a velocity-time graph and calculate distance. Part 1(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert acceleration and constant-speed stages into a labelled velocity-time graph; use its area for distance." }, { "id": "cie-2024-s-41-q01b", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Represent motion with a velocity-time graph and calculate distance. Part 1(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert acceleration and constant-speed stages into a labelled velocity-time graph; use its area for distance." }, { "id": "cie-2024-s-41-q02a", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces in two directions and find a resultant. Part 2(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force component consistently and state the resultant direction with a reference axis." }, { "id": "cie-2024-s-41-q02b", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces in two directions and find a resultant. Part 2(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force component consistently and state the resultant direction with a reference axis." }, { "id": "cie-2024-s-41-q03", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply work and energy to a train moving up an incline. Part 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For 9709/41, many candidates made little progress on the work-energy problem and few formed the correct energy equation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Balance engine work against the gain in gravitational potential energy and any resistance." }, { "id": "cie-2024-s-41-q04", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use power and Newton’s second law for a car towing a trailer. Part 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv to find the driving force, then apply Newton’s second law to the system." }, { "id": "cie-2024-s-41-q05a", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Analyse motion on a slope with and without friction. Part 5(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve along and normal to the slope; distinguish the friction and no-friction cases." }, { "id": "cie-2024-s-41-q05b", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Analyse motion on a slope with and without friction. Part 5(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For 9709/41, candidates struggled with the friction case and were often distracted by the missing mass, which was not needed.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve along and normal to the slope; distinguish the friction and no-friction cases." }, { "id": "cie-2024-s-41-q06a", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use a velocity function to find a parameter and total distance. Part 6(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity to find acceleration and integrate velocity piecewise for distance." }, { "id": "cie-2024-s-41-q06bi", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use a velocity function to find a parameter and total distance. Part 6(b)(i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity to find acceleration and integrate velocity piecewise for distance." }, { "id": "cie-2024-s-41-q06bii", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use a velocity function to find a parameter and total distance. Part 6(b)(ii).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity to find acceleration and integrate velocity piecewise for distance." }, { "id": "cie-2024-s-41-q07a", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Combine vertical motion and momentum in a collision problem. Part 7(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use constant-acceleration equations for each particle and conservation of momentum at impact." }, { "id": "cie-2024-s-41-q07b", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Combine vertical motion and momentum in a collision problem. Part 7(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use constant-acceleration equations for each particle and conservation of momentum at impact." }, { "id": "cie-2024-s-41-q07c", "paperId": "cie-2024-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Combine vertical motion and momentum in a collision problem. Part 7(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_41.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_41.pdf#page=16", "qpPages": [ 13, 13 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For 9709/41, the report identifies the final collision-height part as demanding; candidates needed to combine impact time with both particles’ motion.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Use constant-acceleration equations for each particle and conservation of momentum at impact." }, { "id": "cie-2024-s-51-q01a", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Recover mean and standard deviation from coded summary statistics. Part 1(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "For 9709/51, the report says many candidates found the coded mean calculation challenging and some left it blank.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Undo the coding transformation before using the summary statistics; use the correct variance identity." }, { "id": "cie-2024-s-51-q01b", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Recover mean and standard deviation from coded summary statistics. Part 1(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For 9709/51, the report says coded summary statistics were challenging, including the standard-deviation calculation.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Undo the coding transformation before using the summary statistics; use the correct variance identity." }, { "id": "cie-2024-s-51-q02a", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal distribution to determine probabilities and an unknown parameter. Part 2(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the required bounds and solve the second probability statement for the unknown parameter." }, { "id": "cie-2024-s-51-q02b", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal distribution to determine probabilities and an unknown parameter. Part 2(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the required bounds and solve the second probability statement for the unknown parameter." }, { "id": "cie-2024-s-51-q03a", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a histogram and analyse quartiles and interquartile range. 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Part 5(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the binomial model first; for the approximation use a continuity correction." }, { "id": "cie-2024-s-51-q05b", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Model snowy days with a binomial distribution and a normal approximation. Part 5(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the binomial model first; for the approximation use a continuity correction." }, { "id": "cie-2024-s-51-q05c", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Model snowy days with a binomial distribution and a normal approximation. Part 5(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the binomial model first; for the approximation use a continuity correction." }, { "id": "cie-2024-s-51-q06a", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Construct a discrete probability distribution and determine an unknown parameter. Part 6(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Ensure probabilities sum to one, then use the requested probability condition to solve for the parameter." }, { "id": "cie-2024-s-51-q06b", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Construct a discrete probability distribution and determine an unknown parameter. Part 6(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For 9709/51, this part was the less successful follow-up in the discrete-random-variable question; candidates often could not start.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Ensure probabilities sum to one, then use the requested probability condition to solve for the parameter." }, { "id": "cie-2024-s-51-q07a", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements and find a probability by considering digit cases. Part 7(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count repeated-digit arrangements without overcounting, then divide favourable cases by all outcomes." }, { "id": "cie-2024-s-51-q07b", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements and find a probability by considering digit cases. Part 7(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count repeated-digit arrangements without overcounting, then divide favourable cases by all outcomes." }, { "id": "cie-2024-s-51-q07c", "paperId": "cie-2024-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements and find a probability by considering digit cases. Part 7(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_51.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_51.pdf#page=16", "qpPages": [ 13, 13 ], "msPages": [ 16, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For 9709/51, most candidates found this probability part demanding; successful solutions enumerated the possible cases without omissions or repeats.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Count repeated-digit arrangements without overcounting, then divide favourable cases by all outcomes." }, { "id": "cie-2024-s-61-q01", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model independent passenger arrivals with a Poisson distribution. Part 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the independent Poisson rates for the shared interval, then show the probability expression." }, { "id": "cie-2024-s-61-q02", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Combine independent normal random variables and standardise. Part 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the mean and variance of the linear combination before standardising." }, { "id": "cie-2024-s-61-q03a", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Work with a confidence interval and conditional probability. Part 3(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the interval using the known standard deviation; distinguish overlap from conditional probability." }, { "id": "cie-2024-s-61-q03b", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Work with a confidence interval and conditional probability. 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Part 3(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "For 9709/61, the report says this conditional-probability part was not well attempted; many used an unconditional probability instead.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the interval using the known standard deviation; distinguish overlap from conditional probability." }, { "id": "cie-2024-s-61-q04a", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates and carry out a one-sided significance test. Part 4(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the unbiased sample-variance denominator and state hypotheses, test statistic and contextual conclusion." }, { "id": "cie-2024-s-61-q04bi", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)(i)", "marks": 5, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates and carry out a one-sided significance test. Part 4(b)(i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the unbiased sample-variance denominator and state hypotheses, test statistic and contextual conclusion." }, { "id": "cie-2024-s-61-q04bii", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates and carry out a one-sided significance test. Part 4(b)(ii).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the unbiased sample-variance denominator and state hypotheses, test statistic and contextual conclusion." }, { "id": "cie-2024-s-61-q05a", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use Poisson assumptions and calculate probabilities over different time periods. Part 5(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "For 9709/61, many candidates could not state the additional constant-rate condition for the Poisson model.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "State the constant-rate assumption, scale λ to the time period and use continuity correction for a normal approximation." }, { "id": "cie-2024-s-61-q05b", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use Poisson assumptions and calculate probabilities over different time periods. Part 5(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the constant-rate assumption, scale λ to the time period and use continuity correction for a normal approximation." }, { "id": "cie-2024-s-61-q05c", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use Poisson assumptions and calculate probabilities over different time periods. Part 5(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the constant-rate assumption, scale λ to the time period and use continuity correction for a normal approximation." }, { "id": "cie-2024-s-61-q06a", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use a probability density function defined by a circular arc and evaluate an expectation. Part 6(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the total density area to one, use the circle equation and integrate xf(x) for expectation." }, { "id": "cie-2024-s-61-q06b", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use a probability density function defined by a circular arc and evaluate an expectation. Part 6(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "For 9709/61, many candidates did not recognise that the circle equation was needed to express the density function.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the total density area to one, use the circle equation and integrate xf(x) for expectation." }, { "id": "cie-2024-s-61-q06c", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use a probability density function defined by a circular arc and evaluate an expectation. Part 6(c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "For 9709/61, candidates often did not complete the expectation integral after identifying xf(x).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the total density area to one, use the circle equation and integrate xf(x) for expectation." }, { "id": "cie-2024-s-61-q07a", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out a binomial significance test, assess a Type I error and approximate with Poisson. Part 7(a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses in terms of p, use the instructed binomial model and explain the error probability in context." }, { "id": "cie-2024-s-61-q07b", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out a binomial significance test, assess a Type I error and approximate with Poisson. Part 7(b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "For 9709/61, few candidates fully justified the Type I error probability.", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "tagConfidence": "reviewed", "answerGuide": "State hypotheses in terms of p, use the instructed binomial model and explain the error probability in context." }, { "id": "cie-2024-s-61-q07ci", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)(i)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out a binomial significance test, assess a Type I error and approximate with Poisson. Part 7(c)(i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses in terms of p, use the instructed binomial model and explain the error probability in context." }, { "id": "cie-2024-s-61-q07cii", "paperId": "cie-2024-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)(ii)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out a binomial significance test, assess a Type I error and approximate with Poisson. Part 7(c)(ii).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_61.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_61.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses in terms of p, use the instructed binomial model and explain the error probability in context." }, { "id": "cie-2024-m-12-q01", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Question 1: Evaluate an improper integral exactly.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate, substitute the finite limit, and evaluate the limit at infinity with correct signs." }, { "id": "cie-2024-m-12-q02a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Question 2(a): Read the minimum of a sine curve in radians, then apply a translation and reflection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 1: the report calls both parts the paper’s most challenging; many candidates misread the graph period or minimum.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the period and amplitude to locate the minimum. Apply each transformation in the stated order to the equation and point." }, { "id": "cie-2024-m-12-q02b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Question 2(b): Read the minimum of a sine curve in radians, then apply a translation and reflection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 1: the report calls both parts the paper’s most challenging; candidates often applied the transformation order incorrectly and rarely gave both requested results.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the period and amplitude to locate the minimum. Apply each transformation in the stated order to the equation and point." }, { "id": "cie-2024-m-12-q03", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Question 3: Integrate a given derivative and use two points on the curve to find an unknown coordinate.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate with a constant, use the known point to find it, then substitute the second x-coordinate." }, { "id": "cie-2024-m-12-q04a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Question 4(a): Prove a trigonometric identity and use it to solve an equation in a restricted interval.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand and use sin²θ + cos²θ = 1; reduce the equation to tan θ, solve every branch, and retain angles in range." }, { "id": "cie-2024-m-12-q04b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Question 4(b): Prove a trigonometric identity and use it to solve an equation in a restricted interval.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand and use sin²θ + cos²θ = 1; reduce the equation to tan θ, solve every branch, and retain angles in range." }, { "id": "cie-2024-m-12-q05", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Question 5: Find the equation of a normal to a curve at a specified point.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate for the tangent gradient, take its negative reciprocal, and use point–gradient form." }, { "id": "cie-2024-m-12-q06", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Question 6: Use a binomial expansion to determine a parameter from a specified coefficient.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the terms that contribute to the required power, combine coefficients, and solve for the allowed parameter." }, { "id": "cie-2024-m-12-q07a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 7(a): Find when a line and curve have one intersection, then find the point of contact.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the curve and set the quadratic discriminant to zero; substitute back to find the point." }, { "id": "cie-2024-m-12-q07b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 7(b): Find when a line and curve have one intersection, then find the point of contact.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the curve and set the quadratic discriminant to zero; substitute back to find the point." }, { "id": "cie-2024-m-12-q08a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Question 8(a): Use arithmetic- and geometric-series information to find finite and infinite sums.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic term and sum formulae. For the even geometric terms identify their first term and ratio, then use the infinite-sum formula." }, { "id": "cie-2024-m-12-q08b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Question 8(b): Use arithmetic- and geometric-series information to find finite and infinite sums.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic term and sum formulae. For the even geometric terms identify their first term and ratio, then use the infinite-sum formula." }, { "id": "cie-2024-m-12-q09a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 9(a): Work with composite functions, ranges, and an inverse function.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compose by substitution, use the minimum of a square to determine the range, and reverse the one-to-one function for its inverse." }, { "id": "cie-2024-m-12-q09b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 9(b): Work with composite functions, ranges, and an inverse function.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compose by substitution, use the minimum of a square to determine the range, and reverse the one-to-one function for its inverse." }, { "id": "cie-2024-m-12-q09c", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 9(c): Work with composite functions, ranges, and an inverse function.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compose by substitution, use the minimum of a square to determine the range, and reverse the one-to-one function for its inverse." }, { "id": "cie-2024-m-12-q10a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 10(a): Use a circle’s centre and radius to find tangent geometry, an angle, perimeter, and area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use perpendicular radius and tangent gradients, centre–radius form, chord geometry, then arc-length and sector/segment formulae." }, { "id": "cie-2024-m-12-q10b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 10(b): Use a circle’s centre and radius to find tangent geometry, an angle, perimeter, and area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use perpendicular radius and tangent gradients, centre–radius form, chord geometry, then arc-length and sector/segment formulae." }, { "id": "cie-2024-m-12-q10c", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 10(c): Use a circle’s centre and radius to find tangent geometry, an angle, perimeter, and area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use perpendicular radius and tangent gradients, centre–radius form, chord geometry, then arc-length and sector/segment formulae." }, { "id": "cie-2024-m-12-q10d", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(d)", "marks": 4, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 10(d): Use a circle’s centre and radius to find tangent geometry, an angle, perimeter, and area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use perpendicular radius and tangent gradients, centre–radius form, chord geometry, then arc-length and sector/segment formulae." }, { "id": "cie-2024-m-12-q11a", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "differentiation", "integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Question 11(a): Find a curve’s minimum, intersections with a horizontal line, and enclosed area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=14", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the fractional-power expression as a quadratic in a substitution; find the required limits and integrate between them." }, { "id": "cie-2024-m-12-q11b", "paperId": "cie-2024-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 7, "topicIds": [ "differentiation", "integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Question 11(b): Find a curve’s minimum, intersections with a horizontal line, and enclosed area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_12.pdf/download#page=15", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_12.pdf/download#page=14", "qpPages": [ 15, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the fractional-power expression as a quadratic in a substitution; find the required limits and integrate between them." }, { "id": "cie-2024-m-22-q01", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials", "numerical-methods" ], "primaryTopicId": "exponentials", "summary": "Question 1: Solve an exponential equation using logarithms to the requested accuracy.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms, use log laws to isolate x, and round only the final value." }, { "id": "cie-2024-m-22-q02a", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 2(a): Sketch a cubic and use intersections to find parameter values giving exactly two real roots.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find and label the intercepts and turning behaviour, then identify the parameter value that gives a tangent intersection." }, { "id": "cie-2024-m-22-q02b", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 2(b): Sketch a cubic and use intersections to find parameter values giving exactly two real roots.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find and label the intercepts and turning behaviour, then identify the parameter value that gives a tangent intersection." }, { "id": "cie-2024-m-22-q03a", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Question 3(a): Factor a polynomial using a known factor, then solve a cosecant equation in an interval.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem to determine the parameter and factor the cubic. Substitute cosec θ, solve for admissible values, and select angles in range." }, { "id": "cie-2024-m-22-q03b", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Question 3(b): Factor a polynomial using a known factor, then solve a cosecant equation in an interval.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem to determine the parameter and factor the cubic. Substitute cosec θ, solve for admissible values, and select angles in range." }, { "id": "cie-2024-m-22-q04a", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Question 4(a): Find an area under an exponential curve and the exact volume when the region is revolved.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve for area; integrate πy² over the same limits for the volume." }, { "id": "cie-2024-m-22-q04b", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Question 4(b): Find an area under an exponential curve and the exact volume when the region is revolved.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve for area; integrate πy² over the same limits for the volume." }, { "id": "cie-2024-m-22-q05a", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(a): Set up and use iteration to locate a curve intersection to three significant figures.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the iteration, verify the root lies in the given interval, then show sufficient iterations and round as requested." }, { "id": "cie-2024-m-22-q05b", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(b): Set up and use iteration to locate a curve intersection to three significant figures.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=8", "qpPages": [ 9, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the iteration, verify the root lies in the given interval, then show sufficient iterations and round as requested." }, { "id": "cie-2024-m-22-q05c", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(c): Set up and use iteration to locate a curve intersection to three significant figures.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the iteration, verify the root lies in the given interval, then show sufficient iterations and round as requested." }, { "id": "cie-2024-m-22-q06a", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Question 6(a): Use a curve diagram to find a tangent gradient and exact coordinates of a related point.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and substitute the specified point; use the curve and diagram conditions to determine the requested coordinates." }, { "id": "cie-2024-m-22-q06b", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Question 6(b): Use a curve diagram to find a tangent gradient and exact coordinates of a related point.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=9", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and substitute the specified point; use the curve and diagram conditions to determine the requested coordinates." }, { "id": "cie-2024-m-22-q06c", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Question 6(c): Use a curve diagram to find a tangent gradient and exact coordinates of a related point.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and substitute the specified point; use the curve and diagram conditions to determine the requested coordinates." }, { "id": "cie-2024-m-22-q07a", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "proof", "integration" ], "primaryTopicId": "trigonometry", "summary": "Question 7(a): Prove a trigonometric identity, evaluate an integral, and solve a related equation.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using sine and cosine and standard identities; integrate the simplified form, then use the identity to solve the equation in range." }, { "id": "cie-2024-m-22-q07b", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry", "proof", "integration" ], "primaryTopicId": "trigonometry", "summary": "Question 7(b): Prove a trigonometric identity, evaluate an integral, and solve a related equation.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=10", "qpPages": [ 13, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using sine and cosine and standard identities; integrate the simplified form, then use the identity to solve the equation in range." }, { "id": "cie-2024-m-22-q07c", "paperId": "cie-2024-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry", "proof", "integration" ], "primaryTopicId": "trigonometry", "summary": "Question 7(c): Prove a trigonometric identity, evaluate an integral, and solve a related equation.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_22.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_22.pdf/download#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using sine and cosine and standard identities; integrate the simplified form, then use the identity to solve the equation in range." }, { "id": "cie-2024-m-32-q01", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 1: Divide a quartic by a quadratic and find quotient and remainder.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Perform polynomial long division in descending powers and state both results." }, { "id": "cie-2024-m-32-q02a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Question 2(a): Find a binomial coefficient and state where the expansion is valid.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the general term to collect the requested power and impose the convergence condition." }, { "id": "cie-2024-m-32-q02b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Question 2(b): Find a binomial coefficient and state where the expansion is valid.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 6: part (b) was challenging and relatively few candidates gained its single mark.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the general term to collect the requested power and impose the convergence condition." }, { "id": "cie-2024-m-32-q03a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Question 3(a): Express a complex square in modulus–argument form and determine related complex values.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert to modulus–argument form, square, and use the realness and modulus conditions to find all permitted values." }, { "id": "cie-2024-m-32-q03b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Question 3(b): Express a complex square in modulus–argument form and determine related complex values.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 6: Question 3 was challenging to most candidates and part (b) is specifically singled out.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert to modulus–argument form, square, and use the realness and modulus conditions to find all permitted values." }, { "id": "cie-2024-m-32-q04", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Question 4: Express a logarithmic combination in terms of two given logarithmic parameters.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 6: Question 4 is explicitly described as difficult.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Express log p and log q from the definitions, then apply log laws." }, { "id": "cie-2024-m-32-q05a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 5(a): Shade an Argand-plane region and find its greatest possible argument.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate each complex inequality into a circle or half-plane, sketch the intersection, and locate the boundary point with greatest argument." }, { "id": "cie-2024-m-32-q05b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 5(b): Shade an Argand-plane region and find its greatest possible argument.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate each complex inequality into a circle or half-plane, sketch the intersection, and locate the boundary point with greatest argument." }, { "id": "cie-2024-m-32-q06a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 6(a): Differentiate an implicitly defined curve and show it has no horizontal tangent.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, rearrange for dy/dx, and show the required numerator cannot be zero on the curve." }, { "id": "cie-2024-m-32-q06b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 6(b): Differentiate an implicitly defined curve and show it has no horizontal tangent.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 6: many candidates found the implicit-differentiation work in part (b) challenging.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, rearrange for dy/dx, and show the required numerator cannot be zero on the curve." }, { "id": "cie-2024-m-32-q07a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 7(a): Use a numerical method to bracket and approximate a root.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the iteration, verify the interval by substitution, and iterate to the stated accuracy." }, { "id": "cie-2024-m-32-q07b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 7(b): Use a numerical method to bracket and approximate a root.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the iteration, verify the interval by substitution, and iterate to the stated accuracy." }, { "id": "cie-2024-m-32-q07c", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 7(c): Use a numerical method to bracket and approximate a root.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the iteration, verify the interval by substitution, and iterate to the stated accuracy." }, { "id": "cie-2024-m-32-q08a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 8(a): Rewrite a sine–cosine expression as one sine function, then solve a trigonometric equation.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match coefficients to R sin(x+α), choose the correct phase, then solve over the stated interval." }, { "id": "cie-2024-m-32-q08b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 8(b): Rewrite a sine–cosine expression as one sine function, then solve a trigonometric equation.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 7: part (b) was challenging for most candidates, with few successful solutions.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Match coefficients to R sin(x+α), choose the correct phase, then solve over the stated interval." }, { "id": "cie-2024-m-32-q09a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 9(a): Show a vector quadrilateral is a rectangle and find the angle between its diagonals.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=14", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 7: only a minority established the extra right-angle condition needed to prove a rectangle.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form side vectors and verify opposite sides and perpendicular adjacent sides; use a diagonal-vector dot product for the angle." }, { "id": "cie-2024-m-32-q09b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 9(b): Show a vector quadrilateral is a rectangle and find the angle between its diagonals.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=15", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=15", "qpPages": [ 15, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form side vectors and verify opposite sides and perpendicular adjacent sides; use a diagonal-vector dot product for the angle." }, { "id": "cie-2024-m-32-q10a", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Question 10(a): Express a rational function in partial fractions and evaluate a definite integral.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=16", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find partial-fraction constants, integrate the logarithmic terms, and apply the bounds." }, { "id": "cie-2024-m-32-q10b", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Question 10(b): Express a rational function in partial fractions and evaluate a definite integral.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=17", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=17", "qpPages": [ 17, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find partial-fraction constants, integrate the logarithmic terms, and apply the bounds." }, { "id": "cie-2024-m-32-q11", "paperId": "cie-2024-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11", "marks": 9, "topicIds": [ "integration", "differentiation", "trigonometry" ], "primaryTopicId": "integration", "summary": "Question 11: Solve a separable differential equation with an initial condition and find a trigonometric value.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_32.pdf/download#page=18", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_32.pdf/download#page=18", "qpPages": [ 18, 18 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate, apply the initial condition, then substitute the target value and evaluate tan θ." }, { "id": "cie-2024-m-42-q01a", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics", "data-analysis" ], "primaryTopicId": "kinematics", "summary": "Question 1(a): Interpret a velocity–time data diagram to find motion quantities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the plotted values; use gradient for acceleration and area for displacement or distance." }, { "id": "cie-2024-m-42-q01b", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "kinematics", "data-analysis" ], "primaryTopicId": "kinematics", "summary": "Question 1(b): Interpret a velocity–time data diagram to find motion quantities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the plotted values; use gradient for acceleration and area for displacement or distance." }, { "id": "cie-2024-m-42-q02a", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 2(a): Use constant acceleration for a particle projected vertically upwards.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use signed velocity in v=u+at to find the projection speed, then a constant-acceleration relation for distance." }, { "id": "cie-2024-m-42-q02b", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 2(b): Use constant acceleration for a particle projected vertically upwards.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 9: the distance travelled during the vertical projection is identified as a challenging step.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use signed velocity in v=u+at to find the projection speed, then a constant-acceleration relation for distance." }, { "id": "cie-2024-m-42-q03", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 3: Model a crate pulled up a rough plane at constant speed using power and force balance.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use power = force × speed, resolve weight and friction along the plane, and apply constant speed." }, { "id": "cie-2024-m-42-q04", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 4: Resolve forces on a particle in equilibrium.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces into perpendicular components, set each resultant to zero, and solve." }, { "id": "cie-2024-m-42-q05a", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics", "differentiation", "integration" ], "primaryTopicId": "kinematics", "summary": "Question 5(a): Use a velocity function to find displacement, acceleration, and total distance.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate for displacement, differentiate for acceleration, solve for the required time, and split distance where velocity changes sign." }, { "id": "cie-2024-m-42-q05b", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics", "differentiation", "integration" ], "primaryTopicId": "kinematics", "summary": "Question 5(b): Use a velocity function to find displacement, acceleration, and total distance.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate for displacement, differentiate for acceleration, solve for the required time, and split distance where velocity changes sign." }, { "id": "cie-2024-m-42-q06a", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 6(a): Model a car towing a trailer, then use work–energy to find a speed.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to the combined system and individual bodies; equate net work to change in kinetic energy." }, { "id": "cie-2024-m-42-q06b", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 6(b): Model a car towing a trailer, then use work–energy to find a speed.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 10: candidates had difficulty using work and energy for the towing system in part (b).", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to the combined system and individual bodies; equate net work to change in kinetic energy." }, { "id": "cie-2024-m-42-q07a", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "mechanics", "kinematics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 7(a): Analyse particles on an inclined plane before and after a collision.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces for accelerations, use constant-acceleration equations, apply momentum at coalescence, then find post-collision time." }, { "id": "cie-2024-m-42-q07b", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 7(b): Analyse particles on an inclined plane before and after a collision.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=20", "qpPages": [ 10, 10 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 11: candidates found the second particle’s motion and collision setup in part (b) challenging.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces for accelerations, use constant-acceleration equations, apply momentum at coalescence, then find post-collision time." }, { "id": "cie-2024-m-42-q07c", "paperId": "cie-2024-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "mechanics", "kinematics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 7(c): Analyse particles on an inclined plane before and after a collision.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_42.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_42.pdf/download#page=22", "qpPages": [ 11, 11 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 11: the final post-collision motion in part (c) is identified as challenging.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve forces for accelerations, use constant-acceleration equations, apply momentum at coalescence, then find post-collision time." }, { "id": "cie-2024-m-52-q01a", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 1(a): Find conditional probabilities for successive draws with a colour-dependent replacement rule.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Update the bag after the first draw and add the mutually exclusive routes for the second draw." }, { "id": "cie-2024-m-52-q01b", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 1(b): Find conditional probabilities for successive draws with a colour-dependent replacement rule.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Update the bag after the first draw and add the mutually exclusive routes for the second draw." }, { "id": "cie-2024-m-52-q02a", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 2(a): Use binomial and geometric distributions for independent goal attempts.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model for fixed attempts and a geometric/negative-binomial sequence for the third success." }, { "id": "cie-2024-m-52-q02b", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 2(b): Use binomial and geometric distributions for independent goal attempts.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model for fixed attempts and a geometric/negative-binomial sequence for the third success." }, { "id": "cie-2024-m-52-q02c", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 2(c): Use binomial and geometric distributions for independent goal attempts.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 13: the stopping-time context in part (c) was difficult to interpret.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a binomial model for fixed attempts and a geometric/negative-binomial sequence for the third success." }, { "id": "cie-2024-m-52-q03a", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 3(a): Construct a cumulative-frequency graph, estimate the mean, and identify a quartile class.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use class boundaries and cumulative totals for the graph, class midpoints for the mean, and locate the lower-quartile rank." }, { "id": "cie-2024-m-52-q03b", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 3(b): Construct a cumulative-frequency graph, estimate the mean, and identify a quartile class.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use class boundaries and cumulative totals for the graph, class midpoints for the mean, and locate the lower-quartile rank." }, { "id": "cie-2024-m-52-q03c", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 3(c): Construct a cumulative-frequency graph, estimate the mean, and identify a quartile class.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use class boundaries and cumulative totals for the graph, class midpoints for the mean, and locate the lower-quartile rank." }, { "id": "cie-2024-m-52-q04a", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 4(a): Apply normal and binomial models to small and large bag masses.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable; use a suitable binomial-to-normal approximation with continuity correction." }, { "id": "cie-2024-m-52-q04b", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 4(b): Apply normal and binomial models to small and large bag masses.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable; use a suitable binomial-to-normal approximation with continuity correction." }, { "id": "cie-2024-m-52-q04c", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 4(c): Apply normal and binomial models to small and large bag masses.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable; use a suitable binomial-to-normal approximation with continuity correction." }, { "id": "cie-2024-m-52-q05a", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 5(a): Model game scores from win, draw, and loss probabilities and find moments.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 14: the game-score context and first probability calculation caused difficulty.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Enumerate mutually exclusive outcomes to form the score distribution, then calculate the requested expectation." }, { "id": "cie-2024-m-52-q05b", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 5(b): Model game scores from win, draw, and loss probabilities and find moments.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate mutually exclusive outcomes to form the score distribution, then calculate the requested expectation." }, { "id": "cie-2024-m-52-q05c", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 5(c): Model game scores from win, draw, and loss probabilities and find moments.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate mutually exclusive outcomes to form the score distribution, then calculate the requested expectation." }, { "id": "cie-2024-m-52-q06a", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(a): Count committees and arrangements subject to gender and adjacency conditions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use combinations for the committee; treat adjacent people as a block or fix their positions for arrangements." }, { "id": "cie-2024-m-52-q06b", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(b): Count committees and arrangements subject to gender and adjacency conditions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use combinations for the committee; treat adjacent people as a block or fix their positions for arrangements." }, { "id": "cie-2024-m-52-q06c", "paperId": "cie-2024-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(c): Count committees and arrangements subject to gender and adjacency conditions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_52.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_52.pdf/download#page=16", "qpPages": [ 13, 13 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 15: the arrangements in part (c) are explicitly described as challenging.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use combinations for the committee; treat adjacent people as a block or fix their positions for arrangements." }, { "id": "cie-2024-m-62-q01a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 1(a): Estimate a population mean and variance, then state a reliability condition.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sample mean and corrected variance with n−1; state an appropriate sampling/size condition." }, { "id": "cie-2024-m-62-q01b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 1(b): Estimate a population mean and variance, then state a reliability condition.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sample mean and corrected variance with n−1; state an appropriate sampling/size condition." }, { "id": "cie-2024-m-62-q02a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 2(a): Construct a 98% confidence interval for a proportion and assess a claim.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal interval formula; compare the claimed proportion with the interval and state a conclusion." }, { "id": "cie-2024-m-62-q02b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 2(b): Construct a 98% confidence interval for a proportion and assess a claim.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal interval formula; compare the claimed proportion with the interval and state a conclusion." }, { "id": "cie-2024-m-62-q03a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 3(a): Use a Poisson approximation for rare winning tickets and justify it.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use mean np for the Poisson approximation, calculate the tail, and justify with large n and small p." }, { "id": "cie-2024-m-62-q03b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 3(b): Use a Poisson approximation for rare winning tickets and justify it.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use mean np for the Poisson approximation, calculate the tail, and justify with large n and small p." }, { "id": "cie-2024-m-62-q04a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 4(a): Combine independent normal fuel quantities to calculate amount and cost probabilities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine means and variances for the difference or weighted sum, standardise, and evaluate the normal probabilities." }, { "id": "cie-2024-m-62-q04b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 4(b): Combine independent normal fuel quantities to calculate amount and cost probabilities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine means and variances for the difference or weighted sum, standardise, and evaluate the normal probabilities." }, { "id": "cie-2024-m-62-q05a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Question 5(a): Model independent Poisson arrival counts, then test whether a rate increased.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add Poisson rates over days, state hypotheses, and calculate the relevant upper-tail probability at 5%." }, { "id": "cie-2024-m-62-q05b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Question 5(b): Model independent Poisson arrival counts, then test whether a rate increased.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add Poisson rates over days, state hypotheses, and calculate the relevant upper-tail probability at 5%." }, { "id": "cie-2024-m-62-q05c", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Question 5(c): Model independent Poisson arrival counts, then test whether a rate increased.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 17: the distribution work in part (c) was particularly poorly attempted.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Add Poisson rates over days, state hypotheses, and calculate the relevant upper-tail probability at 5%." }, { "id": "cie-2024-m-62-q05d", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Question 5(d): Model independent Poisson arrival counts, then test whether a rate increased.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add Poisson rates over days, state hypotheses, and calculate the relevant upper-tail probability at 5%." }, { "id": "cie-2024-m-62-q06a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "functions", "integration", "data-analysis" ], "primaryTopicId": "functions", "summary": "Question 6(a): Use symmetry and a probability density to find probabilities, a constant, and a variance.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use symmetry for reflected probabilities, integrate the density for its constant, then calculate E(X), E(X²), and variance." }, { "id": "cie-2024-m-62-q06b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "functions", "integration", "data-analysis" ], "primaryTopicId": "functions", "summary": "Question 6(b): Use symmetry and a probability density to find probabilities, a constant, and a variance.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use symmetry for reflected probabilities, integrate the density for its constant, then calculate E(X), E(X²), and variance." }, { "id": "cie-2024-m-62-q06c", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "functions", "integration", "data-analysis" ], "primaryTopicId": "functions", "summary": "Question 6(c): Use symmetry and a probability density to find probabilities, a constant, and a variance.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=13", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 17: candidates had difficulty applying the domain and moments in the variance calculation.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use symmetry for reflected probabilities, integrate the density for its constant, then calculate E(X), E(X²), and variance." }, { "id": "cie-2024-m-62-q07a", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 7(a): Carry out a one-sided test of a population mean against a historical benchmark.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0 and H1, standardise using the standard error, compare with the 1% critical value, and conclude in context." }, { "id": "cie-2024-m-62-q07b", "paperId": "cie-2024-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 7(b): Carry out a one-sided test of a population mean against a historical benchmark.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_qp_62.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2024-march/9709_m24_ms_62.pdf/download#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2024, physical p. 18: the final test calculation in part (b) is identified as challenging.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2024/9709_m24_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State H0 and H1, standardise using the standard error, compare with the 1% critical value, and conclude in context." }, { "id": "cie-2025-m-12-q01", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Question 1: Find parameter values for which a line and quadratic curve do not intersect.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the curve and require a negative quadratic discriminant." }, { "id": "cie-2025-m-12-q02a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Question 2(a): Differentiate a curve at an x-axis crossing and use its local geometry.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and substitute the point for the gradient; use the curve and minimum-point conditions for the remaining result." }, { "id": "cie-2025-m-12-q02b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Question 2(b): Differentiate a curve at an x-axis crossing and use its local geometry.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and substitute the point for the gradient; use the curve and minimum-point conditions for the remaining result." }, { "id": "cie-2025-m-12-q03a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Question 3(a): Expand a binomial and use it to extract a coefficient in another expansion.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial theorem and identify the products contributing to x²." }, { "id": "cie-2025-m-12-q03b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Question 3(b): Expand a binomial and use it to extract a coefficient in another expansion.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial theorem and identify the products contributing to x²." }, { "id": "cie-2025-m-12-q04a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 4(a): Find perimeter and area of a shaded region partly bounded by a circular arc.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use rθ and the straight segments for perimeter; combine the triangle and sector areas for the shaded area." }, { "id": "cie-2025-m-12-q04b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 4(b): Find perimeter and area of a shaded region partly bounded by a circular arc.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use rθ and the straight segments for perimeter; combine the triangle and sector areas for the shaded area." }, { "id": "cie-2025-m-12-q05", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Question 5: Find the sum of arithmetic-progression terms lying between two bounds.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 2: many candidates rejected the valid negative progression parameter.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Find the first and last term indices satisfying the strict bounds, then use the arithmetic-series sum." }, { "id": "cie-2025-m-12-q06a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 6(a): Use a circle’s distances from the axes and centre distance to find its centre and radius.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the centre coordinates from the axis distances and use Pythagoras with the origin-to-centre distance." }, { "id": "cie-2025-m-12-q06b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 6(b): Use a circle’s distances from the axes and centre distance to find its centre and radius.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the centre coordinates from the axis distances and use Pythagoras with the origin-to-centre distance." }, { "id": "cie-2025-m-12-q06c", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 6(c): Use a circle’s distances from the axes and centre distance to find its centre and radius.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the centre coordinates from the axis distances and use Pythagoras with the origin-to-centre distance." }, { "id": "cie-2025-m-12-q07a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Question 7(a): Prove a trigonometric identity and solve the resulting equation in an interval.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use standard identities to prove the result, then solve all branches and retain angles in range." }, { "id": "cie-2025-m-12-q07b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Question 7(b): Prove a trigonometric identity and solve the resulting equation in an interval.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 2: many candidates missed required solutions in the trigonometric equation.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Use standard identities to prove the result, then solve all branches and retain angles in range." }, { "id": "cie-2025-m-12-q08a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Question 8(a): Determine a geometric progression from one term and its infinite sum, then remove initial terms.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 3: a significant majority rejected the valid negative common ratio.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Solve using ar and a/(1−r), then subtract the first nine terms from the infinite sum." }, { "id": "cie-2025-m-12-q08b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Question 8(b): Determine a geometric progression from one term and its infinite sum, then remove initial terms.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve using ar and a/(1−r), then subtract the first nine terms from the infinite sum." }, { "id": "cie-2025-m-12-q09a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 9(a): Classify a stationary point using the second derivative and evaluate a bounded area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=17", "qpPages": [ 11, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the second derivative sign at the stationary point; find curve limits and integrate for the area." }, { "id": "cie-2025-m-12-q09b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 9(b): Classify a stationary point using the second derivative and evaluate a bounded area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=18", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 3: the report highlights recurring errors with the integration constants and follow-on curve work.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Use the second derivative sign at the stationary point; find curve limits and integrate for the area." }, { "id": "cie-2025-m-12-q10a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Question 10(a): Use a tangent and curve to determine an exact region area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=19", "qpPages": [ 12, 12 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the tangent equation, establish intersection limits, and integrate the difference between curve and line." }, { "id": "cie-2025-m-12-q10b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Question 10(b): Use a tangent and curve to determine an exact region area.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=20", "qpPages": [ 13, 13 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 3: only stronger candidates found the follow-on area calculation straightforward.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Find the tangent equation, establish intersection limits, and integrate the difference between curve and line." }, { "id": "cie-2025-m-12-q11a", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 11(a): Find composite-function properties and transform a curve by translation, stretch, and reflection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=14", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=21", "qpPages": [ 14, 14 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compose by substitution, use the given constant relation, and apply each transformation in order." }, { "id": "cie-2025-m-12-q11b", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 11(b): Find composite-function properties and transform a curve by translation, stretch, and reflection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=15", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=22", "qpPages": [ 15, 15 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compose by substitution, use the given constant relation, and apply each transformation in order." }, { "id": "cie-2025-m-12-q11c", "paperId": "cie-2025-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 11(c): Find composite-function properties and transform a curve by translation, stretch, and reflection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_12.pdf/download#page=15", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_12.pdf/download#page=22", "qpPages": [ 15, 15 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 4: part (c) is called very demanding for most candidates.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Compose by substitution, use the given constant relation, and apply each transformation in order." }, { "id": "cie-2025-m-22-q01", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Question 1: Solve a logarithmic equation using log laws.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms, simplify, and solve while respecting the domain." }, { "id": "cie-2025-m-22-q02", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Question 2: Recover a curve equation from its derivative and a point.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate with a constant and substitute the point to determine it." }, { "id": "cie-2025-m-22-q03a", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Question 3(a): Find the area enclosed by two exponential curves and the y-axis.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find their intersection, identify the upper curve, and integrate their difference over the bounded interval." }, { "id": "cie-2025-m-22-q03b", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Question 3(b): Find the area enclosed by two exponential curves and the y-axis.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find their intersection, identify the upper curve, and integrate their difference over the bounded interval." }, { "id": "cie-2025-m-22-q04a", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Question 4(a): Differentiate a trigonometric curve and find its stationary points.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with the chain rule, solve dy/dx=0, then substitute for each y-coordinate." }, { "id": "cie-2025-m-22-q04b", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Question 4(b): Differentiate a trigonometric curve and find its stationary points.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with the chain rule, solve dy/dx=0, then substitute for each y-coordinate." }, { "id": "cie-2025-m-22-q05a", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "numerical-methods", "functions", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(a): Sketch intersecting exponential/logarithmic graphs and iterate to approximate an intersection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to the stated iteration, sketch and label intersections, then iterate with sufficient precision." }, { "id": "cie-2025-m-22-q05b", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "numerical-methods", "functions", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(b): Sketch intersecting exponential/logarithmic graphs and iterate to approximate an intersection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to the stated iteration, sketch and label intersections, then iterate with sufficient precision." }, { "id": "cie-2025-m-22-q05c", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods", "functions", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(c): Sketch intersecting exponential/logarithmic graphs and iterate to approximate an intersection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to the stated iteration, sketch and label intersections, then iterate with sufficient precision." }, { "id": "cie-2025-m-22-q05d", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "numerical-methods", "functions", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(d): Sketch intersecting exponential/logarithmic graphs and iterate to approximate an intersection.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to the stated iteration, sketch and label intersections, then iterate with sufficient precision." }, { "id": "cie-2025-m-22-q06a", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra", "integration", "exponentials" ], "primaryTopicId": "algebra", "summary": "Question 6(a): Divide a polynomial and use the result to evaluate an exact logarithmic integral.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use polynomial division to simplify the integrand, integrate each term, and apply the limits." }, { "id": "cie-2025-m-22-q06b", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "algebra", "integration", "exponentials" ], "primaryTopicId": "algebra", "summary": "Question 6(b): Divide a polynomial and use the result to evaluate an exact logarithmic integral.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use polynomial division to simplify the integrand, integrate each term, and apply the limits." }, { "id": "cie-2025-m-22-q07a", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 7(a): Express a trig sum in single-angle form, solve it, and maximise another expression.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match to R sin(θ−α), choose the correct phase, solve in range, and use amplitude/phase form for the maximum." }, { "id": "cie-2025-m-22-q07b", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 7(b): Express a trig sum in single-angle form, solve it, and maximise another expression.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match to R sin(θ−α), choose the correct phase, solve in range, and use amplitude/phase form for the maximum." }, { "id": "cie-2025-m-22-q07c", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 7(c): Express a trig sum in single-angle form, solve it, and maximise another expression.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match to R sin(θ−α), choose the correct phase, solve in range, and use amplitude/phase form for the maximum." }, { "id": "cie-2025-m-22-q08a", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Question 8(a): Differentiate a curve involving exponentials and show there are no stationary points.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply product and chain rules, factor the derivative, and show dy/dx=0 has no solution." }, { "id": "cie-2025-m-22-q08b", "paperId": "cie-2025-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Question 8(b): Differentiate a curve involving exponentials and show there are no stationary points.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_22.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_22.pdf/download#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply product and chain rules, factor the derivative, and show dy/dx=0 has no solution." }, { "id": "cie-2025-m-32-q01", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "numerical-methods" ], "primaryTopicId": "exponentials", "summary": "Question 1: Solve an exponential equation to four decimal places.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 6: a significant number struggled with the logarithmic equation.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential, take logarithms, and round only at the end." }, { "id": "cie-2025-m-32-q02", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 2: Find the gradient of an implicitly defined curve at x=0.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 6: relatively few candidates gained full marks on this question.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, collect dy/dx terms, and substitute x=0 and the corresponding curve point." }, { "id": "cie-2025-m-32-q03a", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 3(a): Interpret a complex-number region and determine a greatest argument.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 7: part (a) was challenging and relatively few candidates gained full marks.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Sketch the loci and their intersection, identify the boundary point with extremal argument, and calculate its argument." }, { "id": "cie-2025-m-32-q03b", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 3(b): Interpret a complex-number region and determine a greatest argument.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 7: part (b) was challenging, with many candidates not attempting it and only very few gaining full marks.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Sketch the loci and their intersection, identify the boundary point with extremal argument, and calculate its argument." }, { "id": "cie-2025-m-32-q04", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Question 4: Solve a trigonometric equation by forming a quadratic in tan x.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the double-angle identity, solve the quadratic, and select angles in the stated interval." }, { "id": "cie-2025-m-32-q05", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 5: Find both square roots of a complex number in exact Cartesian form.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set (x+iy)² equal to the given number, equate real and imaginary parts, and solve for both roots." }, { "id": "cie-2025-m-32-q06", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Question 6: Solve a separable differential equation with an initial condition.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate, use the initial condition, and rearrange for x." }, { "id": "cie-2025-m-32-q07a", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 7(a): Set up an iteration for a root, verify a bracket, and approximate the root.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Obtain the iteration, check the endpoints, and iterate to the requested precision." }, { "id": "cie-2025-m-32-q07b", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 7(b): Set up an iteration for a root, verify a bracket, and approximate the root.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=15", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Obtain the iteration, check the endpoints, and iterate to the requested precision." }, { "id": "cie-2025-m-32-q07c", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Question 7(c): Set up an iteration for a root, verify a bracket, and approximate the root.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=16", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Obtain the iteration, check the endpoints, and iterate to the requested precision." }, { "id": "cie-2025-m-32-q08a", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 8(a): Find where two 3D lines intersect and the angle between their directions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=17", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 8: the report records many arithmetic errors when candidates tried to prove the two lines were skew.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Equate components to find line parameters, then use a direction-vector dot product." }, { "id": "cie-2025-m-32-q08b", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 8(b): Find where two 3D lines intersect and the angle between their directions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=18", "qpPages": [ 13, 13 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate components to find line parameters, then use a direction-vector dot product." }, { "id": "cie-2025-m-32-q09a", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Question 9(a): Use factor and derivative remainder conditions to find coefficients, factor a polynomial, and solve an inequality.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=14", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=18", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p(3)=0 and the derivative remainder condition; factor the result and form a sign chart." }, { "id": "cie-2025-m-32-q09b", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Question 9(b): Use factor and derivative remainder conditions to find coefficients, factor a polynomial, and solve an inequality.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=15", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=19", "qpPages": [ 15, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p(3)=0 and the derivative remainder condition; factor the result and form a sign chart." }, { "id": "cie-2025-m-32-q09c", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Question 9(c): Use factor and derivative remainder conditions to find coefficients, factor a polynomial, and solve an inequality.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=15", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=19", "qpPages": [ 15, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 8: very few candidates were fully successful in the final inequality part.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Use p(3)=0 and the derivative remainder condition; factor the result and form a sign chart." }, { "id": "cie-2025-m-32-q10a", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Question 10(a): Use partial fractions to evaluate a definite integral exactly.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=16", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the constants, integrate the resulting logarithmic terms, and substitute the limits." }, { "id": "cie-2025-m-32-q10b", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Question 10(b): Use partial fractions to evaluate a definite integral exactly.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=17", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=21", "qpPages": [ 17, 17 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 8: common problems occurred in the follow-on integral in part (b).", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Find the constants, integrate the resulting logarithmic terms, and substitute the limits." }, { "id": "cie-2025-m-32-q11", "paperId": "cie-2025-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Question 11: Evaluate a trigonometric integral exactly.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_32.pdf/download#page=18", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_32.pdf/download#page=22", "qpPages": [ 18, 18 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use integration by parts or an identity, then evaluate the exact antiderivative at the endpoints." }, { "id": "cie-2025-m-42-q01", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 1: Resolve three coplanar forces to determine an unknown force or direction.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces into perpendicular components and set both resultant components to zero." }, { "id": "cie-2025-m-42-q02a", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 2(a): Use a staged-motion model for a cyclist who accelerates, travels steadily, and decelerates.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v²=u²+2as for the changing-speed stages and s=vt for uniform motion." }, { "id": "cie-2025-m-42-q02b", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 2(b): Use a staged-motion model for a cyclist who accelerates, travels steadily, and decelerates.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v²=u²+2as for the changing-speed stages and s=vt for uniform motion." }, { "id": "cie-2025-m-42-q03a", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 3(a): Use power and resistance for an aircraft in level flight, then account for a climb.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P=Fv to find speed; balance engine work against resistance work and the gain in gravitational potential energy." }, { "id": "cie-2025-m-42-q03b", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 3(b): Use power and resistance for an aircraft in level flight, then account for a climb.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P=Fv to find speed; balance engine work against resistance work and the gain in gravitational potential energy." }, { "id": "cie-2025-m-42-q04a", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 4(a): Analyse a two-mass pulley system and infer the initial height from later motion.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to both masses, then use constant-acceleration relations and the maximum-height condition." }, { "id": "cie-2025-m-42-q04b", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 4(b): Analyse a two-mass pulley system and infer the initial height from later motion.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 11: part (b) is described as the most challenging part of the paper.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to both masses, then use constant-acceleration relations and the maximum-height condition." }, { "id": "cie-2025-m-42-q05a", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Question 5(a): Track particles through successive collisions and find when one meets the combined particle.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of momentum at each collision, then solve the constant-acceleration equations for the meeting time." }, { "id": "cie-2025-m-42-q05b", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Question 5(b): Track particles through successive collisions and find when one meets the combined particle.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of momentum at each collision, then solve the constant-acceleration equations for the meeting time." }, { "id": "cie-2025-m-42-q05c", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Question 5(c): Track particles through successive collisions and find when one meets the combined particle.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 12: the final timing and collision calculation was difficult for a significant number of candidates.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Use conservation of momentum at each collision, then solve the constant-acceleration equations for the meeting time." }, { "id": "cie-2025-m-42-q06a", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 6(a): Resolve forces on a block on a rough inclined plane for limiting equilibrium.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and perpendicular to the plane, use limiting friction μR in the correct direction, and solve each case." }, { "id": "cie-2025-m-42-q06b", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 6(b): Resolve forces on a block on a rough inclined plane for limiting equilibrium.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=16", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and perpendicular to the plane, use limiting friction μR in the correct direction, and solve each case." }, { "id": "cie-2025-m-42-q07a", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "kinematics", "differentiation", "integration" ], "primaryTopicId": "kinematics", "summary": "Question 7(a): Use acceleration and displacement data to find constants in a velocity function, then find distance between events.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=17", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and integrate velocity to determine constants; find event times and integrate speed between them." }, { "id": "cie-2025-m-42-q07b", "paperId": "cie-2025-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics", "differentiation", "integration" ], "primaryTopicId": "kinematics", "summary": "Question 7(b): Use acceleration and displacement data to find constants in a velocity function, then find distance between events.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_42.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_42.pdf/download#page=18", "qpPages": [ 13, 13 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and integrate velocity to determine constants; find event times and integrate speed between them." }, { "id": "cie-2025-m-52-q01a", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 1(a): Build the head-count distribution from three differently biased coins and calculate moments.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate outcomes and combine equal head counts, then calculate E(X) and Var(X)." }, { "id": "cie-2025-m-52-q01b", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 1(b): Build the head-count distribution from three differently biased coins and calculate moments.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=2", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate outcomes and combine equal head counts, then calculate E(X) and Var(X)." }, { "id": "cie-2025-m-52-q01c", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 1(c): Build the head-count distribution from three differently biased coins and calculate moments.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate outcomes and combine equal head counts, then calculate E(X) and Var(X)." }, { "id": "cie-2025-m-52-q02a", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 2(a): Use multinomial probabilities for three customers and a binomial approximation for a large sample.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 14: the online-store context and outcome setup were challenging.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Multiply category probabilities for the small sample; use a binomial model and normal approximation with continuity correction for the large sample." }, { "id": "cie-2025-m-52-q02b", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 2(b): Use multinomial probabilities for three customers and a binomial approximation for a large sample.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply category probabilities for the small sample; use a binomial model and normal approximation with continuity correction for the large sample." }, { "id": "cie-2025-m-52-q02c", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 2(c): Use multinomial probabilities for three customers and a binomial approximation for a large sample.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply category probabilities for the small sample; use a binomial model and normal approximation with continuity correction for the large sample." }, { "id": "cie-2025-m-52-q03a", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 3(a): Draw a cumulative-frequency graph, estimate a percentile, and calculate a grouped mean.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 14: the report says cumulative-frequency graph quality was disappointing.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Plot cumulative frequencies at class boundaries, interpolate the percentile, and use midpoints for the mean." }, { "id": "cie-2025-m-52-q03b", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 3(b): Draw a cumulative-frequency graph, estimate a percentile, and calculate a grouped mean.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative frequencies at class boundaries, interpolate the percentile, and use midpoints for the mean." }, { "id": "cie-2025-m-52-q03c", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 3(c): Draw a cumulative-frequency graph, estimate a percentile, and calculate a grouped mean.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative frequencies at class boundaries, interpolate the percentile, and use midpoints for the mean." }, { "id": "cie-2025-m-52-q04a", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 4(a): Find probabilities for three draws without replacement under colour restrictions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use combinations or sequential conditional probabilities for each allowed colour outcome." }, { "id": "cie-2025-m-52-q04b", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 4(b): Find probabilities for three draws without replacement under colour restrictions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use combinations or sequential conditional probabilities for each allowed colour outcome." }, { "id": "cie-2025-m-52-q05a", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 5(a): Apply normal and geometric models to daily fruit sales.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable, infer the standard deviation, and use the geometric waiting-time formula over seven days." }, { "id": "cie-2025-m-52-q05b", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 5(b): Apply normal and geometric models to daily fruit sales.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable, infer the standard deviation, and use the geometric waiting-time formula over seven days." }, { "id": "cie-2025-m-52-q05c", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 5(c): Apply normal and geometric models to daily fruit sales.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=15", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 15: some candidates did not attempt the geometric waiting-time part.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable, infer the standard deviation, and use the geometric waiting-time formula over seven days." }, { "id": "cie-2025-m-52-q05d", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 5(d): Apply normal and geometric models to daily fruit sales.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=15", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 15: the report notes a substantial non-attempt rate on the final normal-distribution part.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Standardise the normal variable, infer the standard deviation, and use the geometric waiting-time formula over seven days." }, { "id": "cie-2025-m-52-q06a", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(a): Count colour arrangements of repeated books and selections under colour restrictions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use multinomial counts, fix the required pair positions for the spacing condition, and count selections by cases." }, { "id": "cie-2025-m-52-q06b", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(b): Count colour arrangements of repeated books and selections under colour restrictions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use multinomial counts, fix the required pair positions for the spacing condition, and count selections by cases." }, { "id": "cie-2025-m-52-q06c", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(c): Count colour arrangements of repeated books and selections under colour restrictions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=17", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 16: many candidates failed to handle the repeated-colour arrangement correctly.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Use multinomial counts, fix the required pair positions for the spacing condition, and count selections by cases." }, { "id": "cie-2025-m-52-q06d", "paperId": "cie-2025-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 6(d): Count colour arrangements of repeated books and selections under colour restrictions.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_52.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_52.pdf/download#page=18", "qpPages": [ 13, 13 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 16: a significant number did not attempt the final constrained selection question.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Use multinomial counts, fix the required pair positions for the spacing condition, and count selections by cases." }, { "id": "cie-2025-m-62-q01", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 1: Find a probability involving the difference of two independent normal variables.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=3", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the difference distribution with the means subtracted and variances added, then standardise." }, { "id": "cie-2025-m-62-q02a", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 2(a): Estimate population parameters, test a mean, and assess the need for the Central Limit Theorem.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=4", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate unbiased estimates, perform the stated one-sided test, and explain whether the sample size/distribution requires the CLT." }, { "id": "cie-2025-m-62-q02b", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 2(b): Estimate population parameters, test a mean, and assess the need for the Central Limit Theorem.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate unbiased estimates, perform the stated one-sided test, and explain whether the sample size/distribution requires the CLT." }, { "id": "cie-2025-m-62-q02c", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 2(c): Estimate population parameters, test a mean, and assess the need for the Central Limit Theorem.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=5", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 18: many candidates misunderstood when the Central Limit Theorem was needed.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Calculate unbiased estimates, perform the stated one-sided test, and explain whether the sample size/distribution requires the CLT." }, { "id": "cie-2025-m-62-q03a", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 3(a): Use Poisson probabilities and a normal approximation to determine an unknown count.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the Poisson probabilities, use additivity for independent variables, then solve for n from the approximation." }, { "id": "cie-2025-m-62-q03b", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 3(b): Use Poisson probabilities and a normal approximation to determine an unknown count.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=6", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the Poisson probabilities, use additivity for independent variables, then solve for n from the approximation." }, { "id": "cie-2025-m-62-q03c", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 3(c): Use Poisson probabilities and a normal approximation to determine an unknown count.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=7", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2025, physical p. 18: the normal approximation and solving for n are described as demanding.", "difficultySourceUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_er.pdf/download", "tagConfidence": "reviewed", "answerGuide": "Calculate the Poisson probabilities, use additivity for independent variables, then solve for n from the approximation." }, { "id": "cie-2025-m-62-q04ai", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 2, "topicIds": [ "integration", "distributions" ], "primaryTopicId": "integration", "summary": "Question 4(a)(i): Use a probability density graph and formula to find a parameter and probabilities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use total area to find the parameter, then integrate the requested interval or event." }, { "id": "cie-2025-m-62-q04aii", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 5, "topicIds": [ "integration", "distributions" ], "primaryTopicId": "integration", "summary": "Question 4(a)(ii): Use a probability density graph and formula to find a parameter and probabilities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=8", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use total area to find the parameter, then integrate the requested interval or event." }, { "id": "cie-2025-m-62-q04b", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration", "distributions" ], "primaryTopicId": "integration", "summary": "Question 4(b): Use a probability density graph and formula to find a parameter and probabilities.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=9", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use total area to find the parameter, then integrate the requested interval or event." }, { "id": "cie-2025-m-62-q05a", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 5(a): Set up a binomial hypothesis test and calculate a Type II error probability.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0 and H1, find the critical region, assess the observation, then compute the non-rejection probability at the alternative value." }, { "id": "cie-2025-m-62-q05b", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 5(b): Set up a binomial hypothesis test and calculate a Type II error probability.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=10", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0 and H1, find the critical region, assess the observation, then compute the non-rejection probability at the alternative value." }, { "id": "cie-2025-m-62-q05c", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 5(c): Set up a binomial hypothesis test and calculate a Type II error probability.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=11", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0 and H1, find the critical region, assess the observation, then compute the non-rejection probability at the alternative value." }, { "id": "cie-2025-m-62-q06a", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 6(a): Assess sampling bias and compare confidence-interval widths.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain representativeness, relate widths to critical z-values, and solve for the confidence level." }, { "id": "cie-2025-m-62-q06b", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 6(b): Assess sampling bias and compare confidence-interval widths.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=12", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain representativeness, relate widths to critical z-values, and solve for the confidence level." }, { "id": "cie-2025-m-62-q06c", "paperId": "cie-2025-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Question 6(c): Assess sampling bias and compare confidence-interval widths.", "qpUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_qp_62.pdf/download#page=13", "msUrl": "https://xtrapapers.co/papers/caie/as-and-a-level/mathematics-9709/2025-march/9709_m25_ms_62.pdf/download#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain representativeness, relate widths to critical z-values, and solve for the confidence level." }, { "id": "ial-2025-6-WMA12-01A-q01", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove the divisibility result by listing every valid positive-integer case.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the three valid triples, calculate each product, and state that all three products are divisible by 6." }, { "id": "ial-2025-6-WMA12-01A-q02a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the third and sixth terms of a geometric series to show its common ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the sixth-term equation by the third-term equation to obtain r³ = −1/8, hence r = −1/2." }, { "id": "ial-2025-6-WMA12-01A-q02b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the sum to infinity of the geometric series.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use ar² = 64 to find a = 256, then apply S∞ = a/(1 − r) with r = −1/2 to obtain 512/3." }, { "id": "ial-2025-6-WMA12-01A-q03a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State the remainder when f(x) is divided by x − k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = k into the definition of f; the remainder is −5." }, { "id": "ial-2025-6-WMA12-01A-q03b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the remainder theorem to determine k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set f(−2) equal to the given remainder 25 and solve the resulting equation to get k = −4." }, { "id": "ial-2025-6-WMA12-01A-q03c", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide the resulting cubic by 3x − 1 and give both quotient and remainder.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute k = −4, expand to 3x³ + 8x² − 21x − 25, then divide by 3x − 1: quotient x² + 3x − 6 and remainder −31." }, { "id": "ial-2025-6-WMA12-01A-q04a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use logarithm laws to find the exact value of a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the two logarithms into log₃((a + 1)/a) = 4, so (a + 1)/a = 81 and a = 1/80." }, { "id": "ial-2025-6-WMA12-01A-q04b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Apply the trapezium rule with four equal strips and simplify the estimate to log₃ k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": { "label": "Q4(b)", "marks": 4 }, "studyPriority": 3, "difficultyEvidence": "Pearson describes this part as substantially more demanding, requiring both a correct trapezium estimate and careful logarithm simplification.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Use strip width 1 and the ordinates at x = 1, 2, 3, 4, 5; combine the logarithms with log laws to obtain log₃√15." }, { "id": "ial-2025-6-WMA12-01A-q04c", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Explain how to improve the trapezium-rule estimate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Increase the number of strips, which makes each strip narrower." }, { "id": "ial-2025-6-WMA12-01A-q05a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Show that the iterative sequence is periodic with order 3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 15 ], "hardestPart": { "label": "Q5(a)–(b)", "marks": 7 }, "studyPriority": 5, "difficultyEvidence": "Pearson notes that many candidates found the period proof difficult and often failed to justify the return to the first term.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Find u₂ and u₃, then show that u₄ = u₁; conclude that the sequence repeats every three terms." }, { "id": "ial-2025-6-WMA12-01A-q05b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Evaluate the finite sum formed from an arithmetic term and the repeating sequence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 15 ], "hardestPart": { "label": "Q5(a)–(b)", "marks": 7 }, "studyPriority": 5, "difficultyEvidence": "Pearson notes that many candidates did not recognize the expression as a combination of an arithmetic series and a period-3 sum.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the arithmetic sum from the repeating three-term sum, evaluate each over 180 terms, and add them to obtain 82,255." }, { "id": "ial-2025-6-WMA12-01A-q06ia", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(i)(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use an identity and the angle range to find the exact value of sin θ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": { "label": "Q6(i)", "marks": 4 }, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Q6(i) as highly discriminating; many candidates gained only limited credit on its exact-identity work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Use sin²θ + cos²θ = 1, then choose the negative value because θ lies in the third quadrant: sin θ = −2√6/5." }, { "id": "ial-2025-6-WMA12-01A-q06ib", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(i)(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the exact sine and cosine values to find tan θ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": { "label": "Q6(i)", "marks": 4 }, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Q6(i) as highly discriminating; many candidates gained only limited credit on its exact-identity work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Use tan θ = sin θ/cos θ with the value from part (a), giving tan θ = −2√6." }, { "id": "ial-2025-6-WMA12-01A-q06iia", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(ii)(a)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the minimum height in the sinusoidal tide model.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The cosine reaches −1, so the minimum height is 4 − 3 = 1 metre." }, { "id": "ial-2025-6-WMA12-01A-q06iib", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(ii)(b)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the exact first time the tide reaches its minimum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set 30t − 40° = 180°, giving t = 22/3 hours, or 7:20 am." }, { "id": "ial-2025-6-WMA12-01A-q06iic", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(ii)(c)", "marks": 4, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Use the graph and tide equation to find the specified later time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve cos(30T − 40°) = −1/6 and select the solution shown by the graph; T = 16.65 hours to 2 decimal places." }, { "id": "ial-2025-6-WMA12-01A-q07a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Form and solve two coefficient equations to find n and k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 19 ], "hardestPart": { "label": "Q7(a)", "marks": 6 }, "studyPriority": 1, "difficultyEvidence": "Pearson reports that forming and solving the two coefficient equations was mainly achieved by higher-attaining candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate coefficients to get nk = −24 and n(n − 1)k²/2 = 270; solve to obtain n = 16 and k = −3/2." }, { "id": "ial-2025-6-WMA12-01A-q07b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use n and k to find the coefficient p.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the coefficient of x³, C(16,3)(−3/2)³, giving p = −1890." }, { "id": "ial-2025-6-WMA12-01A-q08a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use the derivative to determine where the curve is decreasing.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=20", "qpPages": [ 24, 27 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check the sign of 2(3x − 2)(x + 5); the derivative is negative for −5 < x < 2/3." }, { "id": "ial-2025-6-WMA12-01A-q08b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State an integer-coefficient factor using the given x-intercept.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=20", "qpPages": [ 24, 27 ], "msPages": [ 20, 21 ], "hardestPart": { "label": "Q8(b)", "marks": 1 }, "studyPriority": 7, "difficultyEvidence": "Pearson says this factor question was more demanding than expected; candidates often gave a non-integer factor when integer coefficients were required.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "The root x = 7/2 gives the factor 2x − 7." }, { "id": "ial-2025-6-WMA12-01A-q08c", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the derivative and factorise the polynomial using the intercept.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=20", "qpPages": [ 24, 27 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate to get f(x) = 2x³ + 13x² − 20x + C; use f(7/2) = 0 to find C = −175, then factor as (2x − 7)(x + 5)²." }, { "id": "ial-2025-6-WMA12-01A-q09a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch a circle centred on the x-axis and tangent to the y-axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=22", "qpPages": [ 28, 29 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw a circle whose centre lies on the x-axis and whose leftmost point touches the y-axis." }, { "id": "ial-2025-6-WMA12-01A-q09b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle through (5, 6).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=22", "qpPages": [ 28, 29 ], "msPages": [ 22, 23 ], "hardestPart": { "label": "Q9(b)", "marks": 4 }, "studyPriority": 4, "difficultyEvidence": "Pearson describes the circle setup as unexpected and a major separator; the key is using the shared value of the radius and centre x-coordinate.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Let the centre and radius be a and |a|; use (5 − a)² + 6² = a² to get a = 6.1, hence (x − 6.1)² + y² = 6.1²." }, { "id": "ial-2025-6-WMA12-01A-q10a", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use calculus to show the stationary point has x-coordinate 4.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set dy/dx = 0, and solve for the positive stationary-point coordinate x = 4." }, { "id": "ial-2025-6-WMA12-01A-q10b", "paperId": "ial-2025-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use algebraic integration to find the exact shaded area.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2A_MS.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": { "label": "Q10(b)", "marks": 5 }, "studyPriority": 6, "difficultyEvidence": "Pearson identifies the area calculation as the most challenging part of Q10; many candidates omitted the rectangle when combining the regions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01a-pef-20250814.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve over the stated interval and add the rectangle area indicated in the diagram; the exact area is 20." }, { "id": "ial-2019-1-WME02-01-q01", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a particle coordinate from the centre of mass of three weighted particles.", "answerGuide": "Equate the weighted x- and y-coordinates to (k, 2k), eliminate k and obtain a = 4/3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q02a", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the velocity after a two-dimensional impulse acts on the particle.", "answerGuide": "Use m(v − u) = impulse; the velocity is −4i + (19/3)j m s−1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q02b", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "energy-momentum", "trigonometry" ], "primaryTopicId": "energy-momentum", "summary": "Find the deflection angle between the initial and final velocity directions.", "answerGuide": "Use the velocity vectors to find the included angle, about 108.2°.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q03", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 8, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the resistance and speed of a car moving uphill and downhill at constant speed and power.", "answerGuide": "Write force balance in both directions and P = Fv for v and 2v; solve to obtain R = 540 N and v = 15 m s−1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q04a", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the centre of mass of a circular lamina with an offset disc removed is 13a/3 from B.", "answerGuide": "Use the area ratio 16:4 for the discs and take moments about B to recover x = 13a/3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q04b", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the attached mass ratio that gives the stated equilibrium angle when the lamina is suspended at D.", "answerGuide": "Take moments using the lamina centre of mass and attached mass at B, then substitute tan θ = 3/4; k = 10/3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that many candidates stopped after finding the loaded lamina centre of mass; a common error was assuming it lay on DO.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-1-WME02-01-q05", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the distance from the origin when acceleration first becomes zero for a given velocity function.", "answerGuide": "Differentiate v to find t = 4, integrate v from 0 to 4, and take the positive distance 14.4 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson describes this as an unstructured question; candidates often found total distance instead of displacement to A or left a negative distance.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-1-WME02-01-q06a", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction of a pole on a uniform plank in equilibrium.", "answerGuide": "Take moments about A using the plank weight at its midpoint; the reaction at C is 12.8 N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q06b", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction coefficient at the pole when friction is limiting at both contacts.", "answerGuide": "Resolve parallel and perpendicular to the plank, use limiting friction at A and C, and solve for μ = 0.630.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q07a", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show that P reverses direction after colliding with Q.", "answerGuide": "Conserve momentum with the initial direction of P positive; the resulting velocities require P's final velocity to be negative.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q07b", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "energy-momentum", "algebra" ], "primaryTopicId": "energy-momentum", "summary": "Find the restitution values for which Q also reverses direction.", "answerGuide": "Combine momentum and restitution, require Q's final velocity to point opposite its initial direction, and use 0 ≤ e ≤ 1: 1/9 < e ≤ 1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says only a minority reached the correct restitution range; inconsistent assumptions about the particles' directions were common.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-1-WME02-01-q07c", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the kinetic energy lost when the restitution coefficient is 1/2.", "answerGuide": "Solve for final speeds v = −4u/5 and w = 7u/10, then subtract final from initial kinetic energy: 81mu²/20.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q08a", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to find the constant friction force on the ramp.", "answerGuide": "Balance the change in kinetic energy against gain in gravitational potential energy and work 6F; F ≈ 21.2 N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=16", "qpPages": [ 24, 24 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q08bi", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 3, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the speed after the particle leaves the ramp and falls to ground level.", "answerGuide": "Use conservation of mechanical energy from B to C; w ≈ 11.8 m s−1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=16", "qpPages": [ 24, 24 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q08bii", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the direction of the particle's velocity when it reaches C.", "answerGuide": "Resolve the horizontal and vertical velocity components at impact; the direction is about 37.5° below horizontal.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=16", "qpPages": [ 24, 24 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME02-01-q08c", "paperId": "ial-2019-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2019, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the greatest height above ground reached between A and C.", "answerGuide": "Find the extra rise above B from the vertical component at launch, then add B's height above ground; the total is about 2.65 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_que_20190123.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_msc_20190307.pdf#page=17", "qpPages": [ 24, 24 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson notes that some candidates found only the extra rise above B and did not add the ramp height, so they missed the requested height above ground.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-1-WME03-01-q01", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find how far a particle travels from the origin before first coming to rest under acceleration depending on displacement.", "answerGuide": "Use v dv/dx = a(x), apply the initial speed to find the integration constant, and solve v = 0 for the first stopping point; the distance is 4.5 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q02", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 12, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the allowed period range for a rotating ball on two taut strings, with each tension below the breaking limit.", "answerGuide": "Resolve vertically and radially to obtain 2g < aω² < 4g, then use ω = 2π/S to obtain π√(a/g) < S < π√(2a/g), so k = √2.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q03a", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the period of an SHM particle from its position and speed at C.", "answerGuide": "Use v² = ω²(a² − x²) with the given displacement and speed, then T = 2π/ω.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q03b", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the amplitude parameter a from the maximum acceleration.", "answerGuide": "Use maximum acceleration aω² and the value of ω from part (a); a = 5 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q03c", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the maximum speed in the SHM.", "answerGuide": "Use vmax = amplitude × angular frequency; vmax = 15 m s−1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q03d", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 5, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the separation CD when the particle moves from C to D in a quarter period.", "answerGuide": "Use the quarter-period phase change with an SHM displacement equation, track the direction of travel, then substitute a = 5 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson identifies this as the part causing the most problems in Question 3; many candidates found a relevant time but could not convert it to the displacement with correct direction.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-1-WME03-01-q04a", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the elastic modulus for the symmetric string supporting the hanging mass.", "answerGuide": "Resolve the two string tensions and use Hooke's law for each half-string to obtain λ = 20mg/(3l).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q04b", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show that the particle released from the stretched elastic string stops before reaching AB.", "answerGuide": "Compare initial elastic potential energy with the gravitational energy needed to reach AB, or set final kinetic energy to zero and show the turning point occurs below AB.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson reports that few candidates achieved full marks; common errors treated the two string halves as one string or converted all elastic energy directly into gravitational potential energy.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-1-WME03-01-q05a", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 8, "topicIds": [ "integration", "mechanics" ], "primaryTopicId": "integration", "summary": "Use integration to show the x-coordinate of the centre of mass of the solid of revolution.", "answerGuide": "Integrate cross-sectional volume and first moment about the origin with explicit limits; the centre coordinate is 48r/65.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q05b", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 8, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Find the combined centre of mass of the hollow hemispherical bowl and the shallow liquid layer.", "answerGuide": "Find the bowl's centre using volume differences and the result from part (a), determine the liquid's mass-centre position, then combine moments using masses 5M and 2M; the distance is 3.03 cm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson calls this part challenging and a good discriminator; many could find the bowl centre but then failed to connect it to the liquid centre or combine the stated mass ratios.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-1-WME03-01-q06a", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction on a particle at the horizontal point of a smooth sphere's inner surface.", "answerGuide": "Use energy from the lowest point to B to find the speed, then apply the radial equation; R = 3mg/2.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WME03-01-q06b", "paperId": "ial-2019-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2019, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 11, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show that after leaving the sphere the particle returns to contact with the surface at A.", "answerGuide": "At lift-off set the normal reaction to zero and combine energy with radial motion to determine the launch point and speed; resolve projectile motion to show it reaches A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_que_20190110.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_msc_20190307.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes this final part as the paper's challenge and says it was not always attempted; the required motion follows lift-off from a vertical circular path.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME03_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2019-1-WST02-01-q01a", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "State a context-specific assumption for modelling ticket-holder no-shows with a binomial distribution.", "answerGuide": "State that each ticket-holder independently either turns up or does not, with the no-show probability remaining 0.05.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q01bi", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)(i)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that all 50 ticket-holders turn up.", "answerGuide": "For X ~ B(50, 0.05), calculate P(X = 0) = 0.95^50 ≈ 0.0769.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q01bii", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability exactly one of the 50 ticket-holders does not turn up.", "answerGuide": "Calculate P(X = 1) = 50(0.05)(0.95)^49 ≈ 0.2025.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q01c", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the bus company's expected earnings after compensating passengers who cannot get a seat.", "answerGuide": "Use the earnings outcomes £880, £940 and £1000 with their probabilities; the expected total is about £978.62.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson names this as one of the more discriminating parts; many candidates found possible earnings or payouts but did not form the expected-value calculation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2019-1-WST02-01-q02a", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability fewer than five employees arrive in ten minutes under a Poisson model.", "answerGuide": "Use X ~ Po(2) and evaluate P(X ≤ 4) ≈ 0.947.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q02b", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find n when the 30-minute Poisson probabilities of n and n+1 arrivals are equal.", "answerGuide": "For Y ~ Po(6), equate consecutive probabilities and obtain n = 5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q02c", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the lower count c for a 45-minute inclusive probability from c to 12 of 0.8546.", "answerGuide": "For D ~ Po(9), subtract P(D ≤ c−1) from P(D ≤ 12); c = 4.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q02d", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance exactly four of six non-overlapping intervals each have two arrivals.", "answerGuide": "Find p = P(Po(2)=2) ≈ 0.27067, then use a binomial B(6,p) probability for exactly four; result ≈ 0.043.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson includes this among the more discriminating parts; candidates who omitted the final binomial model or used the wrong Poisson mean did not reach the answer.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2019-1-WST02-01-q03a", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Read an interval probability from the supplied cumulative distribution graph.", "answerGuide": "Subtract F(3) from F(7): P(3 < X < 7) = 0.5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q03b", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the constants in the piecewise linear probability density from the graph.", "answerGuide": "Use each graph segment's gradient; a = 0.2, b = 0 and c = 0.3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q03c", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the expectation of the continuous random variable from its density.", "answerGuide": "Integrate x f(x) over the two non-zero intervals; E(X) = 5.4.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q04a", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% whether the credit-card payment proportion is now above 0.35.", "answerGuide": "Use H0: p = 0.35 and H1: p > 0.35; the one-sided probability is 0.0532, so do not reject H0 and conclude the belief is unsupported.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q04b", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Show 11 is less than two standard deviations above the binomial mean when p remains 0.35.", "answerGuide": "For B(20,0.35), mean = 7 and SD ≈ 2.133; 11 < 7 + 2(2.133) = 11.266.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q05ai", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Show the expectation of a uniform variable on [a,b] is 5a/2 from the given probability.", "answerGuide": "The probability condition gives b = 4a; apply E(X) = (a+b)/2 = 5a/2.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q05aii", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(X > b−4a) for the first uniform variable.", "answerGuide": "Substitute b = 4a so the threshold is zero, below the support; the probability is 1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q05bi", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find c from the variance of a uniform variable on [3,c].", "answerGuide": "Solve (c−3)^2/12 = 3c−9 with c>3; c = 39.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q05bii", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(2Y−7 < 20−Y) for Y uniform on [3,c].", "answerGuide": "Reduce the inequality to Y < 9 and use c = 39; the probability is 1/6.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q05biii", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(iii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(Y²) for the uniform variable on [3,39].", "answerGuide": "Use E(Y²) = Var(Y) + [E(Y)]²; the result is 549.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this as one of the more discriminating parts; common errors omitted the square on the mean or subtracted instead of adding the variance.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-1-WST02-01-q06ia", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(i)(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State when the Poisson distribution can approximate a binomial distribution.", "answerGuide": "State that n is large and p is small.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q06ib", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(i)(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a Poisson approximation to find the chance of more than seven defective tyres.", "answerGuide": "Use λ = np = 7.5 and calculate P(Y>7) ≈ 0.475.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q06iia", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe a sampling frame for selecting employees who cycle to work.", "answerGuide": "Use a complete list, register or database of all employees.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q06iib", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)(b)", "marks": 1, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Explain the sampling distribution of C.", "answerGuide": "It is the probability distribution of the number who cycle to work across all possible samples of 150.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q06iic", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Recover α from an uncorrected normal probability for the number of cycling employees.", "answerGuide": "Approximate with N(60,36), standardise to z = −1.5, and solve α = 51.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q06iid", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Recalculate the probability for at most α employees using a continuity correction.", "answerGuide": "Use P(C≤51) ≈ P(N≤51.5) for N(60,36); the corrected probability is about 0.078.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q07a", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the normalising constant in the piecewise probability density.", "answerGuide": "Integrate both density pieces over their supports and set total area to one; c = 1/12.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=12", "qpPages": [ 20, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q07bi", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the piecewise probability density function.", "answerGuide": "Draw the increasing line on [−3,0) and decreasing line on [0,3], with the correct endpoint heights and support.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=12", "qpPages": [ 20, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q07bii", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain why zero is the mode of the density.", "answerGuide": "The density reaches its greatest value at x = 0, so the mode is zero.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=12", "qpPages": [ 20, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q07c", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the cumulative distribution function across the full support.", "answerGuide": "Integrate each density segment, use F(−3)=0 and match the pieces at zero and one; outside the support F is 0 or 1.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=12", "qpPages": [ 20, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WST02-01-q07d", "paperId": "ial-2019-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2019, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find d such that P(X>d | X>0)=2/5.", "answerGuide": "Cancel the nested event in the conditional probability, use the right-hand CDF branch and solve; d ≈ 1.10.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20190124.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_msc_20190307.pdf#page=12", "qpPages": [ 20, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls this a challenge for many candidates; errors included the wrong conditional event denominator, the wrong CDF branch and failure to solve the resulting equation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-1-WDM01-01-q01a", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Define a bipartite graph.", "answerGuide": "State that vertices form two sets and edges connect across the sets, never within a set.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q01b", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw the worker-task bipartite graph from the eligibility table.", "answerGuide": "Create one vertex set for workers and one for tasks, then join each permitted pair.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q01c", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the initial worker-task matching using augmenting paths.", "answerGuide": "Use alternating paths to update the matching until all six workers have distinct tasks.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q02a", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra's algorithm to find the shortest route from A to L.", "answerGuide": "Label the network and trace back the minimum path A–B–C–E–D–K–L, length 44 km.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q02b", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain how the shortest route is recovered from Dijkstra labels.", "answerGuide": "Work backwards from L; include an arc when the difference in final labels equals its weight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q02c", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the shortest route from J to F that passes through A.", "answerGuide": "Join the shortest J-to-A and A-to-F routes; the route length is 61 km.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q03a", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete early and late event times on the activity network.", "answerGuide": "Apply forward and backward passes through the network to fill the event-time diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q03b", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify the critical activities.", "answerGuide": "The critical activities are E, F and K.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q03c", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw a cascade chart for the project.", "answerGuide": "Schedule each activity within its earliest/latest bounds and show its duration and float on the time grid.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_que_20190118.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_msc_20190307.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WDM01-01-q03d", "paperId": "ial-2019-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2019, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the cascade chart to find the minimum workforce for the shortest project duration.", "answerGuide": "Identify the overlap requiring four simultaneous workers, for activities K, I, J and L during 18 0.", "answerGuide": "Use the positive value of p, find two perpendicular side lengths, and calculate half their product to three significant figures.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_MS.pdf#page=22", "qpPages": [ 22, 25 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson identifies part (c) as the least successfully answered part and notes avoidable vector and area-method errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01A-q09a", "paperId": "ial-2025-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the exact volume formed by rotating the given curve region about the x-axis.", "answerGuide": "Set up the disk-method integral with the curve squared, include π, and integrate using the prescribed substitution or equivalent algebraic method.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_MS.pdf#page=25", "qpPages": [ 26, 29 ], "msPages": [ 25, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01A-q09b", "paperId": "ial-2025-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Use the vase’s remaining capacity to determine the water depth a.", "answerGuide": "Relate the water volume to the complementary portion of the full vase, use the antiderivative from part (a), and solve for a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_MS.pdf#page=25", "qpPages": [ 26, 29 ], "msPages": [ 25, 28 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls part (b) highly discriminating and reports that only 8.4% of candidates scored full marks on Question 9.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01A-q10", "paperId": "ial-2025-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": null, "marks": 5, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove by contradiction that the stated integer expression cannot equal zero.", "answerGuide": "Assume the expression is zero, rearrange to show p² is even, infer that p is even, then substitute p = 2k and reach a contradiction using the integer q.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4A_MS.pdf#page=29", "qpPages": [ 30, 31 ], "msPages": [ 29, 31 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes Question 10 as one of the more challenging questions; complete, logically sound contradictions were rare.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q01ai", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)(i)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the image of the turning point after translating the graph up by 7.", "answerGuide": "Apply the vertical translation to the y-coordinate of P; keep its x-coordinate unchanged.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q01aii", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the image of the turning point after multiplying the graph values by 3.", "answerGuide": "Apply the vertical scale factor to the y-coordinate of P; keep its x-coordinate unchanged.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q01b", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the values of k for which the line y = k meets the curve exactly once.", "answerGuide": "Use the curve’s turning point, intercept and horizontal asymptote to identify the horizontal levels with one intersection.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q01c", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Solve the shifted root equation f(x + 4) = 0.", "answerGuide": "Use the given x-intercept of f and account for the horizontal shift inside the function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q02a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Determine the constants in the square-root model for the leaking vessel.", "answerGuide": "Substitute the two measured time-volume pairs into V = a√t + b and solve the simultaneous equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q02bi", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)(i)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the modelled initial volume of water in the vessel.", "answerGuide": "Set t = 0 in the model and evaluate V.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q02bii", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)(ii)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find to the nearest minute when the vessel empties.", "answerGuide": "Set V = 0 in the model and solve for t, checking the positive time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q03a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Write an equation for the cosine curve shown in the diagram.", "answerGuide": "Read the amplitude, orientation and period from the graph, then state an equivalent sine or cosine expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q03bi", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count the solutions of f(x) = 2 on the stated interval from −720° to 720°.", "answerGuide": "Count every intersection of the graph with y = 2, including endpoints where applicable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q03bii", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count the solutions of f(x) = −3 on the stated interval from −720° to 720°.", "answerGuide": "Count every intersection of the graph with y = −3 over the full interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q04i", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the non-calculator equation involving powers and indices.", "answerGuide": "Rewrite each term as a power of 2, equate exponents and solve for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says part (i) was challenging, with many candidates failing to express all terms as powers of 2.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q04ii", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the surd equation exactly and give the result as a + b√3.", "answerGuide": "Rationalise or cross-multiply, collect the x terms and constants, then isolate x in the requested exact form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q05a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the given function y with respect to x and simplify.", "answerGuide": "Integrate each power term separately, including the reciprocal-power term, and add the constant of integration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q05bi", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given function y and simplify dy/dx.", "answerGuide": "Differentiate each power term using the power rule, including the negative power.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q05bii", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact solutions of the equation dy/dx = 3.", "answerGuide": "Set the derivative equal to 3, substitute a suitable power of x to expose the quadratic, solve it without calculator-only work, and recover x.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this final part was more challenging; many candidates lost marks by jumping from a hidden quadratic to calculator-derived values of x³.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q06", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": null, "marks": 5, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find algebraically the x-coordinates where the cubic and quadratic curves intersect.", "answerGuide": "Equate the two expressions, expand and simplify, factor the resulting cubic, and retain every root including x = 0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes this non-calculator curve-intersection and cubic-solving question as a strong discriminator; losing the x = 0 root was a common error.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q07a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch the translated reciprocal curve and label its vertical asymptote and y-intercept.", "answerGuide": "Sketch both branches of y = 1/(x + 6), state x = −6, and mark the y-intercept (0, 1/6).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q07bi", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Show that the intersection condition leads to the stated quadratic inequality in m.", "answerGuide": "Equate the line and curve, obtain the quadratic in x, then use its discriminant condition for at least one real intersection.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q07bii", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the range of m satisfying the intersection condition.", "answerGuide": "Factor or solve the quadratic inequality in m and select the outside intervals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q08a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the line through the two given points.", "answerGuide": "Calculate the gradient from the coordinates and use point-gradient form to determine the intercept.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q08b", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "algebra", "summary": "Find the equation of the quadratic curve from its two intersections and y-intercept.", "answerGuide": "Substitute the three known points into y = ax² + bx + c and solve for the coefficients.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q08c", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Use inequalities to describe the bounded region R.", "answerGuide": "Identify the x-interval bounded by the y-axis and intersections, then place y between the line and quadratic in the correct order.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q09a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the radius of the sector using the triangle side lengths and included angle.", "answerGuide": "Apply the cosine rule to triangle CPQ and give the radius to three decimal places.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q09b", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Show that the angle at the sector centre is 0.236 radians.", "answerGuide": "Use the sine rule or cosine rule in the triangle, following through the radius from part (a).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q09c", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the total area of the sign.", "answerGuide": "Add the sector area and triangle area, using the correct central angle and the radius from part (a).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q09d", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the total perimeter of the sign.", "answerGuide": "Add the sector arc length to the exposed straight sides, excluding the shared radius.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q10a", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at P from the given normal.", "answerGuide": "Use the negative reciprocal of the normal gradient and the coordinates of P in point-gradient form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q10b", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use the gradient condition at P to determine k.", "answerGuide": "Evaluate the given derivative expression at x = 4 and equate it to the tangent gradient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01A-q10c", "paperId": "ial-2025-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(c)", "marks": 5, "topicIds": [ "integration", "differentiation", "algebra" ], "primaryTopicId": "integration", "summary": "Recover the curve equation f(x) from its derivative and the point P.", "answerGuide": "Integrate f′(x), substitute the known point to determine the constant, and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1A_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q01a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the curve’s expression term by term and simplify.", "answerGuide": "Rewrite the fractional terms as powers of x, then apply the power rule to each term.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q01b", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal to the curve at the specified point and express its equation with integer coefficients.", "answerGuide": "Evaluate the tangent gradient at x=4, take its negative reciprocal, then use the point-slope form through (4,3).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q02ai", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the given function value to determine the positive constant a.", "answerGuide": "Substitute x=-1 and f(-1)=32 to form a quadratic in a; solve it and use a>0 to select the valid root.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q02aii", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the cubic equation after substituting the value of a.", "answerGuide": "Factor out x, factor the remaining quadratic, and report all three roots.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q02bi", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)(i)", "marks": 3, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Find all real solutions of a cubic equation in y^(1/3).", "answerGuide": "Set u=y^(1/3), solve the same cubic, then cube each real u-root.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q02bii", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)(ii)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation by substituting u=9^z.", "answerGuide": "Set u=9^z and solve the cubic; keep the positive admissible root and convert it back to z.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q03a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the cosine rule to establish a relation between the two unknown sides.", "answerGuide": "Apply c²=p²+q²−2pq cos(60°) with BC=2√2, then simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q03bi", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the exact length p given that q is 2 metres longer than p.", "answerGuide": "Substitute q=p+2 into the relation from part (a), solve the quadratic, and retain the positive length.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q03bii", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the exact length q from the relation between p and q.", "answerGuide": "Add 2 to the positive value of p from part (b)(i).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q03c", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the exact area of the flower bed.", "answerGuide": "Use one half pq sin(60°) with the exact side lengths from part (b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q04", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate a sum of algebraic terms and present each term in simplest form.", "answerGuide": "Separate the integrand into powers of x, integrate each term with the power rule, and include the constant of integration.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q05a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Compare the two share-value models at the start of the monitoring period.", "answerGuide": "Set t=0 in both models and subtract the two values to find the difference.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q05b", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the maximum value of company A’s share model over the stated period.", "answerGuide": "Identify the vertex of the quadratic model; its maximum is the vertex value.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q05c", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find when company A’s share value exceeds company B’s over the 15-year interval.", "answerGuide": "Set the two model expressions equal to locate the boundary times, solve the resulting quadratic, then use the sign of the difference and the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q05d", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Explain why company A’s model should not be used at t=20.", "answerGuide": "The model is stated to cover only 15 years; its extrapolated share value would also be negative.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q06a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent to the curve at P(8,2).", "answerGuide": "Substitute x=8 into f′(x) for the gradient, then use point-slope form through (8,2).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q06b", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Recover f(x) from its derivative and the point P(8,2).", "answerGuide": "Integrate f′(x) term by term, use f(8)=2 to determine the constant, and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q07a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Read the y-coordinate where line l1 meets the y-axis.", "answerGuide": "Set x=0 in 4y+3x=48.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=18", "qpPages": [ 22, 22 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q07b", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the y-intercept of the line through D perpendicular to l1.", "answerGuide": "Find l1’s gradient, take the negative reciprocal, then use D(8,6) in point-slope form and set x=0.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=18", "qpPages": [ 22, 22 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q07c", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the arc length for the sector centred at C.", "answerGuide": "Use r=CE and s=rθ with θ=1.8 radians; the radius follows from the y-coordinates of C and E.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=18", "qpPages": [ 22, 22 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q07d", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact area of the shaded sector-and-triangle region.", "answerGuide": "Add the sector area ½r²θ to the area of triangle CDE.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=18", "qpPages": [ 22, 22 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q08a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Write the quadratic in completed-square form.", "answerGuide": "Complete the square in 3x²+6x+9 to obtain 3(x+1)²+6.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=20", "qpPages": [ 26, 26 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q08b", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Deduce the minimum point of the quadratic.", "answerGuide": "Read the vertex from the completed-square form.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=20", "qpPages": [ 26, 26 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q08c", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Determine the four coefficients of a cubic through P with three specified roots.", "answerGuide": "Use the roots to write y=k(x+4)(x+2)(x−3), substitute P to find k, then expand and compare coefficients.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=20", "qpPages": [ 26, 26 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q09a", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "State the vertical asymptote shown for y=tan x.", "answerGuide": "Use the indicated branch of tan x to identify the asymptote x=π/2.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=22", "qpPages": [ 30, 30 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q09bi", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(i)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch y=1/x+1 and state its horizontal asymptote.", "answerGuide": "Translate y=1/x upward by 1; the horizontal asymptote is y=1.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=22", "qpPages": [ 30, 30 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q09bii", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count the intersections of tan x and 1/x+1 on the given interval.", "answerGuide": "Use the two sketches to count intersection points in −2π≤x≤2π and explain the count in terms of solutions.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=22", "qpPages": [ 30, 30 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q09ci", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(i)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count solutions on the first extended interval.", "answerGuide": "Use the repeating branches of tan x together with the translated reciprocal curve over 0≤x≤40π.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=22", "qpPages": [ 30, 30 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA11-01-q09cii", "paperId": "ial-2021-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(ii)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count solutions on the second extended interval.", "answerGuide": "Count the intersections over −10π≤x≤52π, accounting for the endpoint and repeating periods.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/33/e3/dc/ec/7e/5f/92/94/76/2d/f4/88/b2/83/18.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/d3/88/f8/dd/d3/42/f9/1f/e8/9a/f4/f2/c4/e2/57/63.pdf#page=22", "qpPages": [ 30, 30 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q01ai", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(i)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Choose the appropriate sequence model for the weekly savings pattern.", "answerGuide": "Compare the increments and state that the fixed increase makes the savings an arithmetic sequence.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q01aii", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(ii)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Give an expression for the amount saved in week n.", "answerGuide": "Use first term £5 and common difference £0.25 in the arithmetic-term formula.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q01b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find how many weeks it takes for the total savings to reach £350.", "answerGuide": "Set the arithmetic-series sum equal to £350, solve the resulting quadratic, and take the positive integer week count.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q02ai", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Sketch y=2^x and mark its coordinate-axis intersection.", "answerGuide": "Draw the increasing exponential through (0,1), approaching but not meeting the x-axis.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q02aii", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Sketch y=4^x−6 and mark its coordinate-axis intersections.", "answerGuide": "Shift y=4^x down by 6; find the intercepts by setting x=0 and y=0.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q02b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the exact intersection of y=2^x and y=4^x−6.", "answerGuide": "Set 2^x=4^x−6, let u=2^x, solve the quadratic, reject a nonpositive u, and substitute the valid root back.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q03i", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove p³+p is divisible by 10 for every single-digit prime p.", "answerGuide": "Check the possible primes 2, 3, 5 and 7, or show each gives a multiple of 10.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q03ii", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Show (n+1)³−n³ is not divisible by 3 for natural n.", "answerGuide": "Expand and simplify to 3n²+3n+1; the first two terms are divisible by 3, leaving remainder 1.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q04a", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the given expression in ascending powers through x³.", "answerGuide": "Use the binomial expansion through the cubic term and simplify each coefficient.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q04b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "binomial", "numerical-methods" ], "primaryTopicId": "binomial", "summary": "Use the expansion to approximate the stated nearby power to 3 decimal places.", "answerGuide": "Substitute the specified small increment into the expansion from part (a), retaining terms through x³.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q04c", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Decide whether the approximation is an overestimate or underestimate.", "answerGuide": "Use the sign of the first omitted binomial term at the stated positive increment.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q05a", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-analysis", "integration" ], "primaryTopicId": "data-analysis", "summary": "Estimate the bounded area under C1 with the trapezium rule.", "answerGuide": "Use the equal 0.25 x-intervals and all seven listed ordinates in the trapezium-rule formula.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q05b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the stated expression for curve C2.", "answerGuide": "Rewrite the terms as powers of x, integrate term by term, and include the constant of integration if required.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q05c", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "integration", "data-analysis" ], "primaryTopicId": "integration", "summary": "Estimate the shaded area between C1 and C2.", "answerGuide": "Subtract the C2 area estimate from the trapezium estimate for C1, using the intersection limits.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q06a", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find k when the circle is tangent to the x-axis.", "answerGuide": "Complete squares to locate the centre and radius, then set the distance from the centre to the x-axis equal to the radius.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q06b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of triangle RST formed by the circle and coordinate axes.", "answerGuide": "Find the circle’s axis intersections, identify the triangle’s base and height, and apply half base times height.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q07a", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "algebra", "algebra" ], "primaryTopicId": "algebra", "summary": "Transform the logarithmic equation into the stated cubic.", "answerGuide": "Combine logarithms using log laws, convert to a power equation, expand, and collect terms.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q07b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Verify that −4 is a root of the cubic.", "answerGuide": "Substitute x=−4 into the cubic and show that the result is zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q07c", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "algebra", "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the original logarithmic equation using algebra.", "answerGuide": "Factor the cubic, test its roots in the original logarithms, and enforce the logarithm domain restrictions.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q08i", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on the stated degree interval.", "answerGuide": "Combine the sine and cosine terms using a tangent ratio, then list all solutions in 0°<θ<360°.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q08iia", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)(a)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the trigonometric equation in the required factored form.", "answerGuide": "Use sin(3x) and cos(2x)=1−2sin²x identities, factor out sin x, and collect powers of cos x.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q08iib", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)(b)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over the stated radian interval.", "answerGuide": "Set each factor from part (a) to zero, solve for cos x, and list every angle in the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q09a", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Show the surface-area model for the open-top box in terms of h.", "answerGuide": "Use l²h=250000 to express l² in terms of h, then form S=l²+4lh and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q09b", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the condition for a stationary surface area and identify the stated height form.", "answerGuide": "Differentiate S(h), set dS/dh=0, and solve for h in terms of the rational constant.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA12-01-q09c", "paperId": "ial-2021-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Justify that the stationary point gives a minimum.", "answerGuide": "Evaluate the second derivative at the stationary height and show it is positive.", "qpUrl": "https://www.alevelcopilot.com/whfile/b3/5f/a3/87/be/c2/f5/89/e0/a7/ff/52/7b/00/ac/ab.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/97/c5/4b/5c/e3/87/e3/be/68/39/fa/44/53/b4/78/4a.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q01a", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Use calculus to obtain the equation satisfied by the stationary point’s x-coordinate.", "answerGuide": "Differentiate x²cos(x/2), set the derivative to zero, rearrange to a tangent equation, and isolate x.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q01b", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration to find x2 and x6 to 3 decimal places.", "answerGuide": "Start from x1=2 and repeatedly substitute into the given arctangent iteration, rounding only at the end.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q02a", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Show the trigonometric identity and determine the constant k.", "answerGuide": "Use the double-angle identity for sin 2x, rewrite the numerator as a constant multiple of sin x cos x, and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q02b", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on 0°<θ<90°.", "answerGuide": "Use the identity from part (a), convert to an equation in tan θ or sec θ, then select the solution in the required interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q03i", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the rational expression in x in simplest form.", "answerGuide": "Rewrite the quotient as powers of x and integrate each term.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q03iia", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(a)", "marks": 2, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Rewrite the quadratic denominator in the required completed-square form.", "answerGuide": "Complete the square in the denominator and identify the constants A and B.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q03iib", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral using algebraic integration.", "answerGuide": "Use the completed-square expression to split the integrand into standard terms, integrate, and apply the given limits.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q04a", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the composite-function equation f(g(x))=3.", "answerGuide": "Substitute g(x) into f, solve the resulting quadratic, and keep only values in the stated domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q04b", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find f⁻¹(3).", "answerGuide": "Set y=f(x), rearrange to make x the subject, then substitute y=3.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q04c", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch g and its inverse, showing the axis intercepts.", "answerGuide": "Sketch the restricted parabola and reflect it in y=x for the inverse; label the four intercepts.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q05a", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Determine p and q from the straight-line relationship between log10 A and t.", "answerGuide": "Use the intercept to find log10 p and the gradient to find log10 q, then evaluate both constants.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q05b", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the duckweed area’s rate of increase after six days.", "answerGuide": "Differentiate A=pq^t using dA/dt=A ln(q), then substitute t=6 and the constants.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q06ai", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch y=k−2|x| and label its axis intercepts.", "answerGuide": "Draw the inverted V with vertex (0,k) and solve k−2|x|=0 for the x-intercepts.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q06aii", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch y=|2x−k/3| and label its axis intercepts.", "answerGuide": "Draw the V with vertex where 2x−k/3=0, then find the y-intercept by setting x=0.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q06b", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the equation equating the two modulus expressions.", "answerGuide": "Use the intersection points of the graphs or split by the signs inside the modulus bars, then solve each linear case.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q07", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": null, "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate x=6sin²(2y) and express dy/dx in the required form.", "answerGuide": "Differentiate implicitly with the chain rule, then use sin²(2y)=x/6 and the double-angle identity to rewrite the result.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=15", "qpPages": [ 24, 24 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q08a", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find the population at the start of the study.", "answerGuide": "Substitute t=0 into the population model.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q08b", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find the upper limit of the population model.", "answerGuide": "Take the horizontal asymptote as t increases.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q08c", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find when the model predicts 500 fish, in years and months.", "answerGuide": "Set N=500, rearrange the logistic equation to isolate e^(0.3t), take natural logs, and convert the fractional year to months.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q08d", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate the population model and identify its constant A.", "answerGuide": "Apply the quotient rule to the logistic expression and simplify the numerator and denominator.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q08e", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(e)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find T when the rate of change equals 8 fish per year.", "answerGuide": "Set the derivative from part (d) equal to 8, solve the resulting exponential equation, and take logarithms.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q09a", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write 12sin x−5cos x as Rsin(x−α).", "answerGuide": "Match coefficients using Rcos α=12 and Rsin α=5, then find exact R and α in radians.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q09bi", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(i)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the minimum value of g(θ).", "answerGuide": "Use the amplitude form from part (a) and add the vertical shift.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q09bii", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the smallest θ at which g reaches its minimum.", "answerGuide": "Set the sine term to −1, solve the phase equation, and choose the smallest positive θ.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA13-01-q09c", "paperId": "ial-2021-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the range of h(β).", "answerGuide": "Use the range of 12sin β−5cos β from part (a), square its extrema, and apply the negative sign and vertical shift.", "qpUrl": "https://www.alevelcopilot.com/whfile/93/de/c9/72/2f/15/bf/21/e6/ed/5d/96/5c/a1/0b/f4.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/fd/5e/51/75/38/58/f6/88/4d/a9/af/b5/a3/b8/b1/cd.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q01ai", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(i)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find k from the coefficient of x in the given binomial expansion.", "answerGuide": "Match the linear coefficient in the generalized binomial expansion to 18.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q01aii", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(ii)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the coefficients A and B of x² and x³.", "answerGuide": "Use the second- and third-order binomial terms with the k-value from part (i), then simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q01b", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial", "numerical-methods" ], "primaryTopicId": "binomial", "summary": "Use the expansion to approximate the stated nearby value to six decimal places.", "answerGuide": "Substitute the specified x into the truncated expansion and retain the requested precision.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q02", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 7, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the exact volume formed by rotating the bounded region about the x-axis.", "answerGuide": "Solve the curve-boundary intersection for the lower limit, integrate the squared curve radius, and subtract the contribution from the horizontal boundary.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q03a", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the time taken to fill the bowl.", "answerGuide": "Evaluate V at h=20, then divide by the constant inflow rate.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q03b", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find dh/dt when h=5.", "answerGuide": "Differentiate V(h), use dV/dt=160, and apply the chain rule dV/dt=(dV/dh)(dh/dt).", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q04", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 8, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the exact logarithmic integral using the stated substitution.", "answerGuide": "Set u=√x, transform dx and the denominator, integrate the resulting rational function, then apply the limits and express the answer in the requested logarithmic form.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q05a", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the implicit curve equation to obtain dy/dx.", "answerGuide": "Differentiate both sides with respect to x, collect the terms containing dy/dx, and rearrange.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q05b", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find where the tangents at the two y-axis intersections meet.", "answerGuide": "Find the non-origin y-axis intersection, determine both tangent gradients by implicit differentiation, form their tangent equations, and solve simultaneously.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q06ai", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 3, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Show the required integral expression for the area of R.", "answerGuide": "Use x=2cos²t and y=4sin t in the parametric area integral and change to the stated t-limits.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q06aii", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact area of R.", "answerGuide": "Integrate the trigonometric expression from part (i) algebraically and substitute the limits.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q06b", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that every point on the parametric curve satisfies a linear relation between x and y.", "answerGuide": "Eliminate t using sin²t+cos²t=1, then simplify to the required y=ax+b form.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q06c", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the range of f.", "answerGuide": "Use the domain −2≤x≤2 and the linear function found in part (b) to identify its endpoint values.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q07a", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the unit vector OA parallel to the given line.", "answerGuide": "Take the line’s direction vector, divide it by its magnitude, and orient it parallel to l.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=14", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q07b", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the point X on the line that is closest to the origin.", "answerGuide": "Parameterize X, use the perpendicularity condition OX·direction=0 to solve for the parameter, then obtain its coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=14", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q07c", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of triangle OXA.", "answerGuide": "Use one half the magnitude of the vector cross product OA×OX, or the equivalent base-height method.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=14", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q08a", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 7, "topicIds": [ "differentiation", "integration", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the differential equation with the given initial value.", "answerGuide": "Separate y and x, integrate both sides, apply y(0)=1, and rearrange to y²=g(x).", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=16", "qpPages": [ 26, 26 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q08b", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find the horizontal asymptote of y²=g(x).", "answerGuide": "Take the limit of g(x) as x tends to the end of its domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=16", "qpPages": [ 26, 26 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q09i", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Prove the position-vector relation for collinear A, B and C.", "answerGuide": "Write AB and AC in terms of a,b,c and use AB:AC=1:3 with B between A and C.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=17", "qpPages": [ 30, 30 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WMA14-01-q09ii", "paperId": "ial-2021-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)", "marks": 5, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove by contradiction that 3 dividing n² implies 3 divides n.", "answerGuide": "Assume n is not divisible by 3, write n=3k+1 or 3k+2, and show neither square is divisible by 3.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/8c/15/e8/8a/27/2f/dc/9a/eb/e0/64/16/15/7d/20.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/6e/13/35/d7/37/cd/fa/1f/19/e6/ec/2b/2e/a6/b7/db.pdf#page=17", "qpPages": [ 30, 30 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q01a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the magnitude of the impulse exerted on Q in the collision.", "answerGuide": "Use impulse=change in momentum for Q, taking care with the reversal of its direction.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q01b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find both possible values of k.", "answerGuide": "Apply conservation of momentum for the two possible post-collision directions, then solve both equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q02", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 8, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the travel time from A to B under constant acceleration.", "answerGuide": "Use the first-three-second displacement and fourth-second displacement to form equations in u and a; solve them, then use v=u+at with v=20.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q03a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "mechanics" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the angle between F3 and −j when the forces are in equilibrium.", "answerGuide": "Set the vector sum to zero to find F3, then use its components to determine the angle from −j.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q03b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "mechanics" ], "primaryTopicId": "coordinate-geometry", "summary": "Find λ from the acceleration magnitude after F3 is replaced.", "answerGuide": "Find the resultant of F1,F2,F4, equate its magnitude to ma with a=3.25, and solve for positive λ.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q04a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension in the cable while the loaded bucket accelerates upward.", "answerGuide": "Apply Newton’s second law to the combined bucket and cement mass, with upward acceleration 0.2 m s⁻².", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q04b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction on the tools box during downward deceleration.", "answerGuide": "Apply Newton’s second law to the 12 kg box with upward acceleration 0.1 m s⁻².", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q05a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "coordinate-geometry", "kinematics" ], "primaryTopicId": "coordinate-geometry", "summary": "Find T when particle P moves in the specified vector direction.", "answerGuide": "Find P’s constant acceleration from the two velocity vectors, form vP(T), and require its components to have ratio −1:2.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q05b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "kinematics" ], "primaryTopicId": "coordinate-geometry", "summary": "Find μ when the two particles have equal velocities at an instant.", "answerGuide": "Equate the i- and j-components of vP(t) and vQ(t), then solve the simultaneous equations for μ.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q06a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction magnitude while the particle moves up the plane.", "answerGuide": "Find the normal reaction from the perpendicular component of weight, then use F=μR.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q06b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the distance AB before the particle first comes to rest.", "answerGuide": "Resolve forces along the plane to get the upward-motion acceleration, then use v²=u²+2as with final speed zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q06c", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find BC when the particle slides down and returns to speed 5 m s⁻¹.", "answerGuide": "Find the downward acceleration from gravity and friction, then use v²=u²+2as from rest at B.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=13", "qpPages": [ 20, 20 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q07a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass distance x for the non-uniform beam.", "answerGuide": "Use vertical force balance to relate the two support reactions, then take moments about one support and solve for x.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=14", "qpPages": [ 24, 24 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q07b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the mass M that equalizes the support reactions after the box is added.", "answerGuide": "Use vertical force balance and moments for the beam-plus-box system, with the box acting at E.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=14", "qpPages": [ 24, 24 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q08a", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Sketch both trams’ speed-time graphs up to the overtake.", "answerGuide": "Plot each constant-acceleration ramp followed by a horizontal constant-speed segment, with B starting at t=20 s.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=15", "qpPages": [ 28, 28 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q08b", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the lead d when the trams have equal speeds.", "answerGuide": "Find the equal-speed time from their speed-time descriptions, then subtract the distances travelled by the two trams.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=15", "qpPages": [ 28, 28 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME01-01-q08c", "paperId": "ial-2021-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the distance from the depot when tram B overtakes tram A.", "answerGuide": "Equate their displacement expressions after B starts and solve for the later meeting time; substitute into either displacement.", "qpUrl": "https://www.alevelcopilot.com/whfile/d3/a1/4b/78/5f/b8/ff/ef/13/e3/47/1e/a0/1f/1c/9f.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/0c/ac/9b/d0/b3/e3/d5/2c/e2/0a/f1/4d/7d/37/50/00.pdf#page=15", "qpPages": [ 28, 28 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q01a", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the probability tree for the two counter selections.", "answerGuide": "Use the initial red/blue/yellow proportions; after a non-yellow first draw, update the remaining counts before the second draw.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q01b", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability that Gina has two counters given that a yellow counter was selected.", "answerGuide": "Compare the path probability for a yellow second counter with the total probability of selecting yellow on either draw.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q02a", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Identify the mutually exclusive pair of events in the Venn diagram.", "answerGuide": "Find the pair whose regions do not overlap in the diagram.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q02b", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use independence of A and C and P(A)=0.65 to determine r and s.", "answerGuide": "Apply P(A∩C)=P(A)P(C), then use the diagram totals and complements to solve for the two regions.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q02c", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 6, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find exact fractions p and q from the second independence condition.", "answerGuide": "Express P(A∩C′) and P(B∪C) in terms of the labelled regions, equate their product to the intersection probability, and solve with total probability.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q03a", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the histogram bar dimensions for the 15≤L<20 class.", "answerGuide": "Scale bar area to frequency density using the given bar, then multiply density by class width for the height.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q03b", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median carrot length by linear interpolation.", "answerGuide": "Locate the 50th observation in the cumulative frequencies and interpolate within its class.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q03ci", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean carrot length from the grouped table.", "answerGuide": "Use the class midpoints weighted by frequency and divide by the total frequency.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q03cii", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the standard deviation of the grouped lengths.", "answerGuide": "Use the grouped-data formula based on Σfx and Σfx², then take the square root of the variance.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q03d", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Estimate the proportion of carrots the supermarket will buy.", "answerGuide": "Add the class-frequency proportions within 9≤L≤22, interpolating within boundary classes as needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q03e", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Estimate the farm’s mean profit per carrot.", "answerGuide": "Weight each profit or loss rate by the estimated proportion in its length range and add the contributions.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q04a", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Estimate the proportion of letters weighing less than 120 g.", "answerGuide": "Standardize 120 using the given normal mean and standard deviation, then read the lower-tail probability.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q04b", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the two weight limits that divide the letters into three equal categories.", "answerGuide": "Use the 1/3 and 2/3 normal quantiles and transform each z-score back to weight.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q04c", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Given a heavy letter, find the probability it weighs less than 200 g.", "answerGuide": "Use the heavy-category lower cutoff from part (b) and compute the conditional probability within that truncated upper tail.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q04d", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a sample of three contains one letter from each category.", "answerGuide": "Use the multinomial count 3! and the category probabilities 1/3, 1/3, 1/3.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q05a", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Use E(X)=0.5 to find a.", "answerGuide": "Use the symmetry of the probability table to form the expectation equation, together with total probability.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q05b", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Use Var(X)=5.01 to find b and c.", "answerGuide": "Combine total probability with E(X²)−[E(X)]²=5.01 and solve for b and c.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q05ci", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(Y) for Y=5−8X.", "answerGuide": "Use linearity: E(5−8X)=5−8E(X).", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q05cii", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Var(Y) for Y=5−8X.", "answerGuide": "Use Var(a+bX)=b²Var(X), so multiply Var(X) by 64.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q05d", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find P(4X²>Y).", "answerGuide": "Substitute Y=5−8X, solve the inequality for each possible X value, then add the corresponding probabilities.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06a", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show that Syy has the stated value.", "answerGuide": "Use Syy=Σy²−(Σy)²/n with the supplied summaries.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06b", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find Sxx.", "answerGuide": "Use the regression intercept a=ȳ−b x̄ to find x̄ and Σx; then calculate Sxx=Σx²−(Σx)²/n.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06c", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the product-moment correlation coefficient.", "answerGuide": "Use the regression gradient and the summary statistics to obtain r, with its sign matching the negative slope.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06d", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain how Behrouz concluded wages would rise by more than 3% at x=2%.", "answerGuide": "Substitute x=2 into the regression line and interpret the predicted y-value.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06e", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the minimum and maximum x using mean±3 standard deviations.", "answerGuide": "Recover the mean and standard deviation of x from the summaries, then calculate both endpoints.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06f", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(f)", "marks": 2, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Comment on the reliability of Behrouz’s claim.", "answerGuide": "Use the correlation strength and the sample’s scatter or fit to judge whether the prediction is reliable.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST01-01-q06g", "paperId": "ial-2021-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(g)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Assess Andi’s suggestion to use the y-on-x regression line to estimate x.", "answerGuide": "To estimate x from y, the regression of x on y is needed; the supplied y-on-x regression line is not the correct model.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/94/10/9b/16/68/ba/ba/39/e0/b6/cb/a5/ba/3d/27.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/47/84/03/ef/6c/a0/9d/a0/0a/58/a9/cc/18/1c/6f/8b.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q01", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate an indefinite integral in simplest form.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q02a", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the sine rule to find the sine of an angle in a triangle.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q02b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the longest-side condition to select the obtuse angle and find its size.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q03i", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Show a radical expression is an integer and evaluate it.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q03ii", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Simplify a quotient involving surds into rational and surd terms.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q04i", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch a vertical translation of a curve, showing its extrema and asymptote.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q04ii", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch the reflection of a curve in the vertical axis, showing its extrema and asymptote.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q05a", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Determine a quadratic from its turning point and negative root.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": { "label": "5(a)", "marks": 4 }, "studyPriority": 4, "difficultyEvidence": "Only a minority gained full marks on Question 5; many ignored the quadratic scale factor or failed to use all the given conditions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WMA11-01-q05b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a perpendicular line through a given point.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q05c", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Describe the shaded region using inequalities.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": { "label": "5(c)", "marks": 2 }, "studyPriority": 5, "difficultyEvidence": "The examiner notes that the negative quadratic confused some candidates when choosing the inequality for the curved boundary.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WMA11-01-q06a", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Eliminate a variable to derive the stated quartic equation.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q06b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the simultaneous equations exactly and give simplified surd answers.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "hardestPart": { "label": "6(b)", "marks": 5 }, "studyPriority": 3, "difficultyEvidence": "Many candidates used calculator equation-solvers despite the non-calculator instruction, then omitted one sign/root or the paired y-values.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2022-6-WMA11-01-q07a", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use a second-derivative condition to determine a constant.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 18, 18 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q07b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate the derivative and use a point on the curve to determine the function.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 18, 18 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q08a", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use an arc length and the central angle to find another arc length.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 22, 22 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q08b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of a three-section fan face.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 22, 22 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q08c", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the area of the modelled fan face.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 22, 22 ], "msPages": [ 17, 17 ], "hardestPart": { "label": "8(c)", "marks": 5 }, "studyPriority": 2, "difficultyEvidence": "The area work was described as very discriminating; candidates often struggled to identify the second area that had to be combined.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2022-6-WMA11-01-q09ai", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "State a sine value using the supplementary-angle identity.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 26, 26 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q09aii", "paperId": 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null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q09b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Sketch the graph of y = sin(2x).", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 26, 26 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, 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"label": "9(c)", "marks": 3 }, "studyPriority": 1, "difficultyEvidence": "Most candidates scored zero on this part; only the strongest found both solutions using sine symmetry.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2022-6-WMA11-01-q10a", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the cubic curve.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 28, 28 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q10b", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Use the shared normal and tangent condition to derive the stated quadratic.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 28, 28 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q10c", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the quadratic and justify the root representing the second point.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 28, 28 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA11-01-q10d", "paperId": "ial-2022-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(d)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the normal line intercept to determine the curve constant.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/f4/8f/f3/5e/f4/db/9e/3b/b9/d3/e8/b0/46/91/10.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/54/98/ea/69/7b/d0/76/f8/c7/c4/60/e7/1b/41/61/6a.pdf", "qpPages": [ 28, 28 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q01", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand the binomial expression through four terms with integer coefficients.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q02a", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Complete a table of function values to the required accuracy.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q02b", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the trapezium rule to estimate a definite integral.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q02c", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the first integral estimate to find a related integral.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": { "label": "2(c)", "marks": 3 }, "studyPriority": 2, "difficultyEvidence": "This part was extremely discriminating: most candidates could not rewrite the new integral using the earlier estimate, and the negative sign was often lost.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2022-6-WMA12-01-q03i", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "proof", "sequences" ], "primaryTopicId": "proof", "summary": "Give a counterexample to disprove a claim about consecutive cubes.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q03ii", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Prove that a chord is a diameter using the three points on a circle.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q04", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Solve simultaneous equations involving logarithms and positive constants in exact form.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q05", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over the stated degree interval.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q06a", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Show the common ratio satisfies a quadratic equation.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q06b", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the first term given convergence of the geometric series.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q06c", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum to infinity.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": { "label": "6(c)", "marks": 2 }, "studyPriority": 4, "difficultyEvidence": "Many candidates chose r=2 even though convergence was required, so they did not continue with the valid ratio.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WMA12-01-q07", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": null, "marks": 7, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Determine two cubic coefficients from a factor condition and a definite-integral condition.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": 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predicted remaining amount for Item A.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 24, 24 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q09d", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use logarithms to estimate the age of Item B.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 24, 24 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q10a", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Show the condition for a line to intersect the given circle.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 28, 28 ], "msPages": [ 24, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA12-01-q10b", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the tangency condition to express the radius and line constant relation.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 28, 28 ], "msPages": [ 24, 25 ], "hardestPart": { "label": "10(b)", "marks": 3 }, "studyPriority": 3, "difficultyEvidence": "Many candidates did not recognize that tangency can be established by setting the quadratic discriminant to zero; scalar-product approaches often failed.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2022-6-WMA12-01-q10c", "paperId": "ial-2022-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the length AB condition to determine the line constant.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/9b/51/05/9e/58/46/58/af/1d/b4/a3/c0/17/69/52/89.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/ed/22/c5/31/81/36/ec/33/73/af/2b/98/13/43/50/7e.pdf", "qpPages": [ 28, 28 ], "msPages": [ 25, 26 ], "hardestPart": { "label": "10(c)", "marks": 6 }, "studyPriority": 1, "difficultyEvidence": "The examiner calls this the most difficult part of the paper; it was often blank or answered only with minimal attempts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2022-6-WMA13-01-q01a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a power of a linear expression.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q01b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation at the given point in integer-coefficient form.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q02a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate a composite function.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q02b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse function.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q02c", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve an equation involving the inverse and a linear function.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q03a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate an expression involving a logarithm.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q03b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the given integral equality to determine the positive constant.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q04a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a straight line through two plotted points.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q04b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use the logarithmic graph to find the model parameters.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q04c", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use the model to find when the subscriber count reaches the target.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q05ai", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the y-intercept of the graph.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q05aii", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the x-coordinate of the minimum in terms of k.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q05b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the range of x satisfying the inequality.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q05c", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Use the two-intersection condition to determine possible values of k.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": { "label": "5(c)", "marks": 3 }, "studyPriority": 1, "difficultyEvidence": "The examiner describes this as the most challenging part of Question 5 and one of the most difficult parts of the paper; many candidates had no valid approach or left it blank.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WMA13-01-q06a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the composite function and determine the integer constant.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 18, 18 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q06b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the stationary x-values.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 18, 18 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q06c", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of the local maximum.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 18, 18 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q06d", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the range of the transformed function on the stated interval.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 18, 18 ], "msPages": [ 19, 19 ], "hardestPart": { "label": "6(d)", "marks": 3 }, "studyPriority": 2, "difficultyEvidence": "Correct range answers were rare; candidates often identified only one boundary of the range.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2022-6-WMA13-01-q07a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the trigonometric equation in the required form without calculator-only work.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 22, 22 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q07b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the equation over the stated radian interval.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 22, 22 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q08a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Use calculus to find the sprinter model’s maximum velocity.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 26, 26 ], "msPages": [ 22, 23 ], "hardestPart": { "label": "8(a)", "marks": 5 }, "studyPriority": 3, "difficultyEvidence": "Pearson lists this among the paper's most challenging parts; candidates struggled to solve the exponential stationary-point equation and were required to show calculus.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2022-6-WMA13-01-q08b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Integrate the velocity model and derive the equation for race time.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 26, 26 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q08ci", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)(i)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Apply one iteration to calculate the second estimate of race time.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 26, 26 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q08cii", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)(ii)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration to estimate the total race time.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 26, 26 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q09a", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show the x-coordinate of the curve’s minimum satisfies the stated equation.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 30, 30 ], "msPages": [ 24, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA13-01-q09b", "paperId": "ial-2022-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve for the minimum point’s x-coordinate to the required accuracy.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/72/38/12/46/c2/7a/b2/ac/9a/e6/fa/fc/38/a8/9b/75.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/94/8d/8c/04/fb/a1/50/00/53/e3/d0/c3/0b/2c/12/2a.pdf", "qpPages": [ 30, 30 ], "msPages": [ 25, 27 ], "hardestPart": { "label": "9(b)", "marks": 5 }, "studyPriority": 4, "difficultyEvidence": "This final trigonometric work was demanding and often unattempted; squaring approaches could introduce extraneous roots.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2022-6-WMA14-01-q01a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the constant coefficient in the binomial expansion.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q01b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", 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"WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(ii)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use k to find the cubic coefficient.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q02a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the rational function as partial fractions.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q02b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A 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stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": { "label": "3(b)", "marks": 4 }, "studyPriority": 3, "difficultyEvidence": "Question 3 was poorly answered; candidates often reused the single dx/dt value from part (a) rather than carrying forward the rate expression for general x.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WMA14-01-q04a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate the curve and obtain the stated derivative.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q04b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use the turning-point condition to find k.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q05a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use a trigonometric substitution to transform the integral.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q05b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the resulting definite integral exactly.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q06a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Write a vector equation for the line through A and B.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q06b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use perpendicularity to determine p.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q06c", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the area of triangle ABC in simplest surd form.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q07a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Verify the derivative for the parametric curve at the specified parameter.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q07b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent equation at point P.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q07c", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Derive the parameter equation for the second intersection.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q07d", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact y-coordinate at the second intersection.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q08a", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Set up the volume of revolution integral and identify its constant factor.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 26, 26 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WMA14-01-q08b", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the integral involving x times an exponential.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 26, 26 ], "msPages": [ 15, 15 ], "hardestPart": { "label": "8(b)", "marks": 3 }, "studyPriority": 5, "difficultyEvidence": "The examiner describes Question 8 as discriminating and identifies integration by parts as a stumbling block, with sign and setup errors common.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2022-6-WMA14-01-q08c", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the volume results and the joined-solid model to calculate density.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 26, 26 ], "msPages": [ 15, 15 ], "hardestPart": { "label": "8(c)", "marks": 5 }, "studyPriority": 4, "difficultyEvidence": "Full marks on the final density calculation were very rare; candidates often omitted a factor from the previous part or failed to double the volume.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2022-6-WMA14-01-q09", "paperId": "ial-2022-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": null, "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Use contradiction to prove the divisibility statement for every integer n.", "answerGuide": "Set up the relevant identity or equation, show the exact algebra or calculus steps, and check the result against the stated conditions.", "qpUrl": "https://www.alevelcopilot.com/whfile/f4/b4/6f/07/45/3c/b4/89/59/8d/d5/3d/2d/6d/e9/a3.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b1/86/02/cf/90/c2/de/4d/02/cd/41/5c/1a/c6/7a/c9.pdf", "qpPages": [ 30, 30 ], "msPages": [ 16, 18 ], "hardestPart": { "label": "Question 9", "marks": 4 }, "studyPriority": 1, "difficultyEvidence": "The report says the paper's clearest weakness was Question 9: it had a zero modal score and a much larger zero-score share than the other zero-mode questions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2022-6-WME01-01-q01a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use impulse and momentum to find Q’s post-collision speed.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q01b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "State the direction of Q after the collision.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q02a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the motorbike’s acceleration from its displacement and endpoint speed.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q02b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled during the fifth second.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": { "label": "2(b)", "marks": 4 }, "studyPriority": 4, "difficultyEvidence": "Only a few candidates interpreted the fifth second as the interval from t=4 to t=5; 95 was the common wrong answer.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2022-6-WME01-01-q03a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Use the tractor and block equations of motion to find the resistance.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q03b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "State the consequence of modelling the cable as inextensible.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q04a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find friction and determine whether the block moves for the applied force.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q04b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the force when the block is on the point of slipping.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q05a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the reaction at support C in terms of M.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 14, 14 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q05b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the reaction at support D in terms of M.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 14, 14 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q05c", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Use the non-negative support reactions to find the possible range of M.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": { "label": "5(c)", "marks": 3 }, "studyPriority": 1, "difficultyEvidence": "This final range part challenged all but the strongest candidates and attracted many zero scores.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2022-6-WME01-01-q06", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": null, "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the particle’s speed at the specified time from two velocity vectors.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 18, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q07a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch both children’s speed-time graphs on the same axes.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": { "label": "7(a)", "marks": 3 }, "studyPriority": 5, "difficultyEvidence": "Many candidates missed that B's graph needed a steeper gradient and that the two graphs should cross on shared axes.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WME01-01-q07b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find Ajaz’s time to finish the race.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 20, 20 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q07c", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Use the equal finishing times to find Beth’s acceleration duration.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 20, 20 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q07d", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the difference between finishing speeds.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 20, 20 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q08a", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the direction of Q’s velocity as a bearing.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 24, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q08b", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Derive the relative position vector of the two boats.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 24, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME01-01-q08c", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the bearing of P from Q at the stated time.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 24, 24 ], "msPages": [ 17, 17 ], "hardestPart": { "label": "8(c)", "marks": 2 }, "studyPriority": 2, "difficultyEvidence": "The correct bearing was rarely seen; 90 degrees was the most common wrong answer and many candidates spent time finding unnecessary positions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WME01-01-q08d", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance between the boats when Q is northeast of P.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 24, 24 ], "msPages": [ 17, 17 ], "hardestPart": { "label": "8(d)", "marks": 5 }, "studyPriority": 3, "difficultyEvidence": "The examiner described this as a good discriminator; common errors were finding t incorrectly or making arithmetic slips.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WME01-01-q08e", "paperId": "ial-2022-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(e)", "marks": 3, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the times when the boats are 200 metres apart.", "answerGuide": "Choose the relevant force or motion model, show the equations clearly, and state the result with units and direction where needed.", "qpUrl": "https://www.alevelcopilot.com/whfile/f6/7f/b6/1b/ab/3c/40/22/3d/da/fd/a1/44/34/30/1b.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/d8/c3/ce/df/24/3c/5c/69/00/dc/52/36/d3/59/7f/fa.pdf", "qpPages": [ 24, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q01a", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the missing values in the stem-and-leaf summary and quartile table.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q01b", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify outliers using the stated quartile rule.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q01c", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a box plot for the data.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q01d", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the quartile-based skewness.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q01e", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain why the employer may prefer the mean to the median.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q02a", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product moment correlation coefficient.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q02b", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the correlation coefficient in context.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q02c", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the least-squares regression line of GDP on population.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q02d", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the regression slope.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q02ei", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate GDP for a population of seven million.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q02eii", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Comment on the reliability of the estimate in context.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", 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"May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the histogram bar dimensions using the supplied scale.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q03b", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median log length by linear interpolation.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q03c", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show how many logs meet the stove length limit.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q03d", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, 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"year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)(ii)", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the coding relation to find the variance of log weight.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q04a", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use independence to show the probability of event W.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 14, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q04b", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability involving N and H.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 14, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q04c", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Draw a Venn diagram with the exact probabilities for H, W and N.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 14, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q05a", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the second moment of the red-spinner score.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 18, 18 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q05b", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation of the red-spinner score.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 18, 18 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q05c", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Read the final cumulative probability from the distribution table.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 18, 18 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q05d", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability Jessie wins on her second spin.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 18, 18 ], "msPages": [ 11, 11 ], "hardestPart": { "label": "5(e)", "marks": 2 }, "studyPriority": 2, "difficultyEvidence": "This was among the paper's most challenging parts; very few candidates recognized that three event probabilities had to be multiplied.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WST01-01-q05f", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(f)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability both first spins give the same score.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 18, 18 ], "msPages": [ 11, 11 ], "hardestPart": { "label": "5(f)", "marks": 3 }, "studyPriority": 3, "difficultyEvidence": "This part was discriminating; the common 0.195 error came from failing to stop the game when Jessie first scored 2.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-6-WST01-01-q06a", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the normal probability that a bottle contains more than the stated volume.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 22, 22 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q06b", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected count above the volume threshold in 150 bottles.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 22, 22 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST01-01-q06c", "paperId": "ial-2022-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the threshold from a conditional normal probability.", "answerGuide": "State the data or probability model, show the calculation, and interpret the result in the context given.", "qpUrl": "https://www.alevelcopilot.com/whfile/92/3c/53/86/49/c4/67/59/a6/1b/c0/0b/68/2a/f6/11.pdf", "msUrl": "https://www.alevelcopilot.com/whfile/b0/4e/b4/f2/b3/da/01/b8/ce/b9/47/40/cc/c9/f9/91.pdf", "qpPages": [ 22, 22 ], "msPages": [ 12, 12 ], "hardestPart": { "label": "6(c)", "marks": 6 }, "studyPriority": 1, "difficultyEvidence": "Pearson calls this the most discriminating part; candidates often used the wrong numerator or failed to respect the conditional event.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220818.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2021-6-WDM11-01-q01a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound for the number of containers.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q01b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Allocate parcels using first-fit bin packing.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q01c", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Bubble-sort the parcel weights into descending order.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q01d", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Allocate parcels using first-fit decreasing.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q02a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the early and late event times.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q02b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw a cascade chart for the activity network.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q02c", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the chart to state the minimum workers, time window and activities.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q03a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "numerical-methods", "decision-mathematics" ], "primaryTopicId": "numerical-methods", "summary": "Iterate the supplied flowchart to find the root to five decimal places.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q03b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "numerical-methods", "decision-mathematics" ], "primaryTopicId": "numerical-methods", "summary": "Identify non-negative inputs that cannot be used by the root-finding flowchart.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q04a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the least-distance tables for the road network.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q04b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a nearest-neighbour upper-bound route from A.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q04c", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Translate the route into the towns visited.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q04d", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound by deleting A and its arcs.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q04e", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the route-inspection length and repeated edges by pairing odd nodes.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q05a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain why Dijkstra’s algorithm should start at A.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q05b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find shortest routes from A to J and K.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q05c", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give a shortest route from F to H that passes through A.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q06a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network and include required dummies.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q06b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify a guaranteed critical activity and justify it.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q06c", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the project completion time and critical path.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q07a", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the minimum spanning tree property to justify a stated inequality.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q07b", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Derive further constraints on x and y.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q07c", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Plot the constraints on the feasible-region diagram.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q07d", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find integer coordinate pairs satisfying the constraints.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WDM11-01-q07e", "paperId": "ial-2021-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the minimum spanning tree weight to determine x and y.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202021%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q01", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the constant speed of a powered motorcycle descending a slope against resistance.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q02a", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the force vector from the particle’s velocity at t = 2.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q02b", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the position vector when the particle moves in the j direction.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q03a", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a composite lamina.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q03b", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the centre of mass to determine a constant in the tension ratio.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q04", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the magnitude of an impulse from the change in a particle’s velocity.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q05a", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cable tension holding a hinged uniform rod in equilibrium.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q05b", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the hinge reaction on the rod.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q06a", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the gravitational potential energy lost by the two-particle system.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q06b", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Calculate the work done against friction.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q06c", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed immediately before impact using work and energy.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q06d", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance travelled after the specified impact event.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q07a", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the horizontal range of a projectile.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q07b", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the least height above which the projectile speed is below 10 m s−1.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q07c", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile velocity when it is perpendicular to its terminal velocity.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q08ai", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)(i)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show the speed of B after its collision with A.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q08aii", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)(ii)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed of A after colliding with B.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME02-01-q08b", "paperId": "ial-2021-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show that A and B collide again after the later B–C collision.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q01", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use surface-area masses and moments to find the shell’s mass-density ratio.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q02a", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension in the rotating ride’s arm.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 4, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q02b", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the ride’s angular speed.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 4, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q03a", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Show the volume of the solid formed by rotation.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q03b", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "integration", "mechanics" ], "primaryTopicId": "integration", "summary": "Find the centre-of-mass x-coordinate of the solid.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q04a", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cable tension immediately after release.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q04b", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the greatest tension is four times the least tension.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q05a", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Derive the speed as a function of displacement.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q05b", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use integration to find the time at the stated displacement.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q06a", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed after the particle first reaches B.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q06b", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the total distance travelled before coming to rest.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q06c", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle remains at rest at C.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q07a", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the equilibrium position between the two elastic strings.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q07bi", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Show that the displaced particle performs SHM about equilibrium.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q07bii", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the SHM amplitude after the impulse.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q07c", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the speed when one string becomes slack.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WME03-01-q07d", "paperId": "ial-2021-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the travel time from equilibrium to the slack point.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202021%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q01ai", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a packet probability for a binomial count of ungerminated seeds.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q01aii", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that more than 18 seeds germinate.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q01b", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance all five packets exceed the germination threshold.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q01c", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses for the one-tailed claim about the failure rate.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q01d", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a Poisson approximation to test the seed supplier’s claim.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q02a", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of exactly five faults in a four-square-metre rug.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q02b", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance of more than five faults in a six-square-metre rug.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q02c", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Give the next rug’s probability and explain independence.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q02d", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the smallest number of small rugs needed for an expected profit target.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q02e", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether Rhiannon has a higher fault rate than Luis.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q03a", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the piecewise probability density function.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q03b", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "differentiation" ], "primaryTopicId": "distributions", "summary": "Find the mode of the continuous distribution.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q03c", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use algebraic integration to find E(Y²).", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q03d", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Calculate Var(Y) from the given mean and second moment.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q03e", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the 90th percentile from the tail-probability condition.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q04i", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the sample range is four.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q04ii", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of the sample range.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q05ai", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use continuity of the CDF at zero to find its parameter.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q05aii", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use the CDF endpoint to find its second parameter.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q05b", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "distributions", "differentiation" ], "primaryTopicId": "distributions", "summary": "Differentiate the CDF to obtain the density.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q05c", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use a probability comparison to solve for the uniform interval parameter.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q05d", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance of winning from six independent spins.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST02-01-q06i", "paperId": "ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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"ial-2021-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Solve for p and justify the admissible root.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202021%20MS.pdf", "qpPages": [ 22, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q01a", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 5, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Calculate Spearman’s rank coefficient for coverage and moisture.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q01b", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test the claimed positive rank correlation at 5%.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q02", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 9, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Use a chi-squared test to assess independence of diet and health.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q03a", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find a 90% confidence interval from the supplied 95% interval.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q03b", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly three of four intervals cover the mean.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q04a", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe a stratified sample of 70 students.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q04b", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 7, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the vocational mean score exceeds the academic mean.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q04c", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain the Central Limit Theorem’s role in the test.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q04d", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the variance assumption used for the test.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q04e", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson 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"paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "State two assumptions for a binomial model of medicine success.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q05b", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the mean success count from the grouped table.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q05c", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Fit a binomial model and find expected grouped frequencies.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q05d", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 8, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a chi-squared goodness-of-fit test for the binomial model.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q06a", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability a bun weighs below 110% of a roll.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 16, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q06b", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance two bun weights differ by more than one gram.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 16, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2021-6-WST03-01-q06c", "paperId": "ial-2021-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance a bag of buns or rolls exceeds the stated weight threshold.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202021%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202021%20MS.pdf", "qpPages": [ 16, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q01a", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound for the number of trucks.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q01b", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Bubble-sort the crate weights into descending order.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q01c", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Pack the crates using first-fit decreasing.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q02a", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find x and show y from the stated critical path and float relation.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q02b", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the early and late event times.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q02c", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw a cascade chart and find the minimum workers and project duration.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03a", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct a minimum spanning tree using Kruskal’s algorithm.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03b", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct a minimum spanning tree using Prim’s algorithm.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03c", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the MST weight.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03d", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the MST to give an initial route upper bound.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03e", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a nearest-neighbour upper bound from W.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03f", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Choose the best upper bound and explain.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03g", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(g)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a route lower bound by deleting W.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q03h", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(h)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Interpret the chosen route as towns visited.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q04ai", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Eliminate z to obtain simplified inequalities in x and y.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q04aii", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the maximum value of the objective P.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q04bi", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)(i)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the maximum possible x at the optimal objective value.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q04bii", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give a solution (x,y,z) to the linear programme.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q05a", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network with the minimum number of dummies.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q05bi", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the minimum project completion time.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q05bii", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify the critical activities.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q05biii", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(iii)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the total floats on activities G and K.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q06a", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra’s algorithm to find the fastest A-to-H route.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q06b", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use route inspection to bound x from the minimum inspection time.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "In the June 2022 Principal Examiner Feedback, only the most able candidates connected the provided value at J in part (a) to the upper bound for x in part (b).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q06c", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give a possible inspection route.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q06d", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the new D–H road and exact route length to determine x.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q07ai", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find coordinates of the two labelled intersections.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q07aii", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine inequalities defining the feasible region.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WDM11-01-q07b", "paperId": "ial-2022-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the largest k that leaves the feasible region unchanged.", "answerGuide": "Use the stated algorithm and show the intermediate states and interpretation.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/June%202022%20MS.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q01a", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration vector at t = 4.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q01b", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time when velocity is parallel to the given vector.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q02a", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total time for two wall bounces and return to A.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q02b", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the next cycle time after the particle’s speed changes.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q03", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use impulse and momentum to find the two possible initial speeds.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q04", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 8, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use power and equations of motion to determine the engine/resistance quantity.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q05a", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cable tension holding the rod in equilibrium.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q05b", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction at the rod’s contact with the ground.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q06ai", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use the impact law and impulse data to relate y to v.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q06aii", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use the impact law and momentum to relate x to v.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q06aiii", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(iii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the mass ratio k.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q06b", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the kinetic energy lost in the collision.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q07a", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass distance of the composite lamina from QT.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q07b", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Use the centre of mass to find the equilibrium angle.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q08a", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Calculate work done against friction on the ramp.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q08b", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the initial speed needed to reach the top.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME02-01-q08c", "paperId": "ial-2022-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time to stop and the horizontal distance travelled.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M2/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q01a", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the maximum speed in the SHM.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q01b", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the travel time from the centre to the stated midpoint.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q02", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 9, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Use circular-motion equations to find the particle’s distance OC.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The June 2022 Principal Examiner Feedback says this unfamiliar circular-motion question provided significant challenge to candidates at all levels.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q03a", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the deceleration at the given displacement.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q03b", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Integrate the motion relation to find the time at the stated position.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q04a", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use energy to establish the elastic-string modulus.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q04b", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 7, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed during the ball’s motion.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q05a", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "integration", "mechanics" ], "primaryTopicId": "integration", "summary": "Use algebraic integration to locate a solid hemisphere’s centre of mass.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q05b", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cone-to-hemisphere proportion for equilibrium.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The June 2022 Principal Examiner Feedback says part (b) provided significant challenge for candidates at lower grades.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q06a", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the tension as a function of angle.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q06b", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show the speed when the string becomes slack.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q06c", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the maximum height above O after the string becomes slack.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The June 2022 Principal Examiner Feedback says only high-achieving candidates accessed marks in part (c) of this question, which it calls one of the paper’s most challenging.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q07a", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the modulus of the second elastic string from equilibrium.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q07b", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Show the particle performs SHM while both strings are taut.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q07c", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed when the second string becomes slack.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WME03-01-q07d", "paperId": "ial-2022-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the duration for which both strings remain taut in one oscillation.", "answerGuide": "Set up the appropriate model and equations first; show working and check the result against the stated conditions.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Mechanics/M3/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q01a", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the variance of the Poisson variable.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q01b", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the binomial mode with supporting working.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q01c", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the joint probability for one observation from each independent model.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q01d", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that X is less than W.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The June 2022 Principal Examiner Feedback describes Question 1(d) as one of the most challenging parts of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q02a", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Integrate the density to find the mean waiting time.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q02b", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "integration" ], "primaryTopicId": "probability", "summary": "Show the conditional probability for the stated waiting interval.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q02c", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a suitable approximation for the count of calls in the interval.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q03a", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability the uniform point lies right of the origin.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q03b", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability the point is more than 3.5 cm from zero.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q03c", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the cumulative distribution function.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q03di", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability all three points are more than 10 cm from zero.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q03dii", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)(ii)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability at least one point is more than 10 cm from zero.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The June 2022 Principal Examiner Feedback says part 3(d)(ii) was much more challenging and describes the common error in interpreting the event.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q04a", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the smallest n placing zero unfit pears in the lower critical region.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q04b", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a suitable approximation to test whether the overweight-pear proportion changed.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q05ai", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of exactly ten particles in the sample.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q05aii", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of between six and eleven particles inclusive.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q05b", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the binomial and Poisson models to find the sample volume m.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The June 2022 Principal Examiner Feedback says part 5(b) was very challenging for many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q06a", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use total probability to find the density parameter k.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 22, 25 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q06bi", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the endpoint a for the density.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 22, 25 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q06bii", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability over the stated interval.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 22, 25 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q06c", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Integrate to find the piecewise cumulative distribution function.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 22, 25 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q07ai", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the mean number of samples containing a 9.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 26, 28 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The June 2022 Principal Examiner Feedback calls part 7(a) the most challenging for students; this record is one of its separately marked subparts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q07aii", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the variance of that count.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 26, 28 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The June 2022 Principal Examiner Feedback calls part 7(a) the most challenging for students; this record is one of its separately marked subparts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q07b", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "List the samples whose median is seven.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 26, 28 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST02-01-q07c", "paperId": "ial-2022-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of the sample median.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S2/MS/June%202022%20MS.pdf", "qpPages": [ 26, 28 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q01a", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain how to assign tied ranks.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q01b", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate Spearman’s rank coefficient.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q01c", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test for a positive rank correlation.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q01d", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State the stronger significance threshold for the correlation test.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q02a", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find unbiased estimates of the mean and variance for brand A.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 6, 9 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"https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q02c", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain why the large samples support the test method.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q03a", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the sample mean from the confidence interval.", "answerGuide": "Define the variable and parameters, select the relevant 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"variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Name a model for the wait until the next train.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The June 2022 Principal Examiner Feedback identifies Question 4 as the most demanding question and notes that many students incorrectly chose a Poisson model in part (a).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q04b", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of the waiting time.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q04c", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the CLT to estimate the probability for the sample mean.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q04d", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State an assumption for the CLT calculation.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q05a", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain why the given expression is not a statistic.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q05b", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Define an 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"data-analysis", "summary": "Show both estimators are unbiased for the population mean.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q05d", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": 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"May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability a lift load exceeds capacity for the specified occupants.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q06b", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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the initial strategy.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-pef-20220818.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q07a", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses for a uniform model of the die outcomes.", "answerGuide": "Define the variable and parameters, select the relevant distribution or test, and state the result in context.", "qpUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/QP/June%202022%20QP.pdf", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/June%202022%20MS.pdf", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2022-6-WST03-01-q07bi", "paperId": "ial-2022-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 8, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a chi-squared test to determine the count parameter.", "answerGuide": "Define the variable and 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"primaryTopicId": "functions", "summary": "Map P(6, −2) under the transformation y = 2f(3x).", "answerGuide": "Apply the horizontal scale factor and vertical stretch to the point coordinates.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q01b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Map P under the transformation y = f(x − 2) + 8.", "answerGuide": "Translate the point by the horizontal and vertical shifts in the transformed function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Pearson says parts (b) and (c) were the most difficult parts of an otherwise straightforward opening question.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q01c", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the image of P on the inverse curve y = f⁻¹(x).", "answerGuide": "Swap the coordinates of the point on y = f(x).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Pearson says parts (b) and (c) were the most difficult parts of an otherwise straightforward opening question.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q02a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Show the unique root lies in [1, 2].", "answerGuide": "Evaluate f(1) and f(2), then use the sign change and continuity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q02b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Rearrange the root equation into the stated fixed-point form and find k.", "answerGuide": "Rearrange f(x)=0 to isolate x in the form k/(2x+1).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q02ci", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)(i)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration formula to find x₃ to four decimal places.", "answerGuide": "Starting at x₁=1, apply the fixed-point formula twice and round x₃ to four decimal places.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q02cii", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)(ii)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration formula to estimate the root α to four decimal places.", "answerGuide": "Continue the given iteration and round the estimate to four decimal places.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q03a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx in terms of y for the implicitly described curve.", "answerGuide": "Differentiate the relation with respect to y, then invert dx/dy and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q03b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the point on the curve whose tangent gradient is 1/3 and which lies above the x-axis.", "answerGuide": "Set dy/dx=1/3, solve for y, substitute into the curve equation, and keep the point above the x-axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q04ai", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find A, B and C in the rational-expression identity.", "answerGuide": "Divide the numerator by x²+1 and match the quotient and remainder coefficients.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q04aii", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Show that the constant D in the remainder is zero.", "answerGuide": "Use the completed division or compare constant terms in the identity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q04b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the exact area bounded by the curve, x-axis, x=2 and x=3.", "answerGuide": "Integrate the polynomial and rational terms separately, evaluate from 2 to 3, and give the logarithm in the requested exact form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson reports that many candidates gained no marks on the integration after failing to establish the correct rational form.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q05a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the trigonometric curve to find the tangent gradient at x=π/3.", "answerGuide": "Use the chain rule on both trigonometric terms, then substitute x=π/3 and simplify exactly.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q05b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express 5cos2x−12sin2x as Rcos(2x+α) and find R and α.", "answerGuide": "Match coefficients using Rcosα=5 and Rsinα=12, then select α in the stated interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q05c", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find the greatest x-coordinate where the tangent gradient is 6.", "answerGuide": "Use the R-cosine form or the derivative, solve the trigonometric equation, and select the greatest valid x in the interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson describes this unfamiliar final trigonometric step as very discriminating; many candidates found most of the marks but not the greatest solution.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q06a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "exponentials", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate (2x−3)³e^(4x−2) and determine P, Q and n.", "answerGuide": "Apply the product rule and factor the result into the required form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q06bi", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinates of the stationary points.", "answerGuide": "Set the factored derivative equal to zero and solve for both x-values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q06bii", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact range of g on its stated restricted domain.", "answerGuide": "Use the stationary point to identify the minimum, evaluate g exactly there, and include the limiting upper value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says part (b) was more demanding, with one of its three marks a common score and fully correct solutions rare.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q07a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Show sin4θ in the required form using sinθ and cosθ.", "answerGuide": "Start with sin4θ=2sin2θcos2θ and apply the double-angle identities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q07b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the trigonometric equation as the required quadratic relation.", "answerGuide": "Substitute the identity from part (a), then divide by the appropriate sine and cosine factors and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson calls part (b) more demanding: one mark out of three was a common score, and fully correct concise proofs were rare.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q07c", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on the stated interval.", "answerGuide": "Use the relation from part (b), solve for the valid trigonometric values, and check all solutions in the interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q08a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the exact value of fg(2a).", "answerGuide": "Evaluate g(2a) first, then substitute that result into f.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q08b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse function of f and state its domain.", "answerGuide": "Set y=ln(2x−3), make x the subject, then state the domain required for the inverse.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q08c", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch y=½|g(x)| and label its horizontal asymptote.", "answerGuide": "Reflect the negative branch above the x-axis, preserve the vertical asymptote and mark y=3 as the horizontal asymptote.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson says this graph sketch was more demanding than anticipated; many candidates omitted the horizontal asymptote.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q08di", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(d)(i)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use the given larger solution x=8 to find a.", "answerGuide": "Substitute x=8 into the equation with |g(x)|=10 and solve for a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q08dii", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(d)(ii)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the smaller solution after substituting the value of a.", "answerGuide": "Solve the negative branch of |g(x)|=10 and select the smaller solution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q09a", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the antibiotic amount eight hours after the first dose.", "answerGuide": "Substitute the initial dose and elapsed time into the exponential-decay model.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q09b", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show the amount two hours after the second dose is 17.5 mg.", "answerGuide": "Add the decayed contribution from the first dose to the contribution from the second dose at the stated time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01A-q09c", "paperId": "ial-2025-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find when the combined amount from the two differently timed doses reaches 10 mg.", "answerGuide": "Write a separate decay term for each dose using its own elapsed time, set their sum equal to 10, and solve for T.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3A_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes this as probably the most demanding aspect of the paper; handling the different dose times caused the main difficulty.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q01a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rewrite a surd expression in integer-coefficient form, then use it to solve the resulting exponential equation exactly.", "answerGuide": "Use a conjugate or index-law rearrangement to establish the requested form, then substitute it into the equation and isolate the unknown power.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q01b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rewrite a surd expression in integer-coefficient form, then use it to solve the resulting exponential equation exactly.", "answerGuide": "Use a conjugate or index-law rearrangement to establish the requested form, then substitute it into the equation and isolate the unknown power.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q02a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a quadratic curve and find the tangent equation at a specified point.", "answerGuide": "Differentiate term by term, evaluate the gradient at the point, then use point-gradient form and simplify to y = mx + c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q02b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a quadratic curve and find the tangent equation at a specified point.", "answerGuide": "Differentiate term by term, evaluate the gradient at the point, then use point-gradient form and simplify to y = mx + c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q03a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a linear inequality and a quadratic inequality, then find the x-values satisfying both.", "answerGuide": "Solve each inequality separately, preserving the inequality direction, and intersect the two solution sets.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q03b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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"reviewed" }, { "id": "ial-2020-1-WMA01-01-q03c", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a linear inequality and a quadratic inequality, then find the x-values satisfying both.", "answerGuide": "Solve each inequality separately, preserving the inequality direction, and intersect the two solution sets.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q04", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the given rational algebraic integral in simplest form.", "answerGuide": "Use algebraic division or partial fractions to rewrite the integrand, integrate each term and simplify the exact result.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q05i", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve one exponential equation and one logarithmic equation with a variable argument.", "answerGuide": "Use logarithms to isolate the exponent in the first equation. For the second, apply logarithm laws, solve algebraically, and check the argument restrictions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q05ii", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve one exponential equation and one logarithmic equation with a variable argument.", "answerGuide": "Use logarithms to isolate the exponent in the first equation. For the second, apply logarithm laws, solve algebraically, and check the argument restrictions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q06a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find points on a line, construct a perpendicular through one of them, and calculate the area of a quadrilateral.", "answerGuide": "Substitute the stated coordinates into the line equation, use the negative reciprocal gradient for the perpendicular, then use coordinates or triangles to find the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q06b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find points on a line, construct a perpendicular through one of them, and calculate the area of a quadrilateral.", "answerGuide": "Substitute the stated coordinates into the line equation, use the negative reciprocal gradient for the perpendicular, then use coordinates or triangles to find the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q06c", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find points on a line, construct a perpendicular through one of them, and calculate the area of a quadrilateral.", "answerGuide": "Substitute the stated coordinates into the line equation, use the negative reciprocal gradient for the perpendicular, then use coordinates or triangles to find the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q07a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the cosine and sine rules to find a garden diagonal and its area from side lengths and included angles.", "answerGuide": "Apply the cosine rule to the triangle containing the diagonal. Split the quadrilateral into two triangles and add their sine-rule areas.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q07b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the cosine and sine rules to find a garden diagonal and its area from side lengths and included angles.", "answerGuide": "Apply the cosine rule to the triangle containing the diagonal. Split the quadrilateral into two triangles and add their sine-rule areas.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q08a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Substitute a line into a quadratic curve, then determine the line parameters that make it tangent.", "answerGuide": "Substitute the line equation into the curve to obtain the given quadratic. Set its discriminant to zero and solve for the two parameter values.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q08b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Substitute a line into a quadratic curve, then determine the line parameters that make it tangent.", "answerGuide": "Substitute the line equation into the curve to obtain the given quadratic. Set its discriminant to zero and solve for the two parameter values.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q09a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model two sponsored cycling plans, one increasing by a fixed amount and the other by a fixed percentage.", "answerGuide": "Use an arithmetic-series total for the fixed daily increase. Use a geometric-series total for the percentage plan and solve for the initial distance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=17", "qpPages": [ 24, 27 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q09b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model two sponsored cycling plans, one increasing by a fixed amount and the other by a fixed percentage.", "answerGuide": "Use an arithmetic-series total for the fixed daily increase. Use a geometric-series total for the percentage plan and solve for the initial distance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=17", "qpPages": [ 24, 27 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q10a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a cubic using the factor theorem, factor a second polynomial, and simplify a rational expression into A + Bx.", "answerGuide": "Verify the given factor by substitution, divide to find the remaining factors, then factor and divide the rational expression to identify A and B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=19", "qpPages": [ 28, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q10b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a cubic using the factor theorem, factor a second polynomial, and simplify a rational expression into A + Bx.", "answerGuide": "Verify the given factor by substitution, divide to find the remaining factors, then factor and divide the rational expression to identify A and B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=19", "qpPages": [ 28, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q10ci", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a cubic using the factor theorem, factor a second polynomial, and simplify a rational expression into A + Bx.", "answerGuide": "Verify the given factor by substitution, divide to find the remaining factors, then factor and divide the rational expression to identify A and B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=19", "qpPages": [ 28, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q10cii", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a cubic using the factor theorem, factor a second polynomial, and simplify a rational expression into A + Bx.", "answerGuide": "Verify the given factor by substitution, divide to find the remaining factors, then factor and divide the rational expression to identify A and B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=19", "qpPages": [ 28, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q11aiii", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)(i)/(ii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Complete the square to find a circle centre and exact radius, then find two vertical chords of a given length.", "answerGuide": "Complete squares in x and y. Use the circle equation and the chord’s vertical midpoint to solve for the two possible line positions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=21", "qpPages": [ 30, 31 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q11b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Complete the square to find a circle centre and exact radius, then find two vertical chords of a given length.", "answerGuide": "Complete squares in x and y. Use the circle equation and the chord’s vertical midpoint to solve for the two possible line positions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=21", "qpPages": [ 30, 31 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q12a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": "12(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a binomial to three terms and use the first two terms of a related expansion to determine constants.", "answerGuide": "Apply the generalized binomial theorem in ascending powers, match the constant and linear coefficients, and solve successively for a and b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=23", "qpPages": [ 32, 33 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q12b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": "12(b)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a binomial to three terms and use the first two terms of a related expansion to determine constants.", "answerGuide": "Apply the generalized binomial theorem in ascending powers, match the constant and linear coefficients, and solve successively for a and b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=23", "qpPages": [ 32, 33 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q12c", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": "12(c)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a binomial to three terms and use the first two terms of a related expansion to determine constants.", "answerGuide": "Apply the generalized binomial theorem in ascending powers, match the constant and linear coefficients, and solve successively for a and b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=23", "qpPages": [ 32, 33 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q13i", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 13, "questionPart": "13(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find exact solutions of a sine equation in radians, then solve a mixed sine-tangent equation over a full degree interval.", "answerGuide": "Use identities and exact special angles for the first equation. For the second, rewrite tangent as sine over cosine, factor the equation and list all solutions in range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=34", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=25", "qpPages": [ 34, 37 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q13ii", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 13, "questionPart": "13(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find exact solutions of a sine equation in radians, then solve a mixed sine-tangent equation over a full degree interval.", "answerGuide": "Use identities and exact special angles for the first equation. 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"integration" ], "primaryTopicId": "integration", "summary": "Identify a curve intercept, integrate the finite region between a cubic curve and the x-axis, then use scaling to find another area.", "answerGuide": "Read the simple root from the factorization, integrate between the curve’s intercepts, then map the transformed curve’s horizontal and vertical scale factors onto the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=26", "qpPages": [ 38, 41 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q14b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 14, "questionPart": "14(b)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Identify a curve intercept, integrate the finite region between a cubic curve and the x-axis, then use scaling to find another area.", "answerGuide": "Read the simple root from the factorization, integrate between the curve’s intercepts, then map the transformed curve’s horizontal and vertical scale factors onto the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=26", "qpPages": [ 38, 41 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q14c", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 14, "questionPart": "14(c)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Identify a curve intercept, integrate the finite region between a cubic curve and the x-axis, then use scaling to find another area.", "answerGuide": "Read the simple root from the factorization, integrate between the curve’s intercepts, then map the transformed curve’s horizontal and vertical scale factors onto the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=26", "qpPages": [ 38, 41 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q15a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 15, "questionPart": "15(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Derive the perimeter of a stage made from a rectangle and two sectors, minimize it by calculus and justify the minimum.", "answerGuide": "Combine straight-edge and arc lengths, eliminate the angle using the area condition, differentiate the resulting perimeter function and verify its stationary value is a minimum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=42", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=28", "qpPages": [ 42, 45 ], "msPages": [ 28, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q15b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 15, "questionPart": "15(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Derive the perimeter of a stage made from a rectangle and two sectors, minimize it by calculus and justify the minimum.", "answerGuide": "Combine straight-edge and arc lengths, eliminate the angle using the area condition, differentiate the resulting perimeter function and verify its stationary value is a minimum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=42", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=28", "qpPages": [ 42, 45 ], "msPages": [ 28, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q15c", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 15, "questionPart": "15(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Derive the perimeter of a stage made from a rectangle and two sectors, minimize it by calculus and justify the minimum.", "answerGuide": "Combine straight-edge and arc lengths, eliminate the angle using the area condition, differentiate the resulting perimeter function and verify its stationary value is a minimum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=42", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=28", "qpPages": [ 42, 45 ], "msPages": [ 28, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q16a", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 16, "questionPart": "16(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the geometric-sequence ratio condition to obtain a trigonometric quadratic, then find an exact cosine and sequence terms.", "answerGuide": "Equate consecutive ratios, use sin²θ = 1 − cos²θ to form the quadratic, choose the acute-angle root and substitute it into the first term and common ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=46", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=30", "qpPages": [ 46, 48 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q16b", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 16, "questionPart": "16(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the geometric-sequence ratio condition to obtain a trigonometric quadratic, then find an exact cosine and sequence terms.", "answerGuide": "Equate consecutive ratios, use sin²θ = 1 − cos²θ to form the quadratic, choose the acute-angle root and substitute it into the first term and common ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=46", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=30", "qpPages": [ 46, 48 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA01-01-q16ciii", "paperId": "ial-2020-1-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 16, "questionPart": "16(c)(i)/(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the geometric-sequence ratio condition to obtain a trigonometric quadratic, then find an exact cosine and sequence terms.", "answerGuide": "Equate consecutive ratios, use sin²θ = 1 − cos²θ to form the quadratic, choose the acute-angle root and substitute it into the first term and common ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20200305.pdf#page=46", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_rms_20200305.pdf#page=30", "qpPages": [ 46, 48 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q01a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use differentiation to derive an iterative equation for a turning point, run the iteration and justify the root to three decimal places.", "answerGuide": "Differentiate and set f′(x)=0, rearrange into the stated iteration, calculate successive values to four decimals, then use a sign-changing interval or rounding bounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q01b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use differentiation to derive an iterative equation for a turning point, run the iteration and justify the root to three decimal places.", "answerGuide": "Differentiate and set f′(x)=0, rearrange into the stated iteration, calculate successive values to four decimals, then use a sign-changing interval or rounding bounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q01c", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use differentiation to derive an iterative equation for a turning point, run the iteration and justify the root to three decimal places.", "answerGuide": "Differentiate and set f′(x)=0, rearrange into the stated iteration, calculate successive values to four decimals, then use a sign-changing interval or rounding bounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q02a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a negative fractional binomial to four terms and use the expansion to approximate a square root.", "answerGuide": "Factor out the constant term, apply the generalized binomial coefficients through the cubic term, then substitute the specified small x-value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q02b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a negative fractional binomial to four terms and use the expansion to approximate a square root.", "answerGuide": "Factor out the constant term, apply the generalized binomial coefficients through the cubic term, then substitute the specified small x-value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q03a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Estimate a curve-bounded area with four trapezia and use the estimate to approximate two related integrals.", "answerGuide": "Evaluate the function at the five equally spaced ordinates, apply the trapezium rule, and transform each requested integral to the same interval and integrand.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q03biii", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(i)/(ii)", "marks": 4, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Estimate a curve-bounded area with four trapezia and use the estimate to approximate two related integrals.", "answerGuide": "Evaluate the function at the five equally spaced ordinates, apply the trapezium rule, and transform each requested integral to the same interval and integrand.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q04a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch an absolute-value graph, solve its intersection with a line, then find line parameters that prevent intersection.", "answerGuide": "Identify the V-shaped vertex and intercepts, solve piecewise across the absolute-value boundary, then compare line and graph values at the relevant branches.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q04b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch an absolute-value graph, solve its intersection with a line, then find line parameters that prevent intersection.", "answerGuide": "Identify the V-shaped vertex and intercepts, solve piecewise across the absolute-value boundary, then compare line and graph values at the relevant branches.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q04c", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch an absolute-value graph, solve its intersection with a line, then find line parameters that prevent intersection.", "answerGuide": "Identify the V-shaped vertex and intercepts, solve piecewise across the absolute-value boundary, then compare line and graph values at the relevant branches.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q05a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Determine the range and constants of a piecewise continuous function from a root condition, then evaluate a composite value.", "answerGuide": "Read the limiting range from the branches, use the specified root to form the parameter equations, solve them and substitute into the composite expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q05b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Determine the range and constants of a piecewise continuous function from a root condition, then evaluate a composite value.", "answerGuide": "Read the limiting range from the branches, use the specified root to form the parameter equations, solve them and substitute into the composite expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q05c", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Determine the range and constants of a piecewise continuous function from a root condition, then evaluate a composite value.", "answerGuide": "Read the limiting range from the branches, use the specified root to form the parameter equations, solve them and substitute into the composite expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q06", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": null, "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Solve a first-order differential equation with a given initial condition and evaluate the solution at x = e.", "answerGuide": "Separate variables, integrate both sides, apply the initial condition to find the constant, and evaluate the exact expression at e.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q07a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a product of a power and a fourth power, then find where the function is decreasing.", "answerGuide": "Apply the product and chain rules, factor the derivative into the required form, and use its sign to determine the decreasing intervals.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=12", "qpPages": [ 20, 21 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q07b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a product of a power and a fourth power, then find where the function is decreasing.", "answerGuide": "Apply the product and chain rules, factor the derivative into the required form, and use its sign to determine the decreasing intervals.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=12", "qpPages": [ 20, 21 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q08", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": null, "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact area under a parametrically defined curve between the curve and a vertical line.", "answerGuide": "Use area = ∫y(dx/dt)dt with the parameter limits corresponding to x = 0 and x = 2, then integrate using a trigonometric identity.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=13", "qpPages": [ 22, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q09", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": null, "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find constants in two three-dimensional line equations using their intersection parameter and perpendicular direction vectors.", "answerGuide": "Substitute λ = −2 into the first line, equate coordinates with the second line to form equations, and use the zero dot product condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=15", "qpPages": [ 24, 25 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q10a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate a cubic relation and find points where the gradient is zero.", "answerGuide": "Differentiate every term with respect to x, collect dy/dx terms and rearrange; set the numerator to zero and solve simultaneously with the curve equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=16", "qpPages": [ 26, 29 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q10b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate a cubic relation and find points where the gradient is zero.", "answerGuide": "Differentiate every term with respect to x, collect dy/dx terms and rearrange; set the numerator to zero and solve simultaneously with the curve equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=16", "qpPages": [ 26, 29 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q11a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and use it to solve an equation on a specified interval.", "answerGuide": "Rewrite cotangent using sine and cosine and simplify with the Pythagorean identity, then substitute the result and enumerate all roots in range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=18", "qpPages": [ 30, 33 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q11b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and use it to solve an equation on a specified interval.", "answerGuide": "Rewrite cotangent using sine and cosine and simplify with the Pythagorean identity, then substitute the result and enumerate all roots in range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=18", "qpPages": [ 30, 33 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q12", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": null, "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate an exact rational integral by decomposing the integrand into simpler terms.", "answerGuide": "Use polynomial division or partial fractions, integrate powers and logarithmic terms separately, and substitute the limits exactly.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=34", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=21", "qpPages": [ 34, 37 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q13a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 13, "questionPart": "13(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Differentiate a parametric curve, find a tangent at a given parameter and find its other intersection with the curve.", "answerGuide": "Use dy/dx=(dy/dt)/(dx/dt), find the point and tangent at t=2, then substitute the tangent into the parametric curve and solve for the second parameter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=22", "qpPages": [ 38, 41 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q13b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 13, "questionPart": "13(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Differentiate a parametric curve, find a tangent at a given parameter and find its other intersection with the curve.", "answerGuide": "Use dy/dx=(dy/dt)/(dx/dt), find the point and tangent at t=2, then substitute the tangent into the parametric curve and solve for the second parameter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=22", "qpPages": [ 38, 41 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q13c", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 13, "questionPart": "13(c)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Differentiate a parametric curve, find a tangent at a given parameter and find its other intersection with the curve.", "answerGuide": "Use dy/dx=(dy/dt)/(dx/dt), find the point and tangent at t=2, then substitute the tangent into the parametric curve and solve for the second parameter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=22", "qpPages": [ 38, 41 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q14a", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 14, "questionPart": "14(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic population model to find its initial population and limiting value, solve for a target population and state when the growth rate is greatest.", "answerGuide": "Substitute t=0 and take the long-term limit. Solve the target equation using logarithms, then differentiate the logistic model or use its midpoint property for maximum growth.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_que_20200305.pdf#page=42", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA02_01_rms_20200305.pdf#page=24", "qpPages": [ 42, 45 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA02-01-q14b", "paperId": "ial-2020-1-WMA02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA02/01", "variant": "Standard paper", "questionNumber": 14, "questionPart": "14(b)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic population model to find its initial population and limiting value, solve for a target population and state when the growth rate is greatest.", "answerGuide": "Substitute t=0 and take the long-term limit. 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20200305.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The matched Pearson feedback identifies Question 2 among the paper’s most challenging questions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2020-1-WMA11-01-q02c", "paperId": "ial-2020-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20200305.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA11-01-q03b", "paperId": "ial-2020-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find a tangent gradient on a quadratic curve, calculate a 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"answerGuide": "Use the common gradient and the known intercept to form the line, then use opposite sides and coordinate geometry to find the area and missing point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20200305.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies Question 6 as challenging and discusses errors in area and vertex work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 6, 7 ] } }, { "id": "ial-2020-1-WMA11-01-q07aiii", "paperId": "ial-2020-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(i)/(ii)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Read coordinates from a sine graph and determine the minimum point after a vertical translation.", "answerGuide": "Read the marked sine-wave coordinates and shift the original curve vertically so its maximum matches the given value; locate the first minimum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20200305.pdf#page=18", 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"ial-2020-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": null, "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of a curve involving a reciprocal power and a square-root term.", "answerGuide": "Rewrite the curve using fractional indices, differentiate, set dy/dx to zero and solve for the positive x-coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20200305.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20200305.pdf#page=20", "qpPages": [ 22, 23 ], "msPages": [ 20, 21 ], 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20200305.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20200305.pdf#page=22", "qpPages": [ 24, 25 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA11-01-q10biii", "paperId": "ial-2020-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)(i)/(ii)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch a factored cubic, solve an equation involving a 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"tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA11-01-q11a", "paperId": "ial-2020-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the normal to a curve using derivative information and recover the curve from its second derivative and point conditions.", "answerGuide": "Use the tangent gradient to get the negative reciprocal normal slope. 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q02b", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Determine a parameter from a specified binomial coefficient and find a constant term in a related expansion.", "answerGuide": "Match the coefficient of x⁴ using the binomial theorem, solve for the permitted parameter values, then combine 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q04", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact area bounded by a square-root curve, the y-axis and a horizontal line.", "answerGuide": "Invert the curve to express x in terms of y, integrate the horizontal width from y=0 to y=17, and simplify exactly.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q05a", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find an annual percentage growth rate from two observations and use the 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q06b", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Complete the square for a circle, find its centre and radius, and solve tangent and chord conditions for horizontal lines.", "answerGuide": "Complete squares, use the radius for the tangent heights, and apply the circle equation with the specified chord length to find the negative line value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q06c", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Complete the square for a circle, find its centre and radius, and solve tangent and chord conditions for horizontal lines.", "answerGuide": "Complete squares, use the radius for the tangent heights, and apply the circle equation with the specified chord length to find the negative line value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q07a", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q07b", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Convert a trigonometric equation to a quadratic in sine and solve a related double-angle equation over a degree interval.", "answerGuide": "Replace tan θ by sin θ/cos θ and use cos²θ=1−sin²θ; solve the quadratic and enumerate valid angles in the requested range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q08i", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Prove the arithmetic-series sum formula and evaluate a sequence term and finite sum involving alternating signs.", "answerGuide": "Write the sum forwards and backwards and add to establish the formula; substitute n=5, then separate the alternating and linear terms in the sum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The examiner report notes unusually few fully correct attempts at this series proof.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 9 ] } }, { "id": "ial-2020-1-WMA12-01-q08iia", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)(a)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Prove the arithmetic-series sum formula and evaluate a sequence term and finite sum involving alternating signs.", "answerGuide": "Write the sum forwards and backwards and add to establish the formula; substitute n=5, then separate the alternating and linear terms in the sum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q08iib", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Prove the arithmetic-series sum formula and evaluate a sequence term and finite sum involving alternating signs.", "answerGuide": "Write the sum forwards and backwards and add to establish the formula; substitute n=5, then separate the alternating and linear terms in the sum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as the most challenging part of Question 8 and notes incorrect single-series assumptions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 9, 10 ] } }, { "id": "ial-2020-1-WMA12-01-q09a", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Sketch two exponential curves and find the exact x-coordinate where they intersect.", "answerGuide": "Mark the exponential intercepts and asymptote, equate the two functions and use logarithm laws to solve for x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q09b", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure 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"paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Determine a parameter from a stationary-point condition, classify the stationary point and find a second stationary coordinate.", "answerGuide": "Differentiate, use dy/dx=0 at x=1/2 to find k, examine the second derivative for the nature, then solve the derivative equation for its other root.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20200305.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms_20200305.pdf#page=19", "qpPages": [ 26, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA12-01-q10b", "paperId": "ial-2020-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Determine a parameter from a stationary-point condition, classify the stationary point and find a second stationary coordinate.", "answerGuide": "Differentiate, use dy/dx=0 at x=1/2 to find k, examine the second 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Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic-type population model to find the starting population, time to a target population and its unattainable upper bound.", "answerGuide": "Substitute t=0, rearrange and take logarithms for the target time, then use the limiting value or strict range of the model.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q01b", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic-type population model to find the starting population, time to a target population and its unattainable upper bound.", "answerGuide": "Substitute t=0, rearrange and take logarithms for the target time, then use the limiting value or strict range of the model.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q01c", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic-type population model to find the starting population, time to a target population and its unattainable upper bound.", "answerGuide": "Substitute t=0, rearrange and take logarithms for the target time, then use the limiting value or strict range of the model.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q02a", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate a composite function, find an inverse and solve where the inverse and original function are equal.", "answerGuide": "Substitute the exponential input into the logarithmic function, interchange x and y to find the inverse, then solve the equality under the stated domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q02b", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate a composite function, find an inverse and solve where the inverse and original function are equal.", "answerGuide": "Substitute the exponential input into the logarithmic function, interchange x and y to find the inverse, then solve the equality under the stated domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q02c", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate a composite function, find an inverse and solve where the inverse and original function are equal.", "answerGuide": "Substitute the exponential input into the logarithmic function, interchange x and y to find the inverse, then solve the equality under the stated domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q03a", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find a straight-line relation on logarithmic axes and express the original variables in power form.", "answerGuide": "Use the two plotted points to find the gradient and intercept of log₁₀y against log₁₀x, then exponentiate to identify p and q.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls the logarithmic-graph question highly discriminating and notes that many candidates scored zero or one mark.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-2020-1-WMA13-01-q03b", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find a straight-line relation on logarithmic axes and express the original variables in power form.", "answerGuide": "Use the two plotted points to find the gradient and intercept of log₁₀y against log₁₀x, then exponentiate to identify p and q.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls the logarithmic-graph question highly discriminating and notes that many candidates scored zero or one mark.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-2020-1-WMA13-01-q04ia", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a rational function to determine where it increases, then apply a maximum condition to a trigonometric product.", "answerGuide": "Use the quotient rule and factor the derivative; analyze its sign on the domain. For the maximum, differentiate x sin(4x) and set the derivative to zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report highlights missed factorization demands and errors after differentiation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4, 5, 6 ] } }, { "id": "ial-2020-1-WMA13-01-q04ib", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a rational function to determine where it increases, then apply a maximum condition to a trigonometric product.", "answerGuide": "Use the quotient rule and factor the derivative; analyze its sign on the domain. For the maximum, differentiate x sin(4x) and set the derivative to zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q04ii", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a rational function to determine where it increases, then apply a maximum condition to a trigonometric product.", "answerGuide": "Use the quotient rule and factor the derivative; analyze its sign on the domain. For the maximum, differentiate x sin(4x) and set the derivative to zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q05a", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use t=tan x to form a quartic equation and solve the original trigonometric equation over 0° to 360°.", "answerGuide": "Substitute t=tan x, use sec²x=1+tan²x to derive the quartic, solve for t² and list every valid angle.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q05b", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use t=tan x to form a quartic equation and solve the original trigonometric equation over 0° to 360°.", "answerGuide": "Substitute t=tan x, use sec²x=1+tan²x to derive the quartic, solve for t² and list every valid angle.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says solving the quartic was unexpectedly difficult and notes missed or invalid angle solutions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2020-1-WMA13-01-q06a", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Read the vertex of an absolute-value graph, solve its intersection with a line, and find the slope range giving exactly two intersections.", "answerGuide": "Use the translated vertex, solve the piecewise-linear equation, then compare the line gradient with the two branches and exclude boundary tangencies.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q06b", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Read the vertex of an absolute-value graph, solve its intersection with a line, and find the slope range giving exactly two intersections.", "answerGuide": "Use the translated vertex, solve the piecewise-linear equation, then compare the line gradient with the two branches and exclude boundary tangencies.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q06c", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Read the vertex of an absolute-value graph, solve its intersection with a line, and find the slope range giving exactly two intersections.", "answerGuide": "Use the translated vertex, solve the piecewise-linear equation, then compare the line gradient with the two branches and exclude boundary tangencies.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most challenging part of Question 6 and notes that candidates often used only one parameter bound.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2020-1-WMA13-01-q07a", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Bracket and iterate a trigonometric root, then use the curve minima to find a range of parameter values.", "answerGuide": "Check a sign change across the interval, carry out the stated iteration, and use the two local minima inequalities together to find the strict range for k.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q07biii", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)(i)/(ii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Bracket and iterate a trigonometric root, then use the curve minima to find a range of parameter values.", "answerGuide": "Check a sign change across the interval, carry out the stated iteration, and use the two local minima inequalities together to find the strict range for k.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q07c", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Bracket and iterate a trigonometric root, then use the curve minima to find a range of parameter values.", "answerGuide": "Check a sign change across the interval, carry out the stated iteration, and use the two local minima inequalities together to find the strict range for k.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few candidates completed this part correctly, often losing a sign or selecting a maximum.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2020-1-WMA13-01-q08i", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate an algebraic integral and decompose a rational function before integrating it.", "answerGuide": "Integrate the powers directly. For the rational function, use partial fractions in the specified form, determine constants and integrate powers and logarithms.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q08ii", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate an algebraic integral and decompose a rational function before integrating it.", "answerGuide": "Integrate the powers directly. For the rational function, use partial fractions in the specified form, determine constants and integrate powers and logarithms.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q09a", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write a sine-cosine combination in amplitude-phase form, describe graph transformations and determine a transformed function’s range.", "answerGuide": "Match coefficients to R cos(θ+α), read the vertical stretch and translation, then find the range from amplitude and vertical shift.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=19", "qpPages": [ 30, 32 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q09biii", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(i)/(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write a sine-cosine combination in amplitude-phase form, describe graph transformations and determine a transformed function’s range.", "answerGuide": "Match coefficients to R cos(θ+α), read the vertical stretch and translation, then find the range from amplitude and vertical shift.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=19", "qpPages": [ 30, 32 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WMA13-01-q09c", "paperId": "ial-2020-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write a sine-cosine combination in amplitude-phase form, describe graph transformations and determine a transformed function’s range.", "answerGuide": "Match coefficients to R cos(θ+α), read the vertical stretch and translation, then find the range from amplitude and vertical shift.", "qpUrl": "https://www.alevelcopilot.com/whfile/91/ca/a3/d3/d6/86/cc/fd/dd/51/49/bb/0f/cf/49/f5.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20200305.pdf#page=19", "qpPages": [ 30, 32 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report describes Question 9 as highly discriminating and notes that many missed the upper limit after squaring.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4, 8 ] } }, { "id": "ial-2020-1-WDM01-01-q01a", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete a worker-to-delivery matching by finding alternating paths and updating the initial assignment.", "answerGuide": "Trace each alternating path from an unmatched worker, reverse matched and unmatched edges along it, and list the resulting assignment.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q01b", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete a worker-to-delivery matching by finding alternating paths and updating the initial assignment.", "answerGuide": "Trace each alternating path from an unmatched worker, reverse matched and unmatched edges along it, and list the resulting assignment.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q01c", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete a worker-to-delivery matching by finding alternating paths and updating the initial assignment.", "answerGuide": "Trace each alternating path from an unmatched worker, reverse matched and unmatched edges along it, and list the resulting assignment.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q02aiii", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(i)/(ii)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Define a tree and minimum spanning tree, then use Kruskal’s algorithm to build a minimum tree for a weighted network.", "answerGuide": "State the connected, acyclic definition and minimum-total-weight condition; consider edges in increasing weight order, rejecting any that creates a cycle.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q02b", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Define a tree and minimum spanning tree, then use Kruskal’s algorithm to build a minimum tree for a weighted network.", "answerGuide": "State the connected, acyclic definition and minimum-total-weight condition; consider edges in increasing weight order, rejecting any that creates a cycle.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q02c", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Define a tree and minimum spanning tree, then use Kruskal’s algorithm to build a minimum tree for a weighted network.", "answerGuide": "State the connected, acyclic definition and minimum-total-weight condition; consider edges in increasing weight order, rejecting any that creates a cycle.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q03a", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete missing activity durations and event times, draw a cascade chart and give a lower bound for workers.", "answerGuide": "Use float relationships and event-time constraints to find x, y and z, draw each activity over its allowable interval and inspect simultaneous activity demand.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q03b", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete missing activity durations and event times, draw a cascade chart and give a lower bound for workers.", "answerGuide": "Use float relationships and event-time constraints to find x, y and z, draw each activity over its allowable interval and inspect simultaneous activity demand.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q03c", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete missing activity durations and event times, draw a cascade chart and give a lower bound for workers.", "answerGuide": "Use float relationships and event-time constraints to find x, y and z, draw each activity over its allowable interval and inspect simultaneous activity demand.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q04a", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit bin packing, quick sort, first-fit decreasing and bubble sort to the supplied lists.", "answerGuide": "Process items in order for first fit, show each quick-sort pivot pass, reorder by size before first-fit decreasing, and execute bubble-sort passes from the left.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=5", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q04b", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit bin packing, quick sort, first-fit decreasing and bubble sort to the supplied lists.", "answerGuide": "Process items in order for first fit, show each quick-sort pivot pass, reorder by size before first-fit decreasing, and execute bubble-sort passes from the left.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=5", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q04c", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit bin packing, quick sort, first-fit decreasing and bubble sort to the supplied lists.", "answerGuide": "Process items in order for first fit, show each quick-sort pivot pass, reorder by size before first-fit decreasing, and execute bubble-sort passes from the left.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=5", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q04d", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit bin packing, quick sort, first-fit decreasing and bubble sort to the supplied lists.", "answerGuide": "Process items in order for first fit, show each quick-sort pivot pass, reorder by size before first-fit decreasing, and execute bubble-sort passes from the left.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=5", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q05a", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network from a precedence table and identify activities that cannot be critical.", "answerGuide": "Represent every predecessor relation with the minimum dummies, then use the stated critical-path condition to rule out activities that cannot lie on any longest path.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q05b", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network from a precedence table and identify activities that cannot be critical.", "answerGuide": "Represent every predecessor relation with the minimum dummies, then use the stated critical-path condition to rule out activities that cannot lie on any longest path.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q06a", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a shortest route, then solve an open route-inspection problem between two specified endpoints.", "answerGuide": "Use Dijkstra’s algorithm and predecessor labels for the shortest route. 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For inspection, pair the relevant odd vertices at least cost and identify which arcs must be repeated.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=7", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q07a", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve 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Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve a cost-minimizing linear program for the required training-day mix.", "answerGuide": "Define decision variables, translate total, ratio and percentage restrictions into inequalities, draw the feasible region, evaluate the objective at its vertices and interpret the optimal mix.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q07d", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve a cost-minimizing linear program for the required training-day mix.", "answerGuide": "Define decision variables, translate total, ratio and percentage restrictions into inequalities, draw the feasible region, evaluate the objective at its vertices and interpret the optimal mix.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM01-01-q07eiii", "paperId": "ial-2020-1-WDM01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM01", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)(i)/(ii)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve a cost-minimizing linear program for the required training-day mix.", "answerGuide": "Define decision variables, translate total, ratio and percentage restrictions into inequalities, draw the feasible region, evaluate the objective at its vertices and interpret the optimal mix.", "qpUrl": "https://edemat.alevelclouds.com/20_01_QP_D1_Old.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WDM01_01_rms_20200305.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q01a", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a 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null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q02aiii", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(i)/(ii)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Define a tree and minimum spanning tree, then use Kruskal’s algorithm to build a minimum tree for a weighted network.", "answerGuide": "State the connected, acyclic definition and minimum-total-weight condition; consider edges in increasing weight order, rejecting any that creates a cycle.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q02b", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Define a tree and minimum spanning tree, then use Kruskal’s algorithm to build a minimum tree for a weighted network.", "answerGuide": "State the connected, acyclic definition and minimum-total-weight 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q03c", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete missing activity durations and event 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"paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit bin packing, quick sort, first-fit decreasing and bubble sort to the supplied lists.", "answerGuide": "Process items in order for first fit, show each quick-sort pivot pass, reorder by size before first-fit decreasing, and execute bubble-sort passes from the left.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=5", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q04c", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit bin packing, quick sort, first-fit decreasing and bubble sort to the supplied lists.", "answerGuide": "Process items in order for first fit, show each quick-sort pivot pass, reorder by size before first-fit decreasing, and execute bubble-sort passes from the left.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=5", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q04d", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], 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"syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network from a precedence table and identify activities that cannot be critical.", "answerGuide": "Represent every predecessor relation with the minimum dummies, then use the stated critical-path condition to rule out activities that cannot lie on any longest path.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q05b", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network from a precedence table and identify activities that cannot be critical.", "answerGuide": "Represent every predecessor relation with the minimum dummies, then use the stated critical-path condition to rule out activities that cannot lie on any longest path.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q06a", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a shortest route, then solve an open 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8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q07c", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve a cost-minimizing linear program for the required training-day mix.", "answerGuide": "Define decision variables, translate total, ratio and percentage restrictions into inequalities, draw the feasible region, evaluate the objective at its vertices and interpret the optimal mix.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20200305.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WDM11-01-q07d", "paperId": "ial-2020-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2020, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve a cost-minimizing linear 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q01b", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Analyze a one-dimensional collision using momentum and coefficient of restitution, then determine the kinetic-energy change.", "answerGuide": "Use conservation of momentum and the restitution relation for the two velocities, then 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report calls Question 3 the paper’s most challenging question and says many candidates earned only a small fraction of its marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-1-WME01-01-q03c", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Compare two vertical particles’ motions to find a meeting condition and the corresponding time or speed.", "answerGuide": "Use vertical-motion equations with acceleration −g, apply the release delay and equal-height condition, then substitute the meeting time into the required quantity.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The matched examiner report calls Question 3 the paper’s most challenging question and says many candidates earned only a small fraction of its marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20200305.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-1-WME01-01-q04a", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on two rings connected by a string on a rough horizontal wire and determine their motion condition.", "answerGuide": "Resolve parallel and perpendicular to the wire, apply the friction law and Newton’s second law to each ring, and combine the equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q04b", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on two rings connected by a string on a rough horizontal wire and determine their motion condition.", "answerGuide": "Resolve parallel and perpendicular to the wire, apply the friction law and Newton’s second law to each ring, and combine the equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q04c", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on two rings connected by a string on a rough horizontal wire and determine their motion condition.", "answerGuide": "Resolve parallel and perpendicular to the wire, apply the friction law and Newton’s second law to each ring, and combine the equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q05a", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a speed-time graph for a car and accelerating motorcyclist to compare their speeds and positions.", "answerGuide": "Read acceleration from the graph gradient, integrate or use graph areas for displacement, then equate distances or speeds at the required time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q05b", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a speed-time graph for a car and accelerating motorcyclist to compare their speeds and positions.", "answerGuide": "Read acceleration from the graph gradient, integrate or use graph areas for displacement, then equate distances or speeds at the required time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q05c", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a speed-time graph for a car and accelerating motorcyclist to compare their speeds and positions.", "answerGuide": "Read acceleration from the graph gradient, integrate or use graph areas for displacement, then equate distances or speeds at the required time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q06a", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve a vector force into magnitude and direction, then use it to determine a particle’s motion.", "answerGuide": "Use Pythagoras and tan⁻¹ for the force magnitude and direction, then resolve the force along the stated axes and apply F=ma.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q06b", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve a vector force into magnitude and direction, then use it to determine a particle’s motion.", "answerGuide": "Use Pythagoras and tan⁻¹ for the force magnitude and direction, then resolve the force along the stated axes and apply F=ma.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q06c", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve a vector force into magnitude and direction, then use it to determine a particle’s motion.", "answerGuide": "Use Pythagoras and tan⁻¹ for the force magnitude and direction, then resolve the force along the stated axes and apply F=ma.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q07a", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze a two-particle pulley system with friction, find its acceleration and speed, and determine when one particle reaches the floor.", "answerGuide": "Use kinematics to find the initial acceleration, apply the string constraint, resolve Newton’s second law for both particles, and use energy or constant-acceleration equations for later motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q07b", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze a two-particle pulley system with friction, find its acceleration and speed, and determine when one particle reaches the floor.", "answerGuide": "Use kinematics to find the initial acceleration, apply the string constraint, resolve Newton’s second law for both particles, and use energy or constant-acceleration equations for later motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q07c", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze a two-particle pulley system with friction, find its acceleration and speed, and determine when one particle reaches the floor.", "answerGuide": "Use kinematics to find the initial acceleration, apply the string constraint, resolve Newton’s second law for both particles, and use energy or constant-acceleration equations for later motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q07d", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze a two-particle pulley system with friction, find its acceleration and speed, and determine when one particle reaches the floor.", "answerGuide": "Use kinematics to find the initial acceleration, apply the string constraint, resolve Newton’s second law for both particles, and use energy or constant-acceleration equations for later motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME01-01-q07e", "paperId": "ial-2020-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze a two-particle pulley system with friction, find its acceleration and speed, and determine when one particle reaches the floor.", "answerGuide": "Use kinematics to find the initial acceleration, apply the string constraint, resolve Newton’s second law for both particles, and use energy or constant-acceleration equations for later motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20200305.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q01", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a cyclist’s speed from power, gravity, resistance and acceleration along an inclined road.", "answerGuide": "Use P=Fv for the driving force, resolve the weight and resistance along the slope, and apply Newton’s second law.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q02", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of friction from a package projected up a rough ramp and stopping after a known distance.", "answerGuide": "Apply work-energy: initial kinetic energy equals work against friction plus gain in gravitational potential energy; use the normal reaction to express friction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q03iii", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)/(ii)", "marks": 7, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle’s speed and change in direction after an impulse acts at an angle to its original motion.", "answerGuide": "Apply the impulse-momentum relation in components to obtain the new velocity, then use Pythagoras for speed and trigonometry for the direction change.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q04a", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a composite lamina, then determine its position after an additional mass is attached.", "answerGuide": "Use moments of area about the stated axes for the composite shape, then equate the combined weighted moments to total mass times the new centre coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q04b", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a composite lamina, then determine its position after an additional mass is attached.", "answerGuide": "Use moments of area about the stated axes for the composite shape, then equate the combined weighted moments to total mass times the new centre coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q05a", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find acceleration and position for a particle with vector velocity depending on time, then calculate its speed or direction at specified instants.", "answerGuide": "Differentiate velocity for acceleration, integrate each component for displacement using the initial point, and use vector magnitudes or components for speed and direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q05b", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find acceleration and position for a particle with vector velocity depending on time, then calculate its speed or direction at specified instants.", "answerGuide": "Differentiate velocity for acceleration, integrate each component for displacement using the initial point, and use vector magnitudes or components for speed and direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q05c", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find acceleration and position for a particle with vector velocity depending on time, then calculate its speed or direction at specified instants.", "answerGuide": "Differentiate velocity for acceleration, integrate each component for displacement using the initial point, and use vector magnitudes or components for speed and direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q06a", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze the equilibrium of a uniform rod resting on rough ground and a smooth cylinder, then find the friction condition.", "answerGuide": "Resolve forces vertically and horizontally and take moments about the ground contact; use the limiting-friction relation and the angle geometry.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q06b", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyze the equilibrium of a uniform rod resting on rough ground and a smooth cylinder, then find the friction condition.", "answerGuide": "Resolve forces vertically and horizontally and take moments about the ground contact; use the limiting-friction relation and the angle geometry.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q07a", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Analyze two projectile trajectories to find a height, meeting/location condition, minimum speed and time.", "answerGuide": "Resolve initial velocities into horizontal and vertical components, use constant acceleration under gravity, and impose the stated point or intersection conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q07b", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Analyze two projectile trajectories to find a height, meeting/location condition, minimum speed and time.", "answerGuide": "Resolve initial velocities into horizontal and vertical components, use constant acceleration under gravity, and impose the stated point or intersection conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q07c", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Analyze two projectile trajectories to find a height, meeting/location condition, minimum speed and time.", "answerGuide": "Resolve initial velocities into horizontal and vertical components, use constant acceleration under gravity, and impose the stated point or intersection conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q07d", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Analyze two projectile trajectories to find a height, meeting/location condition, minimum speed and time.", "answerGuide": "Resolve initial velocities into horizontal and vertical components, use constant acceleration under gravity, and impose the stated point or intersection conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q07e", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Analyze two projectile trajectories to find a height, meeting/location condition, minimum speed and time.", "answerGuide": "Resolve initial velocities into horizontal and vertical components, use constant acceleration under gravity, and impose the stated point or intersection conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME02-01-q08iii", "paperId": "ial-2020-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2020, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)/(ii)", "marks": 12, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of restitution and impulse in a one-dimensional collision where both particles reverse direction.", "answerGuide": "Use the kinetic-energy loss to form an equation for the post-impact velocities, combine it with conservation of momentum and restitution, then calculate the impulse on A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20200305.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20200305.pdf#page=17", "qpPages": [ 24, 28 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q01", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the minimum coefficient of friction needed to keep a particle stationary on a rotating horizontal disc.", "answerGuide": "Equate the required centripetal force to friction at the limiting condition, using angular speed in radians per second and R=0.2 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q02a", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-position relation to find the distance at a stated time and the resultant force then acting.", "answerGuide": "Separate variables and integrate dx/dt=v(x) to find x(t), then use a= v dv/dx and F=ma.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q02b", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-position relation to find the distance at a stated time and the resultant force then acting.", "answerGuide": "Separate variables and integrate dx/dt=v(x) to find x(t), then use a= v dv/dx and F=ma.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q03a", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on a particle moving in a horizontal circle inside a smooth hemispherical bowl with a taut string.", "answerGuide": "Use the circular geometry to resolve weight, normal reaction and tension vertically and radially; combine equilibrium and centripetal-force equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q03b", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces on a particle moving in a horizontal circle inside a smooth hemispherical bowl with a taut string.", "answerGuide": "Use the circular geometry to resolve weight, normal reaction and tension vertically and radially; combine equilibrium and centripetal-force equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q04a", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the extension and coefficient of friction for an elastic string and particle released down a rough inclined plane.", "answerGuide": "Apply Hooke’s law and resolve forces along and normal to the slope, then use work-energy between release and the point where the particle momentarily stops.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q04b", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the extension and coefficient of friction for an elastic string and particle released down a rough inclined plane.", "answerGuide": "Apply Hooke’s law and resolve forces along and normal to the slope, then use work-energy between release and the point where the particle momentarily stops.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q05a", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Analyze a particle projected on a taut string pendulum to find its speed at the ceiling and the later motion condition.", "answerGuide": "Use conservation of energy between positions, then apply radial tension and tangential motion equations at the specified point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q05b", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Analyze a particle projected on a taut string pendulum to find its speed at the ceiling and the later motion condition.", "answerGuide": "Use conservation of energy between positions, then apply radial tension and tangential motion equations at the specified point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q06a", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show motion on an elastic spring is simple harmonic, find its period and amplitude, then determine a subsequent time or position.", "answerGuide": "Use Hooke’s law to derive a=−ω²x, read the period from ω, use impulse to find the initial speed and relate amplitude to the SHM energy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q06b", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show motion on an elastic spring is simple harmonic, find its period and amplitude, then determine a subsequent time or position.", "answerGuide": "Use Hooke’s law to derive a=−ω²x, read the period from ω, use impulse to find the initial speed and relate amplitude to the SHM energy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q06c", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show motion on an elastic spring is simple harmonic, find its period and amplitude, then determine a subsequent time or position.", "answerGuide": "Use Hooke’s law to derive a=−ω²x, read the period from ω, use impulse to find the initial speed and relate amplitude to the SHM energy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q06d", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show motion on an elastic spring is simple harmonic, find its period and amplitude, then determine a subsequent time or position.", "answerGuide": "Use Hooke’s law to derive a=−ω²x, read the period from ω, use impulse to find the initial speed and relate amplitude to the SHM energy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q07aiii", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(i)/(ii)", "marks": 8, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume and centre of mass of a solid of revolution, then analyze a second shape formed by removing a cylindrical section.", "answerGuide": "Use π∫y²dx for the volume and moments of volume for the centre coordinate; subtract the removed cylinder’s volume and moment from the whole solid.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WME03-01-q07b", "paperId": "ial-2020-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2020, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume and centre of mass of a solid of revolution, then analyze a second shape formed by removing a cylindrical section.", "answerGuide": "Use π∫y²dx for the volume and moments of volume for the centre coordinate; subtract the removed cylinder’s volume and moment from the whole solid.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20200110.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20200305.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q01", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find three probabilities in a discrete distribution using its mean and cumulative probability.", "answerGuide": "Use the total-probability equation and F(1) condition, then use E(X)=1 to solve the remaining simultaneous equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q02a", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete a three-set Venn diagram and answer event, independence and conditional-probability questions about pet ownership.", "answerGuide": "Place the intersection and outside counts consistently, use event definitions for exclusivity, compare P(D∩C) with P(D)P(C), and form conditional probabilities from the diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=5", "qpPages": [ 4, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q02b", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "probability" ], 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"hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q02d", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete a three-set Venn diagram and answer event, independence and conditional-probability questions about pet ownership.", "answerGuide": "Place the intersection and outside counts consistently, use event definitions for exclusivity, compare P(D∩C) with P(D)P(C), and form conditional probabilities from the diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=5", "qpPages": [ 4, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q02e", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete a three-set Venn diagram and answer event, independence and 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"January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(f)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete a three-set Venn diagram and answer event, independence and conditional-probability questions about pet ownership.", "answerGuide": "Place the intersection and outside counts consistently, use event definitions for exclusivity, compare P(D∩C) with P(D)P(C), and form conditional probabilities from the diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=5", "qpPages": [ 4, 7 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q03a", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Analyze correlation and regression data for sales, compare a proportional model with a fitted regression and recommend a model.", "answerGuide": "Calculate Smp and r, interpret strength and direction, obtain the proportionality constant and regression line, then compare their predictions at m=70.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=6", "qpPages": [ 8, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q03b", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Analyze correlation and regression data for sales, compare a proportional model with a fitted regression and recommend a model.", "answerGuide": "Calculate Smp and r, interpret strength and direction, obtain the proportionality constant and regression line, then compare their predictions at m=70.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=6", "qpPages": [ 8, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q03c", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Analyze correlation and regression data for sales, compare a proportional model with a fitted regression and recommend a model.", "answerGuide": "Calculate Smp and r, interpret strength and direction, obtain the proportionality constant and regression line, then compare their predictions at m=70.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20200305.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_rms_20200305.pdf#page=6", "qpPages": [ 8, 11 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST01-01-q03d", "paperId": "ial-2020-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", 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count with the critical region, solve the zero-sighting probability condition for n, and perform the one-sided winter test.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_rms_20200305.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST02-01-q05d", "paperId": "ial-2020-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2020, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test a Poisson claim about whale sightings, find the actual significance, calculate a trip threshold and test a winter rate.", "answerGuide": "Find both tails under the claimed Poisson model, compare the observed count with the critical region, solve the zero-sighting probability condition for n, and perform the one-sided winter test.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_rms_20200305.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST02-01-q05e", "paperId": "ial-2020-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2020, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test a Poisson claim about whale sightings, find the actual significance, calculate a trip threshold and test a winter rate.", "answerGuide": "Find both tails under the claimed Poisson model, compare the observed count with the critical region, solve the zero-sighting probability condition for n, and perform the one-sided winter test.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20200305.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_rms_20200305.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST02-01-q06a", "paperId": "ial-2020-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2020, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of a continuous variable from its density and calculate a two-tail probability from its cumulative distribution.", "answerGuide": "Integrate x²f(x) and subtract E(X)²; integrate the density to obtain F(x), then evaluate the lower and upper tail probabilities and add them.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20200305.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_rms_20200305.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-1-WST02-01-q06b", "paperId": "ial-2020-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2020, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of a continuous variable from its density and calculate a two-tail probability from its cumulative distribution.", "answerGuide": "Integrate x²f(x) and subtract E(X)²; integrate the density to obtain F(x), then evaluate the lower and upper tail probabilities and add them.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_que_20200305.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST02_01_rms_20200305.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2019-m-12-q01-u", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Question 1: Find p from the coefficient of x^3 in (1 − px)^5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial general term, equate the x^3 coefficient to −2160, and solve for p." }, { "id": "cie-2019-m-12-q02-u", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Question 2: Find k when a curve passes through two given points and f′(x) = kx^2 − 2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate f′, use the two coordinates to eliminate the integration constant, and solve for k." }, { "id": "cie-2019-m-12-q03-u", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 3: Find the total area enclosed by a semicircle of radius 7 and an overlapping circular arc of radius 8.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the two circle radii and centre geometry to find the sector and triangle areas, then combine them." }, { "id": "cie-2019-m-12-q04-i", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 4(i): Find the first and second derivatives of y = (2x − 1)^−1 + 2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each term once and twice, keeping the chain rule for the reciprocal term." }, { "id": "cie-2019-m-12-q04-ii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 4(ii): Find the stationary x-coordinates of the curve and classify each point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the first derivative to zero, solve for x, and use the second derivative or a justified gradient sign change." }, { "id": "cie-2019-m-12-q05-i", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 5(i): Find the values of q that make two three-dimensional vectors perpendicular.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form their scalar product, set it to zero, and solve the resulting quadratic." }, { "id": "cie-2019-m-12-q05-ii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 5(ii): Find the angle between the vectors when q = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute q, calculate the dot product and both magnitudes, then use the inverse cosine." }, { "id": "cie-2019-m-12-q06-i", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Question 6(i): Show that a geometric progression with first terms p and 2p has 2^n > 1001 when its sum exceeds 1000p.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric-series sum formula, divide by positive p, and rearrange the inequality." }, { "id": "cie-2019-m-12-q06-ii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Question 6(ii): Find n and p for an arithmetic progression whose first terms are p and 2p, nth term is 336, and sum is 7224.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the nth-term and sum equations, eliminate one unknown, and solve for n and p." }, { "id": "cie-2019-m-12-q07-a", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Question 7(a): Solve 3sin^2(2θ) + 8cos(2θ) = 0 for 0° ≤ θ ≤ 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 2: The 2θ angle caused confusion, and some candidates gave only one solution in the interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=2", "tagConfidence": "reviewed", "answerGuide": "Use sin^2(2θ) = 1 − cos^2(2θ), solve the quadratic in cos(2θ), then retain all angles in range." }, { "id": "cie-2019-m-12-q07-b", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Question 7(b): Find a and b from the x- and y-intercepts of y = a + tan(bx), where x is in radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 2: Finding b was less successful; candidates often treated tan(bπ/6) as b tan(π/6).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=2", "tagConfidence": "reviewed", "answerGuide": "Use the y-intercept for a, then substitute the x-intercept and respect the tangent period to determine b." }, { "id": "cie-2019-m-12-q08-i", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 8(i): Express x^2 − 4x + 7 in completed-square form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square in x." }, { "id": "cie-2019-m-12-q08-ii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Question 8(ii): Find the largest k for which f(x) = x^2 − 4x + 7 is decreasing on x < k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the vertex and monotonicity of the quadratic to identify the greatest allowable endpoint." }, { "id": "cie-2019-m-12-q08-iii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 8(iii): Find the inverse of the restricted quadratic when k = 1 and state its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 3: Only stronger candidates selected the inverse branch consistent with the restricted domain.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=3", "tagConfidence": "reviewed", "answerGuide": "Complete the square, switch x and y, select the branch consistent with the restriction, and state the range of the original as the inverse domain." }, { "id": "cie-2019-m-12-q08-iv", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iv)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Question 8(iv): Find gf(x) and its range for the stated restricted quadratic f and g(x) = 2x − 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 3: Many candidates omitted the lower range bound because they did not recognize the composite function was positive.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=3", "tagConfidence": "reviewed", "answerGuide": "Substitute f into g, simplify, and use the restricted domain to obtain both range bounds." }, { "id": "cie-2019-m-12-q09-i", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Question 9(i): Find the volume formed by rotating the region under y = √(x^3 + x^2) from x = 0 to x = 3 about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use V = π∫y^2 dx over the stated interval and evaluate exactly." }, { "id": "cie-2019-m-12-q09-ii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Question 9(ii): Find where the normal at the curve point with x = 3 meets the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=13", "qpPages": [ 17, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 3: Solutions had varying success, with the chain rule the most common source of error.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=3", "tagConfidence": "reviewed", "answerGuide": "Find the point, differentiate using the chain rule, calculate the normal gradient, and find its y-intercept." }, { "id": "cie-2019-m-12-q10-i", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 10(i): Find the distance between the intersections of y = x + 3 and y = 4√x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=14", "qpPages": [ 18, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Eliminate y, solve the resulting equation for both intersection x-values, then use the distance formula." }, { "id": "cie-2019-m-12-q10-ii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 10(ii): Find the point where the tangent to y = 4√x is parallel to the line joining the intersections in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=15", "qpPages": [ 19, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the curve gradient equal to the line gradient and find the corresponding coordinates." }, { "id": "cie-2019-m-12-q10-iii", "paperId": "cie-2019-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 10(iii): Find where the normal at T meets the line y = x + 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_12.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_12.pdf#page=15", "qpPages": [ 19, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal gradient through T, then solve its line equation simultaneously with y = x + 3." }, { "id": "cie-2019-m-22-q01-u", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Question 1: Solve sec^2θ + tan^2θ = 5tanθ + 4 for 0° < θ < 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use sec^2θ = 1 + tan^2θ, solve the quadratic in tanθ, and give all angles in range." }, { "id": "cie-2019-m-22-q02-u", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 2: Use |2x + 3| = |2x − 1| to evaluate |4x − 3| − |6x|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 4: Many candidates squared the expression incorrectly while trying to handle the modulus signs.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=4", "tagConfidence": "reviewed", "answerGuide": "Solve the absolute-value equation for x and substitute into the requested expression." }, { "id": "cie-2019-m-22-q03-u", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "data-analysis" ], "primaryTopicId": "exponentials", "summary": "Question 3: Find A and p from a straight-line graph of ln y against x when y = A e^(px+p).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the line gradient for p, then substitute a plotted point to solve for A." }, { "id": "cie-2019-m-22-q04-i", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 4(i): Divide 4x^3 + 8x^2 + 11x + 9 by 2x + 1 and verify the remainder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Perform polynomial long division and evaluate or subtract to confirm the remainder is 5." }, { "id": "cie-2019-m-22-q04-ii", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 4(ii): Show that 4x^3 + 8x^2 + 11x + 4 = 0 has exactly one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factorisation from part (i), examine the remaining quadratic discriminant, and state the root count." }, { "id": "cie-2019-m-22-q05-i", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(i): Show that the curve point with y = 10 leads to the stated logarithmic iteration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute y = 10 into the curve equation, rearrange, and take logarithms to obtain the fixed-point form." }, { "id": "cie-2019-m-22-q05-ii", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Question 5(ii): Use the given iteration starting at 2.3 to find the point x-coordinate to four significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 5: The report says candidates often omitted the written iterations needed to justify the final value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=5", "tagConfidence": "reviewed", "answerGuide": "Iterate with the stated rounding, show enough values, and report the converged x-coordinate." }, { "id": "cie-2019-m-22-q05-iii", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Question 5(iii): Find the curve gradient at P to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the quotient defining the curve and substitute the x-value from part (ii) at adequate precision." }, { "id": "cie-2019-m-22-q06-a", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Question 6(a): Show that the given definite rational integral equals ln 48.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split the integrand into logarithmic forms, integrate, apply limits, and combine logarithms." }, { "id": "cie-2019-m-22-q06-b", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Question 6(b): Integrate sin(2x)(cot x + 2cosec x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand using sin(2x) = 2sin x cos x, simplify to cosine terms, and integrate." }, { "id": "cie-2019-m-22-q07-i", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Question 7(i): Show that the parameter at P satisfies 2sin(2t) − 4cos(2t) = 1 when dy/dx = 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x(t) and y(t), form dy/dx, set it to 2, and rearrange." }, { "id": "cie-2019-m-22-q07-ii", "paperId": "cie-2019-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Question 7(ii): Find the coordinates of P from the given parameter equation, to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_22.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_22.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 5: Candidates often did not use the result from part (i); the report stresses radians and completing both coordinates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=5", "tagConfidence": "reviewed", "answerGuide": "Express the left side as R sin(2t − α), solve for the admissible t in radians, and substitute into x(t), y(t)." }, { "id": "cie-2019-m-32-q01-i", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Question 1(i): Show that log10(x − 4) = 2 − log10 x can be written as a quadratic in x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use logarithm laws to combine the logarithms, convert to an exponential equation, and expand." }, { "id": "cie-2019-m-32-q01-ii", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Question 1(ii): Solve the logarithmic equation from part (i) to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 7: Approximately half of candidates with the correct quadratic did not reject the negative root.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=7", "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic and reject any root outside the logarithms’ domain." }, { "id": "cie-2019-m-32-q02-i", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Question 2(i): Use the given iteration from x1 = 2 to find α to four decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate to sufficient precision and show the values required to justify rounding." }, { "id": "cie-2019-m-32-q02-ii", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Question 2(ii): State an equation for α and find its exact value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the fixed point equal to its iteration expression, simplify the polynomial, and solve exactly." }, { "id": "cie-2019-m-32-q03-i", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 3(i): Find an exact surd expression for tan θ from the given shifted-angle equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand the sine and cosine angle sums, collect sinθ and cosθ terms, and divide to obtain tanθ." }, { "id": "cie-2019-m-32-q03-ii", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Question 3(ii): Solve the shifted-angle equation for 0° < θ < 360°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 7: Many candidates lost the second angle by choosing the wrong quadrant after finding tan θ.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=7", "tagConfidence": "reviewed", "answerGuide": "Use the exact tanθ result, identify both quadrants, and report both angles." }, { "id": "cie-2019-m-32-q04-u", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Question 4: Show that the stated definite integral of x^(-3/2) ln x equals 2 − ln 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate by parts, integrate the remaining power term, apply both limits, and preserve exact logarithms." }, { "id": "cie-2019-m-32-q05-u", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Question 5: Implicitly differentiate sin y = tan x and establish the stated expression for dy/dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 7: The report describes this implicit-differentiation question as difficult for nearly all candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=7", "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides, use cos y dy/dx = sec^2 x, then use tan x = sin y and a double-angle identity." }, { "id": "cie-2019-m-32-q06-u", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Question 6: Solve the separable differential equation with two given boundary conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 7: Many candidates struggled with signs and algebra after separating and integrating the differential equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=7", "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate both sides, apply both points to determine constants, and rearrange for y." }, { "id": "cie-2019-m-32-q07-a", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 7(a): Solve the complex quadratic without a calculator and show the roots as x + iy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 8: Many candidates made arithmetic errors while solving and simplifying the complex quadratic.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=8", "tagConfidence": "reviewed", "answerGuide": "Use the quadratic formula, simplify the complex square root, rationalise by the conjugate, and state both roots." }, { "id": "cie-2019-m-32-q07-b", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 7(b): Sketch the Argand region satisfying the modulus and argument inequalities relative to u = −1 − i.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 8: Apart from placing u, few candidates made progress on the required Argand-region sketch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=8", "tagConfidence": "reviewed", "answerGuide": "Plot u, draw the perpendicular bisector for the modulus condition and the two argument boundary rays from u, then shade their intersection." }, { "id": "cie-2019-m-32-q08-i", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Question 8(i): Express the rational function in partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up the linear-factor terms and a constant, then equate coefficients or substitute convenient x-values." }, { "id": "cie-2019-m-32-q08-ii", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Question 8(ii): Expand the partial-fraction form in ascending powers through x^2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use binomial expansions for each linear denominator, retain terms through x^2, and add the constant." }, { "id": "cie-2019-m-32-q09-i", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 9(i): Find the acute angle between two planes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the planes’ normal vectors and the dot-product formula, then choose the acute angle." }, { "id": "cie-2019-m-32-q09-ii", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Question 9(ii): Find a vector equation for the line where two planes intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 8: The report notes frequent arithmetic errors and confusion between a direction vector and a point on the line.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=8", "tagConfidence": "reviewed", "answerGuide": "Find a point common to both planes and a direction perpendicular to both normal vectors, then form r = a + λd." }, { "id": "cie-2019-m-32-q10-i", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Question 10(i): Use u = cos x to find the exact area under y = sin^3 x √(cos x) over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 9: Many candidates made errors in the substitution integral, including sign and bracket errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=9", "tagConfidence": "reviewed", "answerGuide": "Convert dx and the limits to u, simplify using sin^2x = 1 − u^2, integrate, and substitute the limits." }, { "id": "cie-2019-m-32-q10-ii", "paperId": "cie-2019-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Question 10(ii): Find the x-coordinate of the maximum point to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_32.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_32.pdf#page=15", "qpPages": [ 17, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 9: Chain-rule omissions lost marks and poor algebra often prevented an acceptable final equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=9", "tagConfidence": "reviewed", "answerGuide": "Differentiate the product with chain rule, set the derivative to zero, simplify to one trig function, and solve within the interval." }, { "id": "cie-2019-m-42-q01-u", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 1: Find the coefficient of friction for a ring in limiting equilibrium against a rough vertical rod.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces perpendicular and parallel to the rod, use F = μR, and solve for μ." }, { "id": "cie-2019-m-42-q02-i", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 2(i): Show that the particle’s maximum height is 45 m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v^2 = u^2 + 2as at the highest point with v = 0 and a = −g." }, { "id": "cie-2019-m-42-q02-ii", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 2(ii): Find the first time the particle reaches 33.75 m and its speed then.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the vertical displacement equation for both times, select the earlier one, and use v = u + at." }, { "id": "cie-2019-m-42-q03-u", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 3: Find F and α for four coplanar forces in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally and vertically, then combine the components to find the magnitude and direction." }, { "id": "cie-2019-m-42-q04-i", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 4(i): Find the car resistance R when the car-trailer system travels at constant speed under known power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv for the driving force, then balance the total resistances." }, { "id": "cie-2019-m-42-q04-ii", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 4(ii): Find the acceleration and rod tension after engine power increases at speed 25 m/s.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 11: The report identifies this as one of the more difficult questions; missing or extra force terms and premature rounding caused errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=11", "tagConfidence": "reviewed", "answerGuide": "Find the new driving force, write Newton equations for car and trailer or the whole system, then solve for acceleration and tension." }, { "id": "cie-2019-m-42-q05-i", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 5(i): Find the distance travelled during the first eight seconds from a velocity-time graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the velocity at t = 3 and add the absolute areas of the first three graph sections." }, { "id": "cie-2019-m-42-q05-ii", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 5(ii): Find the minimum velocity V given the displacement at t = 16.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use signed areas under the velocity-time graph and solve the final triangular area for V." }, { "id": "cie-2019-m-42-q06-i", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Question 6(i): Show that the acceleration is 16 m/s^2 when the particle next comes to rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration to velocity, use the initial rest condition, solve v = 0 for the later time, then substitute into a(t)." }, { "id": "cie-2019-m-42-q06-ii", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Question 6(ii): Find the displacement from O at t = 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity function, apply s = 0 at t = 0, and evaluate at t = 5 with its sign." }, { "id": "cie-2019-m-42-q07-i", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 7(i): Find the work done by friction while the particle travels along the rough incline PQ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 13: Question 7 was among the most challenging; candidates sometimes used the wrong distance for friction work.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=13", "tagConfidence": "reviewed", "answerGuide": "Resolve normal to the plane, find the friction force, then multiply by the distance moved along its direction." }, { "id": "cie-2019-m-42-q07-ii", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 7(ii): Find the particle speed at Q after moving along the rough incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 13: Question 7 was among the most challenging; errors included wrong acceleration signs and missing or mis-signed energy terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=13", "tagConfidence": "reviewed", "answerGuide": "Use a work-energy equation with initial and final kinetic energy, friction work, and potential-energy gain." }, { "id": "cie-2019-m-42-q07-iii", "paperId": "cie-2019-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 7(iii): Find the height h where the particle comes to rest at R.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_42.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_42.pdf#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 13: Question 7 was among the most challenging; candidates sometimes omitted friction work or the extra height from Q.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=13", "tagConfidence": "reviewed", "answerGuide": "Use energy from Q to R or from P to R and include the height of Q above P." }, { "id": "cie-2019-m-52-q01-u", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 1: Find the speed and direction of a projectile four seconds after launch at 24 m/s and 30°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve initial velocity into horizontal and vertical components, update vertical velocity under gravity, and combine the components." }, { "id": "cie-2019-m-52-q02-i", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Question 2(i): Find the centre of mass height for three vertically stacked uniform cubes of edge lengths 3, 2, and 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use volumes as relative masses and take moments about the base of the largest cube." }, { "id": "cie-2019-m-52-q02-ii", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Question 2(ii): Find the ratio of the masses of the old and replacement unit cubes when the centre of mass is at a specified level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 14: The report places 2(ii) among the harder questions; some candidates used areas instead of volumes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=14", "tagConfidence": "reviewed", "answerGuide": "Set the total moment about the base equal to total mass times the required centre-of-mass height, then solve for the replacement mass." }, { "id": "cie-2019-m-52-q03-i", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Question 3(i): Find the trajectory equation from x = 4t and y = 6t − 5t^2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Eliminate t between the horizontal and vertical displacement equations." }, { "id": "cie-2019-m-52-q03-ii", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Question 3(ii): Find the launch angle and initial speed from the trajectory.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare with the standard projectile trajectory or resolve initial velocity into horizontal and vertical components." }, { "id": "cie-2019-m-52-q04-u", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 4: Find the speed of a particle moving in a horizontal circle on an elastic string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve vertically to find tension, apply Hooke’s law for extension, determine the radius, and apply Newton’s second law horizontally." }, { "id": "cie-2019-m-52-q05-i", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 5(i): Find the greatest speed of a particle attached to a vertical elastic string after release.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 15: The report places 5(i) among the harder questions; candidates often missed the equilibrium position for greatest speed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=15", "tagConfidence": "reviewed", "answerGuide": "Find the equilibrium extension, then equate initial energy to energy at equilibrium using the elastic and gravitational terms." }, { "id": "cie-2019-m-52-q05-ii", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Question 5(ii): Find the greatest distance below O reached by the particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 15: The report places 5(ii) among the harder questions; candidates needed to use zero velocity at maximum extension.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=15", "tagConfidence": "reviewed", "answerGuide": "At the turning point set speed to zero and solve the energy equation for the extension." }, { "id": "cie-2019-m-52-q06-i", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Question 6(i): Find the centre of mass of a cylindrical prism after an off-centre cylindrical hole is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use area subtraction as relative mass and take moments about the original cylinder centre." }, { "id": "cie-2019-m-52-q06-ii", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Question 6(ii): Find the coefficient of friction when the drilled prism is in limiting equilibrium on a rough plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=9", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 15: The report explicitly identifies part 6(ii) as the hardest question on the paper.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=15", "tagConfidence": "reviewed", "answerGuide": "Take moments about the contact point to find the applied force, resolve for friction and reaction, then use F = μR." }, { "id": "cie-2019-m-52-q07-i", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Question 7(i): Show that the particle mass is 0.24 kg when it begins moving again at t = 8.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At the threshold use the applied force equal to limiting friction, with F = μmg." }, { "id": "cie-2019-m-52-q07-ii", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Question 7(ii): Show dv/dt = 0.25t − 2 for 0 ≤ t ≤ 4 and find the projection speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law with the time-varying force and friction, integrate acceleration, and use v = 0 at t = 4." }, { "id": "cie-2019-m-52-q07-iii", "paperId": "cie-2019-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "mechanics", "kinematics", "integration" ], "primaryTopicId": "mechanics", "summary": "Question 7(iii): Find the distance from O where the particle comes to rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_52.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_52.pdf#page=10", "qpPages": [ 13, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity function from t = 0 to t = 4, using the initial position and the constant from part (ii)." }, { "id": "cie-2019-m-62-q01-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 1(i): Find the probability that Tamar wears red socks on a workday.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the total-probability rule over blue and grey suits." }, { "id": "cie-2019-m-62-q01-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 1(ii): Find the probability Tamar wears a grey suit given that he is not wearing red socks.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the joint probability of grey suit and no red socks by the total probability of no red socks." }, { "id": "cie-2019-m-62-q02-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 2(i): Find Σ(x − c) from the given squared-deviation total and standard deviation for 40 observations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 16: The report says most candidates found the coded-data question challenging and many struggled with the rearrangement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=16", "tagConfidence": "reviewed", "answerGuide": "Use the coded-data variance identity, substitute the given values, and rearrange for the required sum." }, { "id": "cie-2019-m-62-q02-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 2(ii): Find the mean when c = 50.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Σx = Σ(x − c) + 40c and divide by 40." }, { "id": "cie-2019-m-62-q03-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 3(i): Find the probability a train journey takes under 132 minutes for a normal model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise using the stated mean and standard deviation and read the lower-tail probability." }, { "id": "cie-2019-m-62-q03-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 3(ii): Find k when the probability journey time exceeds k is 0.675.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the upper-tail probability to a z-value, standardise, and solve for k." }, { "id": "cie-2019-m-62-q04-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 4(i): Complete the probability table for X with probabilities kx^2 and find k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the four possible x-values, set the probabilities to sum to one, and solve for k." }, { "id": "cie-2019-m-62-q04-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 4(ii): Find E(X) and Var(X) for the distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the table to calculate the first and second moments, then subtract the square of the mean." }, { "id": "cie-2019-m-62-q05-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 5(i): Draw a back-to-back stem-and-leaf diagram for the Dolphins and Sharks weights.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Order each team’s data, align leaves to shared stems, and include a key with team labels and units." }, { "id": "cie-2019-m-62-q05-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 5(ii): Find the Dolphins’ median and interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Order the Dolphins’ weights, identify the median and quartiles, and subtract the lower quartile from the upper." }, { "id": "cie-2019-m-62-q06-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 6(i): Find the probability that more than three of six customers shop online.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with n = 6 and p = 0.35; sum the outcomes 4, 5, and 6." }, { "id": "cie-2019-m-62-q06-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "probability", "exponentials" ], "primaryTopicId": "probability", "summary": "Question 6(ii): Find the least sample size for which the probability at least one customer shops online exceeds 0.95.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the complement probability for zero online shoppers, solve the exponential inequality, and round up to an integer." }, { "id": "cie-2019-m-62-q06-iii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Question 6(iii): Use a suitable approximation to find the probability more than 39 of 100 customers shop online.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a normal approximation to the binomial, justify it, apply the continuity correction at 39.5, and evaluate the upper tail." }, { "id": "cie-2019-m-62-q07-i", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 7(i): Count arrangements of all nine letters in CAMERAMAN with no restrictions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide 9! by the factorials for repeated As and Ms." }, { "id": "cie-2019-m-62-q07-ii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 7(ii): Count arrangements when the As occupy positions 1, 5, and 9.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Fix the As and arrange the remaining six letters, correcting for the repeated Ms." }, { "id": "cie-2019-m-62-q07-iii", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 7(iii): Count arrangements with exactly one letter between the two Ms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 18: The report says the CAMERAMAN counting question was challenging; this part required careful counting of the allowed M positions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=18", "tagConfidence": "reviewed", "answerGuide": "Choose one of seven allowable M-pairs, then arrange the remaining letters while dividing out repeated As." }, { "id": "cie-2019-m-62-q07-iv", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 7(iv): Count selections of three letters with exactly one M and one A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=13", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose one of the two Ms, one of the three As, and one of the four other letters." }, { "id": "cie-2019-m-62-q07-v", "paperId": "cie-2019-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(v)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Question 7(v): Count selections of three letters containing at least one M.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_62.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_62.pdf#page=13", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 18: The report records frequent omissions and overcounting in the final constrained selection part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=18", "tagConfidence": "reviewed", "answerGuide": "Count selections with one or two Ms, account for repeated letters as identical, and include the possible A combinations." }, { "id": "cie-2019-m-72-q01-i", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 1(i): Calculate a 98% confidence interval for the population mean plum mass from a sample of 200.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the known population standard deviation and the appropriate two-sided normal critical value." }, { "id": "cie-2019-m-72-q01-ii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 1(ii): State whether the Central Limit Theorem is needed and explain why.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 19: The report says part 1(ii) was less well attempted; answers often failed to identify the population distribution assumption.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=19", "tagConfidence": "reviewed", "answerGuide": "Relate the need for the theorem to the population distribution not being stated to be normal." }, { "id": "cie-2019-m-72-q02-u", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Question 2: Find P(X > 3Y) for independent normal variables X and Y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the normal distribution of X − 3Y using independent means and variances, then standardise at zero." }, { "id": "cie-2019-m-72-q03-u", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 3: Test at 2% whether factory B’s Poisson accident mean is lower than factory A’s known mean of 32.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 20: The report says the hypothesis test was very demanding for many candidates, especially hypotheses, approximation, and continuity correction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=20", "tagConfidence": "reviewed", "answerGuide": "Set one-sided hypotheses, approximate the count distribution by a normal model with continuity correction, compare with the critical value, and state a cautious contextual conclusion." }, { "id": "cie-2019-m-72-q04-i", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Question 4(i): Find an unbiased mean estimate and show the unbiased variance estimate is 490 to 3 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sample mean and n − 1 variance formula; retain enough precision to show the rounding." }, { "id": "cie-2019-m-72-q04-ii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Question 4(ii): Find n such that the sample mean exceeds 5 with probability 0.9377 using the estimates from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the sample mean, use the corresponding normal quantile, solve for n, and give a whole number." }, { "id": "cie-2019-m-72-q05-i", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 5(i): Find the probability that at least two eagles are seen in one hour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the complement of zero or one eagle under the Poisson model with mean 1.8." }, { "id": "cie-2019-m-72-q05-ii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Question 5(ii): Find the probability of fewer than five total eagles and vultures in a half-hour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Halve both Poisson means, add the independent rates, and calculate the Poisson probability through four." }, { "id": "cie-2019-m-72-q05-iii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "probability", "exponentials" ], "primaryTopicId": "probability", "summary": "Question 5(iii): Find the minimum observation time for at least 99% certainty of seeing one or more eagles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge Principal Examiner Report for Teachers, February/March 2019, physical page 20: The report says candidates often could not solve the minimum-time probability inequality and rarely succeeded by trial and improvement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_er.pdf#page=20", "tagConfidence": "reviewed", "answerGuide": "Use the zero-sighting complement for a Poisson rate of 1.8 per hour, solve the logarithmic inequality, and state the time." }, { "id": "cie-2019-m-72-q06-i", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Question 6(i): Give a reason the test of whether the new-task mean differs from 91.4 should be two-tailed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that the alternative allows either an increase or a decrease." }, { "id": "cie-2019-m-72-q06-ii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Question 6(ii): State the Type I error probability and explain the error in context.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the significance level and describe rejecting equal means when the new-task mean is actually unchanged." }, { "id": "cie-2019-m-72-q06-iii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Question 6(iii): Carry out the 5% test of whether the new-task mean differs from 91.4, stating an assumption.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assume the stated population standard deviation applies, set two-sided hypotheses, calculate the test statistic, compare it with the critical value, and conclude in context." }, { "id": "cie-2019-m-72-q07-i", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Question 7(i): Find P(X > π/6) for the stated cosine probability density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density from π/6 to the upper support limit." }, { "id": "cie-2019-m-72-q07-ii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Question 7(ii): Find the median of X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density from zero to the median, set the result to one half, and solve in radians." }, { "id": "cie-2019-m-72-q07-iii", "paperId": "cie-2019-m-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "February/March", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Question 7(iii): Find E(X) for the stated probability density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_qp_72.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m19_ms_72.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x times the density across the support, using integration by parts." }, { "id": "ial-2025-6-WME02-01A-q01a", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Verify that the particle is instantaneously at rest at t = 2.", "answerGuide": "Differentiate the displacement function to obtain velocity, then substitute t = 2 and show that the velocity is zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q01b", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled from t = 0 to t = 3, given that t = 2 is the only time at rest.", "answerGuide": "Evaluate displacement at t = 0, 2 and 3, then add the distances on the two sides of the direction change at t = 2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q01c", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the magnitude of the acceleration when t = 1.5.", "answerGuide": "Differentiate the velocity to obtain acceleration, substitute t = 1.5, and give its magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q02a", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the stated centre-of-mass distance for a square lamina with a semicircle removed and reattached.", "answerGuide": "Use signed moments about AD (or a parallel axis) for the square and both semicircles, then simplify to the required exact expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q02b", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the exact value of k for the loaded lamina suspended by two strings with tensions 4T and T.", "answerGuide": "Use a moments equation and a vertical-force equation for the loaded lamina, then solve the simultaneous equations for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says part (b) was more challenging than part (a); many candidates formed only one of the two required equations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q03a", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the greatest height reached by the projectile.", "answerGuide": "Use vertical motion with zero vertical speed at the maximum height.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q03b", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile’s speed 2.4 seconds after launch.", "answerGuide": "Find the vertical velocity at 2.4 seconds and combine it with the constant horizontal velocity using Pythagoras.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q03c", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the horizontal distance AB when the projectile passes through C, 3 m above B after reaching its maximum height.", "answerGuide": "Use vertical motion to find the later time at height 3 m, then multiply by the horizontal speed to obtain AB.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q04a", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the cyclist’s rate of working when she travels at 5 m s⁻¹ with acceleration 0.4 m s⁻² against 30 N resistance.", "answerGuide": "Use Newton’s second law to find the driving force, then calculate power as force multiplied by speed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q04b", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the work–energy principle to find the change in vertical height as the cyclist travels downhill for 250 m.", "answerGuide": "Account for the change in kinetic energy, work against resistance, and change in gravitational potential energy.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q04c", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the total work done by the cyclist travelling 200 m uphill at constant speed on a 5° slope.", "answerGuide": "Add the work against resistance to the gain in gravitational potential energy; the kinetic energy is unchanged.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q05", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the two possible impulse vectors that deflect the particle’s direction of motion by 45°.", "answerGuide": "Express each possible post-impulse velocity in components, apply impulse = change in momentum, then use the given impulse magnitude to solve for both directions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes Question 5 as probably the most challenging question on the paper; many candidates earned only 2 or 3 of its 8 marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q06a", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the stated expression for the tension in the string supporting the horizontal rod and attached particle.", "answerGuide": "Take moments about A, include the rod’s weight and the attached particle’s weight, and use the string geometry to substitute the trigonometric ratios.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q06b", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the exact value of k from the direction of the resultant force at A.", "answerGuide": "Resolve the forces horizontally and vertically, use tan θ = Y/X, and solve the resulting equation for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that only a minority of candidates reached the final correct value in part (b).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q07a", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show the speed of Q immediately after its first collision with P.", "answerGuide": "Use conservation of linear momentum and Newton’s law of impact, keeping the directions of the velocities consistent.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=16", "qpPages": [ 24, 27 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q07b", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the full range of possible values of e when P reverses direction after the first collision.", "answerGuide": "Express P’s post-collision velocity in terms of e and u, then apply the direction condition and the physical bound on the coefficient of restitution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=16", "qpPages": [ 24, 27 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson says the upper bound e = 1 was rarely shown, although many candidates earned two of the three marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q07c", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show that e = 4/5 from the stated impulse exerted on Q.", "answerGuide": "Apply impulse–momentum to Q using the result from part (a), then show the algebra leading to the stated value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=16", "qpPages": [ 24, 27 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01A-q07d", "paperId": "ial-2025-6-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Decide, with clear reasoning, whether P and Q collide for a second time.", "answerGuide": "Find the velocities after the subsequent collision involving Q and R, then compare the relevant speeds and directions of P and Q.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2A_MS.pdf#page=16", "qpPages": [ 24, 27 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says the challenge increased in part (d), and only a minority gained full marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01a-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-1-WDM11-01-q01a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Pack the listed item lengths into bins using first fit.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Scan the items in order and place each into the earliest bin with enough remaining capacity; open a new bin only when necessary." }, { "id": "ial-2025-1-WDM11-01-q01b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Show the quick-sort passes for the given list.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the specified pivot convention, partition the list at each pass, and record the sublists until ordered." }, { "id": "ial-2025-1-WDM11-01-q01c", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply first-fit decreasing to the same packing list.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sort the item lengths in descending order, then place each item into the first bin with sufficient space." }, { "id": "ial-2025-1-WDM11-01-q02ai", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra’s algorithm to identify the shortest route between the named attractions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Update permanent and temporary labels from the starting vertex, then trace the predecessor links back from the destination." }, { "id": "ial-2025-1-WDM11-01-q02aii", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the length of the shortest route.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the edge lengths along the route found in part (i)." }, { "id": "ial-2025-1-WDM11-01-q02b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the minimum closed inspection route that traverses every network edge.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify odd vertices, compare all pairings by shortest-path totals, and add the least repeated distance to the network total." }, { "id": "ial-2025-1-WDM11-01-q02ci", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)(i)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Choose the best finishing attraction for an inspection route that starts at A but ends elsewhere.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports this as the least successful subpart: many scripts left it blank or chose an endpoint without justification.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Leave the start vertex out of the final odd-node pairing; repeat the shortest eligible connection and justify the resulting endpoint. " }, { "id": "ial-2025-1-WDM11-01-q02cii", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)(ii)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate the difference between the closed and open inspection-route lengths.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the open-route length from the closed-route length, using the repeated-edge totals already found." }, { "id": "ial-2025-1-WDM11-01-q03a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Run the given flowchart on the supplied input and complete its working table.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Follow each branch in order, updating the variables after every iteration and stopping at the stated termination condition." }, { "id": "ial-2025-1-WDM11-01-q03b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify the number produced by the completed flowchart.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the final value from the last completed iteration and express it as the required common multiple." }, { "id": "ial-2025-1-WDM11-01-q04a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw an activity-on-arc network from the precedence information.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=13", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Place each activity after all of its immediate predecessors, using the smallest number of dummy arcs needed to preserve the dependencies." }, { "id": "ial-2025-1-WDM11-01-q04b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify the critical activities in the network.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=13", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Carry out forward and backward passes; activities with zero total float are critical." }, { "id": "ial-2025-1-WDM11-01-q05a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the objective function for the production-planning problem.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=15", "qpPages": [ 6, 6 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Define the decision variables and state both the direction of optimization and the linear objective." }, { "id": "ial-2025-1-WDM11-01-q05b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Translate the product-mix condition into a linear constraint.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=15", "qpPages": [ 6, 6 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says forming this particular inequality was especially challenging for most candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert the stated percentages to fractions of total production, clear denominators, and write the corresponding inequality. " }, { "id": "ial-2025-1-WDM11-01-q05c", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Write the remaining production and order restrictions as inequalities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=15", "qpPages": [ 6, 6 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the limits and ordering conditions into algebraic inequalities, keeping each variable in the required range." }, { "id": "ial-2025-1-WDM11-01-q06a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Distinguish the classical travelling-salesperson problem from the practical version.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the classical version seeks the shortest tour visiting each vertex once and returning to the start; contrast the practical setting’s constraints." }, { "id": "ial-2025-1-WDM11-01-q06b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct a minimum spanning tree using the stated algorithm.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Start at the required vertex and repeatedly add the least-weight eligible edge that does not create a cycle." }, { "id": "ial-2025-1-WDM11-01-q06c", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give an upper bound for the route length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Double the minimum-spanning-tree total to obtain a valid upper bound." }, { "id": "ial-2025-1-WDM11-01-q06d", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a route using the nearest-neighbour method.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "From the current vertex choose the nearest unvisited one at each step, then close the tour and total its length." }, { "id": "ial-2025-1-WDM11-01-q06e", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Compare the route bounds.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the computed lower and upper bounds to state which is smaller and how they constrain the optimum." }, { "id": "ial-2025-1-WDM11-01-q06f", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(f)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound for the travelling-salesperson route.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the specified lower-bound construction, selecting the necessary incident edges and accounting for the degree condition." }, { "id": "ial-2025-1-WDM11-01-q06g", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(g)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the interval containing the optimal route length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the calculated lower and upper bounds to give the inclusive range for the optimum." }, { "id": "ial-2025-1-WDM11-01-q07a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain why a dummy activity is needed in the precedence network.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Describe the dependency that would otherwise be represented incorrectly by the network arcs." }, { "id": "ial-2025-1-WDM11-01-q07b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate early and late event times in the activity network.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Make a forward pass for earliest times and a backward pass for latest times, applying the event-time constraints at merges and splits." }, { "id": "ial-2025-1-WDM11-01-q07c", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify the critical activities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the early and late timings; zero-float activities lie on the critical path." }, { "id": "ial-2025-1-WDM11-01-q07d", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate the float for activity K.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use its latest and earliest start/finish times to find the available delay." }, { "id": "ial-2025-1-WDM11-01-q07e", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw the requested schedule chart from the network timings.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Place each activity on its allowed time interval and show any float clearly with aligned time columns." }, { "id": "ial-2025-1-WDM11-01-q07f", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(f)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound for the number of workers required.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide total worker-hours by the critical completion time and round up to a whole worker." }, { "id": "ial-2025-1-WDM11-01-q08a", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Write the inequalities defining the feasible production region.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=20", "qpPages": [ 9, 9 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert each resource or order limit into a linear inequality and include non-negativity." }, { "id": "ial-2025-1-WDM11-01-q08b", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Derive the objective function for the integer optimization problem.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=20", "qpPages": [ 9, 9 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express each item’s contribution in the common units, combine the contributions, and state the quantity to minimize." }, { "id": "ial-2025-1-WDM11-01-q08c", "paperId": "ial-2025-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the minimum integer solution that satisfies the constraints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_D1_MS.pdf#page=20", "qpPages": [ 9, 9 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says the correct integer optimum was rarely seen by examiners.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Check feasible integer combinations against the constraints and select the one with the smallest objective value. " }, { "id": "ial-2025-1-WMA11-01-q01", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the given algebraic integral.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite each term using index notation, integrate term by term, and simplify the constant and variable terms." }, { "id": "ial-2025-1-WMA11-01-q02a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Rationalize a surd expression used to find a line gradient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply by the conjugate, simplify the surds, and evaluate the gradient from the coordinate differences." }, { "id": "ial-2025-1-WMA11-01-q02b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a line perpendicular to the given line through the specified point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the negative reciprocal gradient from part (a) and substitute the point into point-slope form." }, { "id": "ial-2025-1-WMA11-01-q03a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Determine the constants in the exponential population model from two observations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute both population-time pairs, eliminate the exponential term or take logarithms, and solve for the model parameters." }, { "id": "ial-2025-1-WMA11-01-q03b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Use the fitted model to find the requested future population quantity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the stated model condition and solve for the unknown time or value with logarithms." }, { "id": "ial-2025-1-WMA11-01-q04ia", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(a)", "marks": 1, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Rewrite the first expression in the required power form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply index laws to collect the powers of the base into the requested exponent." }, { "id": "ial-2025-1-WMA11-01-q04ib", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(b)", "marks": 3, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Simplify the second expression using index laws.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson lists this index-law simplification among the paper’s particularly challenging parts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the numerical factors and add or subtract the exponents carefully; retain the specified variable form." }, { "id": "ial-2025-1-WMA11-01-q04iia", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)(a)", "marks": 2, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Use the given substitution to show the equation becomes quadratic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson lists this exponential substitution step among the paper’s particularly challenging parts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Replace the exponential expression by the new variable, expand, and collect the quadratic terms." }, { "id": "ial-2025-1-WMA11-01-q04iib", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)(b)", "marks": 3, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Solve the transformed equation and return to the original variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic, reject inadmissible roots if needed, then solve the exponential equations for the original unknown." }, { "id": "ial-2025-1-WMA11-01-q05a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the curve function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the power and product rules as needed, then simplify the derivative." }, { "id": "ial-2025-1-WMA11-01-q05b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use the tangent condition to obtain an equation for the unknown parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the point and derivative into the tangent-gradient condition and rearrange to the required equation." }, { "id": "ial-2025-1-WMA11-01-q05c", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Solve for the parameter and the corresponding point or tangent condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson lists this parameter-solving part among the paper’s particularly challenging parts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Factor or solve the resulting equation in the squared coordinate, keep valid real solutions, and calculate the associated parameter." }, { "id": "ial-2025-1-WMA11-01-q06a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "differentiation", "integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Show the derivative has the stated linear form at the specified point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the function, substitute the point information, and simplify to the target relationship." }, { "id": "ial-2025-1-WMA11-01-q06b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "differentiation", "integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Recover the function by determining constants and integrating.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given derivative relation to determine constants, integrate the derivative, and apply the point condition to fix the integration constant." }, { "id": "ial-2025-1-WMA11-01-q06c", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "differentiation", "integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Map the specified point under the stated coordinate transformation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the horizontal and vertical changes to the point coordinates in the order implied by the transformed function." }, { "id": "ial-2025-1-WMA11-01-q07a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch the reciprocal-style curve and mark its key features.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the vertical and horizontal asymptotes, intercepts and sign of each branch, then label them on the sketch." }, { "id": "ial-2025-1-WMA11-01-q07b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the parameter range for which the line intersects the curve in the required way.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the curve, form the resulting quadratic and use the discriminant condition for the required number of intersections." }, { "id": "ial-2025-1-WMA11-01-q08a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the obtuse angle in the triangle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sine rule to obtain the possible angles and select the obtuse value consistent with the diagram." }, { "id": "ial-2025-1-WMA11-01-q08b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Calculate the triangle area.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use one half times two sides times the sine of their included angle." }, { "id": "ial-2025-1-WMA11-01-q08c", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the sector angle from the area relationship.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sector-area formula and equate it to the required fraction or difference of the triangle area." }, { "id": "ial-2025-1-WMA11-01-q08d", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 5, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Calculate the perimeter of the composite triangle-and-sector boundary.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the straight exposed edges and the relevant arc length, excluding any shared internal boundary." }, { "id": "ial-2025-1-WMA11-01-q09a", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Complete the square for the given quadratic curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Group the quadratic terms, complete the square, and rewrite the curve in vertex form." }, { "id": "ial-2025-1-WMA11-01-q09b", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "State the maximum point from the completed-square form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the vertex coordinates and identify the maximum according to the sign of the quadratic coefficient." }, { "id": "ial-2025-1-WMA11-01-q09ci", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(i)", "marks": 1, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Determine the first requested feature of the shaded region.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated boundary condition and the vertex or intercept information to identify the required value." }, { "id": "ial-2025-1-WMA11-01-q09cii", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(ii)", "marks": 1, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Determine the next requested region condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this subpart as one of the paper’s particularly challenging items.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the relevant boundary value into the curve and simplify." }, { "id": "ial-2025-1-WMA11-01-q09ciii", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(iii)", "marks": 1, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Determine the final requested coordinate or boundary value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this subpart as one of the paper’s particularly challenging items.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the remaining stated condition and solve for the required coordinate." }, { "id": "ial-2025-1-WMA11-01-q09d", "paperId": "ial-2025-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(d)", "marks": 5, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Describe the region satisfying the curve and line inequalities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P1_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate each inequality into a boundary, determine the side to retain, and state the combined region with strict or inclusive bounds as required." }, { "id": "ial-2025-1-WMA12-01-q01a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the requested terms of the arithmetic sequence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first term and common difference to generate the specified terms." }, { "id": "ial-2025-1-WMA12-01-q01b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Calculate the required sum of the arithmetic series.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic-series sum formula with the term count, first term and last term." }, { "id": "ial-2025-1-WMA12-01-q02a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the given binomial expression to the requested power.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=5", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the generalized binomial expansion in ascending powers and simplify each coefficient." }, { "id": "ial-2025-1-WMA12-01-q02b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Select the expansion term needed for the stated approximation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=5", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that this follow-up approximation was unexpectedly more challenging; many candidates omitted or mishandled it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the relevant power of x and substitute the requested value." }, { "id": "ial-2025-1-WMA12-01-q03a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Express the material area of the open box as a function of its dimensions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the fixed sheet dimensions and cut-out geometry to form the area expression in the chosen variable." }, { "id": "ial-2025-1-WMA12-01-q03b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the stationary value of the area expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, set the derivative to zero and solve for the positive admissible dimension." }, { "id": "ial-2025-1-WMA12-01-q03c", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Show the stationary value is a minimum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies the minimum-proof part as an added challenge after the difficult modelling and stationary-point work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the second derivative test or a sign change in the first derivative to establish the minimum." }, { "id": "ial-2025-1-WMA12-01-q04a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the common ratio from the finite and infinite geometric sums.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write both sum formulae, eliminate the first term, and solve for the admissible ratio." }, { "id": "ial-2025-1-WMA12-01-q04b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Determine the first term of the geometric sequence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the ratio into either given sum relationship and solve for the initial term." }, { "id": "ial-2025-1-WMA12-01-q05a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Apply the factor theorem to the given polynomial.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the stated root and set the polynomial value to zero." }, { "id": "ial-2025-1-WMA12-01-q05b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the unknown polynomial coefficients from the given remainders.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that the remainder-theorem part caused candidates difficulty.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the remainder theorem at each supplied input to obtain simultaneous equations, then solve for the constants. " }, { "id": "ial-2025-1-WMA12-01-q05c", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Factor the polynomial using the known factor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem result to divide the polynomial, then factor the quotient and state all factors." }, { "id": "ial-2025-1-WMA12-01-q06a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the perpendicular bisector of the segment joining the given points.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=15", "qpPages": [ 16, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the midpoint and segment gradient, then use the negative reciprocal slope in a line equation." }, { "id": "ial-2025-1-WMA12-01-q06b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the unknown coordinate of the circle centre.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=15", "qpPages": [ 16, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the point-on-bisector condition or the known centre coordinate to solve for the parameter." }, { "id": "ial-2025-1-WMA12-01-q06c", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the equation of the circle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=15", "qpPages": [ 16, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the centre and a known point to find the radius, then substitute them into the standard circle form." }, { "id": "ial-2025-1-WMA12-01-q07i", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "integration", "algebra", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Apply the trapezium rule to the stated logarithmic integral.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the tabulated or endpoint values into the trapezium rule and simplify the logarithmic expression." }, { "id": "ial-2025-1-WMA12-01-q07ii", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "integration", "algebra", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Solve the resulting logarithmic equation for the unknown base or parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms, convert to an algebraic equation, solve for the admissible value, and check the domain." }, { "id": "ial-2025-1-WMA12-01-q08i", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 1, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Give a counterexample to the stated claim about irrational numbers.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose a simple irrational value and demonstrate that it does not satisfy the asserted result." }, { "id": "ial-2025-1-WMA12-01-q08ii", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove the stated parity property for an integer expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes this parity proof as quite demanding.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Assume the relevant odd or even form, expand it, and show the result is even but cannot be divisible by four." }, { "id": "ial-2025-1-WMA12-01-q09a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "differentiation", "integration", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact stationary point of the curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this calculus part was equally challenging, with recurrent errors in fractional powers and solving the stationary equation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve the stationary equation with the required algebraic steps, and substitute the x-value into the curve." }, { "id": "ial-2025-1-WMA12-01-q09b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "differentiation", "integration", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the non-zero x-intercept.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that candidates found this intercept part comparably challenging and often missed the direct factor-based route.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Set y to zero, factor the equation, and select the non-zero root." }, { "id": "ial-2025-1-WMA12-01-q09c", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "differentiation", "integration", "algebra" ], "primaryTopicId": "differentiation", "summary": "Calculate the area bounded by the curve and the stated line or axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find an antiderivative, evaluate it at the intersection limits, and combine it with any required rectangle or triangle area." }, { "id": "ial-2025-1-WMA12-01-q10a", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the trigonometric identity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite in sine and cosine, use the Pythagorean identity, and simplify both sides to a common expression." }, { "id": "ial-2025-1-WMA12-01-q10b", "paperId": "ial-2025-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over the specified interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P2_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says the final trigonometric solution part was more demanding than the preceding proof.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity to form a quadratic in a trig ratio, solve for the ratio and list every angle in range. " }, { "id": "ial-2025-1-WMA13-01-q01a", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show that the root lies within the stated interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the function at both endpoints, verify a sign change, and use continuity to conclude a root lies between them." }, { "id": "ial-2025-1-WMA13-01-q01b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Rearrange the root equation into the supplied iterative form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the function equal to zero and isolate the variable on the left." }, { "id": "ial-2025-1-WMA13-01-q01ci", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(i)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Carry out one iteration from the stated starting value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the initial estimate into the recurrence and round to four decimal places." }, { "id": "ial-2025-1-WMA13-01-q01cii", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(ii)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Continue iteration to estimate the root.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Repeat the recurrence until the estimate is stable to four decimal places." }, { "id": "ial-2025-1-WMA13-01-q02a", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the initial pond area covered by the weed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set time to zero in the logarithmic growth model, then convert to area." }, { "id": "ial-2025-1-WMA13-01-q02b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find when the covered area reaches the stated amount.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the target area, solve the linear logarithmic equation for time, and round as requested." }, { "id": "ial-2025-1-WMA13-01-q03", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Use calculus to determine the interval where the rational function is increasing.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the quotient, find its critical values and undefined point, then test the sign of the derivative on each interval." }, { "id": "ial-2025-1-WMA13-01-q04ai", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 3, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Find the constants in the partial-fraction decomposition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Clear denominators, compare coefficients or substitute convenient values, and solve for the three constants." }, { "id": "ial-2025-1-WMA13-01-q04aii", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 1, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Show that the remaining partial-fraction constant is zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the coefficient identity or expand the decomposition and compare the relevant coefficient." }, { "id": "ial-2025-1-WMA13-01-q04b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Integrate the decomposed rational function and express the exact result in the requested form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate polynomial terms directly and the reciprocal quadratic term logarithmically; evaluate the limits and combine constants and logarithms." }, { "id": "ial-2025-1-WMA13-01-q05a", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "exponentials", "functions", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find the initial temperature from the cooling model.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute zero for time in the exponential model." }, { "id": "ial-2025-1-WMA13-01-q05b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "exponentials", "functions", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Sketch the temperature curve and state its asymptote.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that parts (b) and (d) provided discrimination, with candidates losing marks on the sketch and asymptote.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Mark the initial value and limiting temperature, then draw the decreasing exponential curve approaching the horizontal asymptote." }, { "id": "ial-2025-1-WMA13-01-q05c", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "exponentials", "functions", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find when the metal reaches the target temperature.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the target temperature, isolate the exponential, take natural logarithms and solve for time." }, { "id": "ial-2025-1-WMA13-01-q05d", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "exponentials", "functions", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Show that time is a linear function of temperature and determine its constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies this derivative-based rearrangement as one of the parts that separated candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the model with respect to time, eliminate the exponential using the original equation, and rearrange to the required linear relation." }, { "id": "ial-2025-1-WMA13-01-q06a", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the stated value of the inverse function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the function equation for the input corresponding to the given output." }, { "id": "ial-2025-1-WMA13-01-q06b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the coefficients in the composite function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the rational function into itself, simplify the nested expression and compare coefficients." }, { "id": "ial-2025-1-WMA13-01-q06c", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Map the given point under the combined function transformation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Undo the horizontal input scaling to find the new x-coordinate, then apply the vertical stretch and shift to y." }, { "id": "ial-2025-1-WMA13-01-q07ai", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch the first absolute-value graph and mark its minimum and y-intercept.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=13", "qpPages": [ 20, 21 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the zero of the inner linear expression for the vertex and set x to zero for the y-intercept." }, { "id": "ial-2025-1-WMA13-01-q07aii", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch the vertically shifted absolute-value graph with its key points.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=13", "qpPages": [ 20, 21 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Keep the vertex x-coordinate, shift its y-value by the constant, and calculate the shifted y-intercept." }, { "id": "ial-2025-1-WMA13-01-q07b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the equation formed by the shifted absolute-value curve and a line.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=13", "qpPages": [ 20, 21 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Consider the positive and negative cases of the absolute value and solve the resulting linear equations." }, { "id": "ial-2025-1-WMA13-01-q08i", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the first trigonometric equation in radians on the specified interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes Question 8 as a good discriminator; this part is one of its two required equation solves.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite cosecant as a reciprocal, use the double-angle identity, solve the resulting trig equation and retain in-range solutions." }, { "id": "ial-2025-1-WMA13-01-q08ii", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the second trigonometric equation in degrees.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes Question 8 as a good discriminator; this part is one of its two required equation solves.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use compound-angle or double-angle identities to isolate a tangent or sine value, then include every solution in the interval. " }, { "id": "ial-2025-1-WMA13-01-q09a", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "differentiation", "functions", "exponentials" ], "primaryTopicId": "differentiation", "summary": "State the x-intercept of the logarithmic curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the function equal to zero and solve the logarithmic factor." }, { "id": "ial-2025-1-WMA13-01-q09b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "differentiation", "functions", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact stationary point of the curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies the final exact coordinate work in this stationary-point part as particularly discriminating.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product rule, solve the stationary equation, and substitute back for both exact coordinates. " }, { "id": "ial-2025-1-WMA13-01-q09c", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "differentiation", "functions", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Determine the range after multiplying the function by a negative constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that many candidates mishandled the range and inequality direction in this final part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stationary-point value and the end behaviour to reverse the original range inequality." }, { "id": "ial-2025-1-WMA13-01-q10a", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find dy/dx for the parametrically defined curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=19", "qpPages": [ 30, 32 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to the parameter and divide dy/dt by dx/dt." }, { "id": "ial-2025-1-WMA13-01-q10b", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Derive the stated expression for the gradient in terms of x.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=19", "qpPages": [ 30, 32 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says candidates struggled most with this part and only a minority obtained a fully correct result.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the trigonometric relation to eliminate the parameter and simplify the derivative." }, { "id": "ial-2025-1-WMA13-01-q10c", "paperId": "ial-2025-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find the exact point where the tangent has the specified gradient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P3_MS.pdf#page=19", "qpPages": [ 30, 32 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative equal to the given slope, solve for the parameter or coordinate, and use the curve relation to find the exact point." }, { "id": "ial-2025-1-WMA14-01-q01", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume generated by revolving the curve about the stated axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the disc or washer method, square the radius function, integrate between the given limits and simplify." }, { "id": "ial-2025-1-WMA14-01-q02a", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the implicit curve equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each term with respect to x, apply the chain rule to y terms, and collect dy/dx." }, { "id": "ial-2025-1-WMA14-01-q02b", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent gradient at the point on the y-axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the point condition to determine y, substitute into dy/dx, and simplify." }, { "id": "ial-2025-1-WMA14-01-q03a", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Determine coefficients in the unfamiliar binomial expansion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson names Question 3 as one of the hardest questions because the binomial expansion was presented in an unfamiliar form.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand to the required power, collect coefficients and use the supplied coefficient conditions to solve for unknown constants. " }, { "id": "ial-2025-1-WMA14-01-q03b", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "State the valid range for the binomial expansion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the convergence condition to the magnitude of the expansion variable." }, { "id": "ial-2025-1-WMA14-01-q03c", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the requested coefficient from the expansion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Select the relevant term and substitute the constants found earlier." }, { "id": "ial-2025-1-WMA14-01-q04i", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Use related rates to find how quickly the sphere’s radius changes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the sphere volume equation with respect to time and substitute the given radius and volume rate." }, { "id": "ial-2025-1-WMA14-01-q04ii", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve the differential equation for the changing fluid depth.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate both sides and use the two stated depth conditions to find the constant and required time." }, { "id": "ial-2025-1-WMA14-01-q05i", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the integral using integration by parts.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=14", "qpPages": [ 12, 15 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the algebraic factor for differentiation and the exponential or logarithmic factor for integration, then apply the limits." }, { "id": "ial-2025-1-WMA14-01-q05ii", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the rational expression using partial fractions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=14", "qpPages": [ 12, 15 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Decompose the integrand into simpler fractions, integrate each term, and apply the specified bounds." }, { "id": "ial-2025-1-WMA14-01-q06", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": null, "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove the integer statement by contradiction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assume the opposite parity condition, express the integers in the required odd/even forms, and derive a contradiction with the stated divisibility." }, { "id": "ial-2025-1-WMA14-01-q07", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": null, "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the prescribed trigonometric substitution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=19", "qpPages": [ 18, 19 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the stated sine expression, transform the limits and differential, simplify with a Pythagorean identity, then integrate." }, { "id": "ial-2025-1-WMA14-01-q08a", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter values for which the point lies on the vector line.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=20", "qpPages": [ 20, 23 ], "msPages": [ 20, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate corresponding coordinates in the vector equation and solve consistently for the parameters." }, { "id": "ial-2025-1-WMA14-01-q08b", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Calculate the vector joining the two specified points.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=20", "qpPages": [ 20, 23 ], "msPages": [ 20, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the initial point coordinates from the terminal point coordinates." }, { "id": "ial-2025-1-WMA14-01-q08c", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the angle between the two relevant vectors.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=20", "qpPages": [ 20, 23 ], "msPages": [ 20, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the direction vectors, use their dot product and magnitudes, and apply the inverse cosine." }, { "id": "ial-2025-1-WMA14-01-q08d", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find both possible points satisfying the stated area condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=20", "qpPages": [ 20, 23 ], "msPages": [ 20, 24 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says this unfamiliar area-geometry part was challenging and often received no credit.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the point on the line with a parameter, form a triangle-area or cross-product equation, solve the resulting quadratic and recover both coordinates. " }, { "id": "ial-2025-1-WMA14-01-q09ai", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)(i)", "marks": 3, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find dy/dx for the parametric curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to the parameter and form the ratio dy/dt over dx/dt." }, { "id": "ial-2025-1-WMA14-01-q09aii", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)(ii)", "marks": 3, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Use the tangent condition to verify the stated tangent equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the parameter at the point, calculate the gradient and substitute into point-slope form." }, { "id": "ial-2025-1-WMA14-01-q09b", "paperId": "ial-2025-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Find the other intersection point of the line and curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_P4_MS.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line equation into the parametric relation, solve for the parameter and calculate both coordinates." }, { "id": "ial-2025-1-WME01-01-q01a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction of the resultant acceleration from three forces.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=7", "qpPages": [ 3, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve all forces into horizontal and vertical components, divide the resultant by the mass and use atan2 for direction." }, { "id": "ial-2025-1-WME01-01-q01b", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the unknown force parameter from the direction of motion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=7", "qpPages": [ 3, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the ratio of the resultant force components to the tangent of the stated direction, then solve for the parameter." }, { "id": "ial-2025-1-WME01-01-q02a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph for the journey.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=8", "qpPages": [ 5, 8 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot the known speed and time points, show constant-acceleration segments as straight lines, and label the changes in motion." }, { "id": "ial-2025-1-WME01-01-q02bi", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time at which the first speed condition is met.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=8", "qpPages": [ 5, 8 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the constant-acceleration velocity equation on the relevant interval." }, { "id": "ial-2025-1-WME01-01-q02bii", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the corresponding time or speed for the second motion stage.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=8", "qpPages": [ 5, 8 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated velocity relation and the acceleration for that stage." }, { "id": "ial-2025-1-WME01-01-q02c", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the speed using the total travel time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=8", "qpPages": [ 5, 8 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the stage times and apply the appropriate suvat relation." }, { "id": "ial-2025-1-WME01-01-q02d", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Calculate the deceleration during the final stage.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=8", "qpPages": [ 5, 8 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the change in velocity over the known time, with the sign representing slowing down." }, { "id": "ial-2025-1-WME01-01-q03a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Calculate the impulse during the collision.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=10", "qpPages": [ 9, 12 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use impulse as change in momentum, with a consistent positive direction." }, { "id": "ial-2025-1-WME01-01-q03b", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Determine the friction coefficient from the motion after impact.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=10", "qpPages": [ 9, 12 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes this loss-of-speed friction part as substantially more demanding than the preceding work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the post-impact speed from momentum, then use work-energy or suvat over the rough distance to solve for friction." }, { "id": "ial-2025-1-WME01-01-q03c", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed after the particles collide.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=10", "qpPages": [ 9, 12 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of linear momentum and the stated coefficient of restitution." }, { "id": "ial-2025-1-WME01-01-q04a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the rod’s weight from its equilibrium conditions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=13", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=12", "qpPages": [ 13, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about a convenient point and use the force balance to determine the weight." }, { "id": "ial-2025-1-WME01-01-q04b", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine where an added particle must be placed for equilibrium.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=13", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=12", "qpPages": [ 13, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write a new moments equation including the particle’s weight and solve for its distance." }, { "id": "ial-2025-1-WME01-01-q04c", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the condition at which the rod is about to tip.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=13", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=12", "qpPages": [ 13, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this final tipping part was less successful than the first two parts, with many candidates misunderstanding the limiting condition.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the contact reaction at the limiting point to zero and solve the resulting moments or force equation." }, { "id": "ial-2025-1-WME01-01-q05a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the velocity vector of the moving particle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=17", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=14", "qpPages": [ 17, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate its position vector with respect to time." }, { "id": "ial-2025-1-WME01-01-q05b", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Calculate the distance from the particle to the target point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=17", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=14", "qpPages": [ 17, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes this distance calculation as more challenging than the preceding vector work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the displacement vector from the particle to the target and find its magnitude." }, { "id": "ial-2025-1-WME01-01-q06a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the frictional force on the particle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=19", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=15", "qpPages": [ 19, 22 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal reaction and coefficient of friction, checking whether the friction is limiting." }, { "id": "ial-2025-1-WME01-01-q06b", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the angle satisfies the stated tangent relation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=19", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=15", "qpPages": [ 19, 22 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces for both connected bodies and eliminate the tension and acceleration using the string constraint." }, { "id": "ial-2025-1-WME01-01-q06c", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Decide whether the system remains at rest.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=19", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=15", "qpPages": [ 19, 22 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this as a good discriminator and notes it was sometimes omitted or answered without the required friction comparison.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the required friction with the limiting friction and state the resulting motion condition." }, { "id": "ial-2025-1-WME01-01-q07a", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the tension and acceleration in the connected-particle system.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=23", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=17", "qpPages": [ 23, 25 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to each mass with a common acceleration and solve the simultaneous equations." }, { "id": "ial-2025-1-WME01-01-q07b", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the speed when one particle reaches the floor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=23", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=17", "qpPages": [ 23, 25 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the acceleration and displacement in a suvat equation." }, { "id": "ial-2025-1-WME01-01-q07c", "paperId": "ial-2025-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total time for both stages of the motion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_QP.pdf#page=23", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M1_MS.pdf#page=17", "qpPages": [ 23, 25 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this two-stage motion part as challenging for many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the first stage time from the initial conditions, then model the second stage with the changed acceleration and combine times. " }, { "id": "ial-2025-1-WME02-01-q01", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Calculate the work done moving the object up a rough inclined plane.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the gain in gravitational potential energy and the work against friction, resolving the normal reaction from the incline geometry." }, { "id": "ial-2025-1-WME02-01-q02a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the towbar tension from the van-and-trailer equations and power.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write equations of motion for the combined system and trailer, use power equals force times speed to find the driving force, then solve for tension." }, { "id": "ial-2025-1-WME02-01-q02b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the steady speed of the vehicle on the incline.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set acceleration to zero, resolve the weight component and driving force along the slope, then use the power relation." }, { "id": "ial-2025-1-WME02-01-q03a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find when velocity is parallel to the stated direction and calculate the speed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the position vector, equate the relevant component to zero for parallel motion, and find the velocity magnitude." }, { "id": "ial-2025-1-WME02-01-q03b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration magnitude at the stated time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the velocity vector and calculate the magnitude using Pythagoras." }, { "id": "ial-2025-1-WME02-01-q03c", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Show that the particle never returns to the origin.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set each position component to zero and show the resulting conditions cannot hold simultaneously for positive time." }, { "id": "ial-2025-1-WME02-01-q04a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the initial velocity components of the projectile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the initial velocity horizontally and vertically using the launch angle." }, { "id": "ial-2025-1-WME02-01-q04b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Calculate the horizontal distance travelled before impact.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the flight time from vertical motion, then multiply by the constant horizontal velocity." }, { "id": "ial-2025-1-WME02-01-q05a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the parameter from the impulse vector during impact.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply impulse-momentum in vector form and equate the squared magnitudes to solve for the parameter." }, { "id": "ial-2025-1-WME02-01-q05b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "State the impulse vector using the parameter value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the parameter into the vector expression." }, { "id": "ial-2025-1-WME02-01-q05c", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the angle of the impulse vector.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the ratio of its vertical and horizontal components in an inverse tangent." }, { "id": "ial-2025-1-WME02-01-q06a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of the composite lamina.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use area-weighted moments about the stated axis, subtracting the cut-out region as needed." }, { "id": "ial-2025-1-WME02-01-q06b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass after adding plates and determine the equilibrium angle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the component moments to obtain the centroid, then use the geometry of the support to calculate the angle." }, { "id": "ial-2025-1-WME02-01-q07a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension in the rod or support at equilibrium.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the hinge and resolve the forces using the given geometry." }, { "id": "ial-2025-1-WME02-01-q07b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the direction of the hinge force.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve horizontal and vertical forces, then use their ratio to calculate the force angle." }, { "id": "ial-2025-1-WME02-01-q08a", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use the impact law to relate the speeds before and after collision.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the coefficient of restitution along the line of motion, keeping the direction signs consistent." }, { "id": "ial-2025-1-WME02-01-q08b", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use conservation of momentum to determine the mass ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the momentum equation before and after impact and substitute the speed relation." }, { "id": "ial-2025-1-WME02-01-q08c", "paperId": "ial-2025-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Determine the restitution parameter from the repeated collision and wall motion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M2_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this as the most challenging task on the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the times or distances for both particles between collisions, include the rebound speed from the wall, and solve for the parameter. " }, { "id": "ial-2025-1-WME03-01-q01a", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration at the specified instant from the velocity function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the piecewise velocity expression and substitute the stated time." }, { "id": "ial-2025-1-WME03-01-q01b", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the displacement or position at the specified instant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each velocity segment, apply continuity and any initial position condition, then evaluate at the target time." }, { "id": "ial-2025-1-WME03-01-q02", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 10, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the equilibrium configuration of two masses connected by an elastic string.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=10", "qpPages": [ 4, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that this equilibrium problem gave an unexpected challenge across grades, with many candidates losing marks by omitting string extension or assuming uniform tension.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the forces on each mass, apply Hooke’s law to the extensions, and use the geometric constraint to solve for the ceiling separation." }, { "id": "ial-2025-1-WME03-01-q03", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed of a ring on an elastic string using energy.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the initial and final elastic potential energies and gravitational energy change, then apply conservation of mechanical energy." }, { "id": "ial-2025-1-WME03-01-q04a", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the vertical coordinate of the centre of mass of a composite lamina.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate each component area and centroid distance, then divide the net first moment by total area." }, { "id": "ial-2025-1-WME03-01-q04b", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the force or reaction needed to hold the lamina in equilibrium about a pivot.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says the geometry in this part brought a surprising challenge; many candidates stopped after identifying the horizontal coordinate.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the pivot and resolve forces using the component weights and their distances." }, { "id": "ial-2025-1-WME03-01-q05a", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Relate the speed and height of a particle moving in a vertical circle with a peg.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy between the stated positions and include the change in gravitational potential energy." }, { "id": "ial-2025-1-WME03-01-q05b", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the string tension at the specified point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says forming the circular equation of motion in this part distinguished weaker candidates from the majority.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the radial equation of motion using the speed from part (a)." }, { "id": "ial-2025-1-WME03-01-q05c", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the height at which the string becomes slack.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the tension to zero in the radial equation, then use energy to obtain the corresponding height." }, { "id": "ial-2025-1-WME03-01-q06a", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 8, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the stated length relation for a particle moving in a horizontal circle on an elastic string.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally for centripetal acceleration, use the geometric triangle and Hooke’s law, and eliminate the tension and angle." }, { "id": "ial-2025-1-WME03-01-q06b", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the reaction from the horizontal surface.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve vertically using the weight and string component." }, { "id": "ial-2025-1-WME03-01-q06c", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the sum of kinetic and elastic potential energies.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the speed from circular motion and the string extension in the kinetic and elastic energy formulae." }, { "id": "ial-2025-1-WME03-01-q07a", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show the particle performs simple harmonic motion and find its period.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=20", "qpPages": [ 24, 28 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the equation of motion at a general displacement, use Hooke’s law to obtain acceleration proportional to negative displacement, then identify angular frequency." }, { "id": "ial-2025-1-WME03-01-q07b", "paperId": "ial-2025-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the time when the oscillating particle reaches the stated position and direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_M3_MS.pdf#page=20", "qpPages": [ 24, 28 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies the final part as a notable challenge, with weaker candidates struggling to complete the oscillation-time calculation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the appropriate displacement or acceleration relation with the amplitude and period; select the time consistent with the direction of travel. " }, { "id": "ial-2025-1-WST01-01-q01a", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Name the probability distribution for the die outcome.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the discrete distribution and its parameter values from the equal outcomes." }, { "id": "ial-2025-1-WST01-01-q01b", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate the probability of the stated event for the red die.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count the equally likely outcomes that satisfy the event and divide by the total." }, { "id": "ial-2025-1-WST01-01-q01ci", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(i)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the expected value of the red-die variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probability-weighted sum of the possible outcomes." }, { "id": "ial-2025-1-WST01-01-q01cii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(ii)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the expected value of the blue-die variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probability-weighted sum for the blue outcomes." }, { "id": "ial-2025-1-WST01-01-q01d", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate the variance of the specified die-based quantity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find its first and second moments from the outcome table, then subtract the square of the mean." }, { "id": "ial-2025-1-WST01-01-q01e", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability of the stated combined die event.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the favourable ordered pairs and divide by the total number of equally likely pairs." }, { "id": "ial-2025-1-WST01-01-q01f", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(f)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Show the stated relationship between the two random variables.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the definitions of the variables and simplify the algebraic relation." }, { "id": "ial-2025-1-WST01-01-q01g", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(g)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the cumulative distribution function or parameter condition for the dice expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls this final part a good discriminator and reports confusion over the outcome combinations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Enumerate the ordered outcomes, group them by the value of the expression, and accumulate the probabilities. " }, { "id": "ial-2025-1-WST01-01-q02a", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the range of the sample.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the smallest observation from the largest." }, { "id": "ial-2025-1-WST01-01-q02b", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the median from the ordered stem-and-leaf data.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the central observation in the ordered sample." }, { "id": "ial-2025-1-WST01-01-q02c", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the interquartile range using the stem-and-leaf key.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the quartile observations from the ordered data and subtract the lower quartile from the upper." }, { "id": "ial-2025-1-WST01-01-q02d", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the sample standard deviation from the supplied data.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the summary totals in the standard deviation formula, applying the required sample convention." }, { "id": "ial-2025-1-WST01-01-q02ei", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the identity relating the sum of squared deviations to the mean and raw totals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand the squared deviations, sum the terms and use the definition of the mean." }, { "id": "ial-2025-1-WST01-01-q02eii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Combine the samples to find the overall summary statistics.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says fully correct solutions to this combined-statistics part were rarely seen.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the combined count, total and sum of squares; recompute the overall mean and standard deviation. " }, { "id": "ial-2025-1-WST01-01-q03a", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Draw a probability tree for the rug and blanket production choices.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the successive material choices with the correct branch probabilities." }, { "id": "ial-2025-1-WST01-01-q03bi", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a selected blanket is made without wool.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the relevant mutually exclusive paths on the tree." }, { "id": "ial-2025-1-WST01-01-q03bii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability for the stated blanket event.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this conditional probability part as more discriminating than the preceding tree work.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide the joint probability of the requested event by the probability of the conditioning event. " }, { "id": "ial-2025-1-WST01-01-q04a", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the Venn diagram for the cat-food flavour survey.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the intersection and total counts to populate each region and the outside area." }, { "id": "ial-2025-1-WST01-01-q04b", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Test whether the two flavour choices are independent.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the intersection probability with the product of the marginal probabilities." }, { "id": "ial-2025-1-WST01-01-q04ci", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a sampled cat prefers only the specified flavour.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the matching single-set region from the Venn diagram and divide by the total." }, { "id": "ial-2025-1-WST01-01-q04cii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability for the second flavour event.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Restrict the sample space to the stated condition and divide the relevant region count by that subtotal." }, { "id": "ial-2025-1-WST01-01-q05ai", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a Normal probability for the daily glass weight.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the weight using the given mean and standard deviation, then read the appropriate tail." }, { "id": "ial-2025-1-WST01-01-q05aii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the nearest-50-kg rounding condition to find a probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson notes that candidates rarely understood the nearest-50 kg interval and many used incorrect bounds or left it blank.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the rounding condition into the correct half-width interval around the target, standardize both bounds and subtract the cumulative probabilities. " }, { "id": "ial-2025-1-WST01-01-q05bi", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Form equations for the unknown Normal mean and standard deviation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate each stated probability into a standard-normal quantile equation." }, { "id": "ial-2025-1-WST01-01-q05bii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Solve for the Normal distribution parameters.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the two equations simultaneously and report the requested mean and standard deviation." }, { "id": "ial-2025-1-WST01-01-q06a", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation coefficient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied summary statistics to calculate Sxx, Sww and Sxw, then evaluate r." }, { "id": "ial-2025-1-WST01-01-q06b", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe the pattern shown by the scatter diagram.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes this interpretation as more challenging; full marks were rare and many responses confused correlation with how the plotted points look.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Describe direction, strength and any notable features in the plotted relationship." }, { "id": "ial-2025-1-WST01-01-q06c", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the regression line of the response variable on the explanatory variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the regression gradient from the summary statistics and use the sample means for the intercept." }, { "id": "ial-2025-1-WST01-01-q06d", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the gradient of the regression line in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain the predicted change in the response for a one-unit increase in the explanatory variable." }, { "id": "ial-2025-1-WST01-01-q06ei", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the consequence of the stated data correction for one summary measure.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the affected observation’s position and the definition of the measure." }, { "id": "ial-2025-1-WST01-01-q06eii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the consequence of the correction for the correlation or regression result.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine whether the correction changes the required summary quantity, using how the data enter its formula." }, { "id": "ial-2025-1-WST01-01-q06eiii", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)(iii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the final consequence of correcting the observation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the correction to the remaining requested statistic or interpretation." }, { "id": "ial-2025-1-WST01-01-q07a", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate a quartile from the cumulative frequency graph.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says many candidates struggled to interpret the quartiles and set up interpolation from the box plot.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the required cumulative frequency and interpolate between the adjacent plotted values." }, { "id": "ial-2025-1-WST01-01-q07b", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate the upper outlier fence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Q3 plus 1.5 times the interquartile range." }, { "id": "ial-2025-1-WST01-01-q07c", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Check whether the distribution is approximately Normal.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the shape or symmetry of the data with the expected Normal pattern." }, { "id": "ial-2025-1-WST01-01-q07d", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the probability of an observation above the stated standard-deviation threshold.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the threshold to a standard Normal z-score and find the upper-tail area." }, { "id": "ial-2025-1-WST01-01-q07e", "paperId": "ial-2025-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the largest possible standard deviation under the given condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S1_MS.pdf#page=14", "qpPages": [ 22, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Normal probability condition to determine the limiting z-value and solve for the standard deviation." }, { "id": "ial-2025-1-WST02-01-q01a", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Calculate an exact binomial probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the trial count and success probability into the binomial probability formula." }, { "id": "ial-2025-1-WST02-01-q01b", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a Poisson approximation for the stated event.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter to np and evaluate the required probability." }, { "id": "ial-2025-1-WST02-01-q01c", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the percentage error in the approximation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the exact and approximate probabilities, divide their difference by the exact value, and convert to a percentage." }, { "id": "ial-2025-1-WST02-01-q01d", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "State the condition that justifies the approximation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Describe the small success probability and large trial count requirement." }, { "id": "ial-2025-1-WST02-01-q01e", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Determine the Poisson parameter from two probability equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the two equations in the parameter, simplify them and solve algebraically, checking admissible values." }, { "id": "ial-2025-1-WST02-01-q02ia", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)(a)", "marks": 1, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Decide whether the stated sample quantity is a statistic and justify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check whether it is calculated solely from observations or depends on an unknown population parameter." }, { "id": "ial-2025-1-WST02-01-q02ib", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)(b)", "marks": 1, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Classify the second stated quantity as a statistic or parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the same distinction between sample-derived values and unknown population properties." }, { "id": "ial-2025-1-WST02-01-q02ii", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Find the probability of the stated ordered sample under independence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the possible outcomes in the specified order and multiply their probabilities." }, { "id": "ial-2025-1-WST02-01-q02iiia", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(iii)(a)", "marks": 1, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "List the possible unordered pairs in the sample.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the distinct combinations without replacement." }, { "id": "ial-2025-1-WST02-01-q02iiib", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(iii)(b)", "marks": 2, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Calculate the possible values of the test statistic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the statistic for each distinct pair." }, { "id": "ial-2025-1-WST02-01-q02iiic", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(iii)(c)", "marks": 3, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Construct the sampling distribution of the statistic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign each statistic value its probability from the pair outcomes and combine repeated values." }, { "id": "ial-2025-1-WST02-01-q03a", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Write the probability density function for the uniform variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the reciprocal of the interval width on the support and zero outside." }, { "id": "ial-2025-1-WST02-01-q03b", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Sketch the density function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the constant density over the support with zero elsewhere and label its height and endpoints." }, { "id": "ial-2025-1-WST02-01-q03c", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the mean of the uniform distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take the midpoint of the support interval." }, { "id": "ial-2025-1-WST02-01-q03d", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Derive the cumulative distribution function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density over the support and state the piecewise values below, within and above the interval." }, { "id": "ial-2025-1-WST02-01-q03e", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Calculate the expectation of the transformed random variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the transformed value against the density or use the derived distribution function." }, { "id": "ial-2025-1-WST02-01-q03f", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the final probability involving the uniform variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson explicitly calls the final part the most difficult item on the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Set up and evaluate the correct integral or interval area, using the bounds from the given transformation. " }, { "id": "ial-2025-1-WST02-01-q04ia", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(a)", "marks": 3, "topicIds": [ "distributions", "integration", "probability" ], "primaryTopicId": "distributions", "summary": "Evaluate the integral needed for the continuous distribution’s second moment.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x-squared times the density over its support and apply the limits." }, { "id": "ial-2025-1-WST02-01-q04ib", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(b)", "marks": 2, "topicIds": [ "distributions", "integration", "probability" ], "primaryTopicId": "distributions", "summary": "Calculate the variance of the continuous random variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Var(X)=E(X²)−[E(X)]² with the expectation from the earlier part." }, { "id": "ial-2025-1-WST02-01-q04iia", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)(a)", "marks": 3, "topicIds": [ "distributions", "integration", "probability" ], "primaryTopicId": "distributions", "summary": "Model the count as binomial and calculate the first quartile-related probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes the second half of Question 4 as substantially more challenging for many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the relevant success probability from the quartile condition and evaluate the requested binomial tail." }, { "id": "ial-2025-1-WST02-01-q04iib", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)(b)", "marks": 2, "topicIds": [ "distributions", "integration", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of the specified count relative to the other quartile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes the second half of Question 4 as substantially more challenging for many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the complementary event or the corresponding binomial cumulative probability. " }, { "id": "ial-2025-1-WST02-01-q05ai", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Model the meteor sightings with the appropriate count distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated rate and time interval to set the Poisson parameter and evaluate the first event probability." }, { "id": "ial-2025-1-WST02-01-q05aii", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Calculate the second requested meteor-count probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the relevant count into the Poisson probability formula." }, { "id": "ial-2025-1-WST02-01-q05b", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State the null and alternative hypotheses for the rate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the hypotheses as a parameter equality and the directional or two-sided alternative required." }, { "id": "ial-2025-1-WST02-01-q05c", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the critical region for the count test.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the null distribution and significance level to identify the smallest tail region whose probability does not exceed the level." }, { "id": "ial-2025-1-WST02-01-q05d", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Make a conclusion from the observed count.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the observation with the critical region and state the contextual decision." }, { "id": "ial-2025-1-WST02-01-q06a", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the variance of the binomial count.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use np(1−p)." }, { "id": "ial-2025-1-WST02-01-q06b", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a Normal approximation to determine the unknown binomial probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the continuity correction, standardize the boundary, use the stated tail probability and solve the resulting equation for p." }, { "id": "ial-2025-1-WST02-01-q06c", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Infer the requested population or sample probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the parameter found in part (b) in the relevant binomial or Normal expression." }, { "id": "ial-2025-1-WST02-01-q07a", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Sketch the given probability density function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=16", "qpPages": [ 22, 24 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the supported curve with its zero regions and mark the relevant boundary points." }, { "id": "ial-2025-1-WST02-01-q07b", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "State the mode of the density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=16", "qpPages": [ 22, 24 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the x-value where the density reaches its maximum." }, { "id": "ial-2025-1-WST02-01-q07ci", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)(i)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Determine the normalizing constant in the density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=16", "qpPages": [ 22, 24 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the total area under the density equal to one and integrate over each piece." }, { "id": "ial-2025-1-WST02-01-q07cii", "paperId": "ial-2025-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)(ii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Verify that the stated value is a quartile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S2_MS.pdf#page=16", "qpPages": [ 22, 24 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the cumulative probability equal to the quartile fraction, integrate the appropriate part of the density and check the boundary." }, { "id": "ial-2025-1-WST03-01-q01a", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Calculate Spearman’s rank correlation coefficient from the paired rankings.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the missing ranks, calculate the rank differences and substitute their squared sum into the coefficient formula." }, { "id": "ial-2025-1-WST03-01-q01b", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test for positive rank correlation at the stated significance level.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State directional hypotheses, use the critical value or p-value for the sample size and conclude in context." }, { "id": "ial-2025-1-WST03-01-q02a", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the mean number of customers from the frequency table.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each value by its frequency, sum the products and divide by the total count." }, { "id": "ial-2025-1-WST03-01-q02b", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Explain the calculated expected frequency for one class.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the estimated Poisson mean into the class probability and multiply by the sample size." }, { "id": "ial-2025-1-WST03-01-q02c", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Explain why the upper classes are combined.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the expected frequencies must be large enough for the chi-squared approximation." }, { "id": "ial-2025-1-WST03-01-q02d", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Carry out a goodness-of-fit test for the Poisson model.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses, calculate the chi-squared statistic, determine degrees of freedom and compare with the critical value." }, { "id": "ial-2025-1-WST03-01-q03a", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the estimated means or variances for the two regions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sample totals and sample sizes to calculate the required unbiased estimates." }, { "id": "ial-2025-1-WST03-01-q03b", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 7, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the mean weight in region A is lower than in region B.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-sided hypotheses, form the standard error for the difference, calculate the test statistic and compare with the critical value." }, { "id": "ial-2025-1-WST03-01-q03c", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Explain why a Normal approximation is appropriate for the test.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this explanation part challenged candidates who did not recognize the large-sample basis.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Refer to the large sample sizes and invoke the central limit theorem. " }, { "id": "ial-2025-1-WST03-01-q04a", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Show the standard error from the stated confidence interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the interval width and the relevant Normal critical value to calculate the standard error." }, { "id": "ial-2025-1-WST03-01-q04b", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find a 95% confidence interval for the population mean.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the same sample mean and standard error with the 95% Normal critical value, then round both endpoints." }, { "id": "ial-2025-1-WST03-01-q04c", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Determine the confidence level from the specified interval width.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson notes that only stronger candidates typically completed this inverse confidence-interval calculation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert the half-width to a Normal z-value using the known standard error, then convert the central probability to a percentage. " }, { "id": "ial-2025-1-WST03-01-q05ai", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 1, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Give an advantage of quota sampling in the survey context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that the advantage of quota sampling caused more difficulty than the disadvantage.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "State that it ensures the relevant subgroups are included or avoids the need for a complete sampling frame. " }, { "id": "ial-2025-1-WST03-01-q05aii", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 1, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Give a disadvantage of quota sampling.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that selection within each quota is non-random and can introduce bias." }, { "id": "ial-2025-1-WST03-01-q05bi", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 1, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the expected frequency for the first location and subject group.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the relevant row and column totals and divide by the grand total." }, { "id": "ial-2025-1-WST03-01-q05bii", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 2, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the expected frequency for the second location and subject group.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the expected-frequency formula to the second cell." }, { "id": "ial-2025-1-WST03-01-q05c", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 7, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Test whether subject preference is independent of location.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses, compute all chi-squared contributions, identify degrees of freedom, compare with the critical value and conclude in context." }, { "id": "ial-2025-1-WST03-01-q06a", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Show that the stated estimator is unbiased.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=11", "qpPages": [ 22, 25 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate its expectation using linearity and compare it with the population parameter." }, { "id": "ial-2025-1-WST03-01-q06b", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Find the bias of the estimator.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=11", "qpPages": [ 22, 25 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the target parameter from its expectation." }, { "id": "ial-2025-1-WST03-01-q06c", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Determine constants that make the new estimator unbiased.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=11", "qpPages": [ 22, 25 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set its expectation equal to the parameter and solve the resulting coefficient equations." }, { "id": "ial-2025-1-WST03-01-q06d", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Find the maximum-likelihood estimate of the parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=11", "qpPages": [ 22, 25 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the likelihood or log-likelihood for the observed sample, differentiate and choose the admissible maximizing value." }, { "id": "ial-2025-1-WST03-01-q07a", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find a probability for the sum of independent Normal variables.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=12", "qpPages": [ 26, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine their means and variances, standardize the specified sum and evaluate the tail probability." }, { "id": "ial-2025-1-WST03-01-q07b", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find a probability involving the difference between two Normal sample means.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=12", "qpPages": [ 26, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the difference distribution using the independent means and variances, then standardize the threshold." }, { "id": "ial-2025-1-WST03-01-q07c", "paperId": "ial-2025-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Determine the sample size from the stated probability condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202025/IAL_MATHS_2025_Jan_S3_MS.pdf#page=12", "qpPages": [ 26, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies this part as the strongest discriminator, with only the most able candidates earning full marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20250306.pdf", "tagConfidence": "reviewed", "answerGuide": "Express the test statistic using the appropriate Normal mean and variance, set it equal to the specified tail condition and solve for n. " }, { "id": "ial-2025-10-WMA12-01A-q01a", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Write the second and third terms of the recurrence in terms of k.", "answerGuide": "Substitute u₁ = 1 into the recurrence to obtain u₂, then substitute that expression to obtain u₃.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q01b", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the possible values of k given that u₃ = 6.", "answerGuide": "Set the expression for u₃ equal to 6, solve the resulting quadratic, and give both roots.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q02", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first three terms in ascending powers of x for the stated binomial expression.", "answerGuide": "Use the first three binomial coefficients, collect powers of x, and simplify each term.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q03a", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use the trapezium rule and all four tabulated ordinates to estimate the stated definite integral to two decimal places.", "answerGuide": "Use equal width 3 and the trapezium rule with the first, interior and final ordinates in their correct positions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q03b", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use the estimate from part (a) to estimate the related definite integral.", "answerGuide": "Rewrite the integrand as 1 plus half the integrand from part (a), then integrate each term over the interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q04ai", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 1, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Show that k = 5 is a possible value when x + 3 is a factor of f(x).", "answerGuide": "Substitute x = −3 into f(x), then demonstrate that k = 5 makes f(−3) equal to zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q04aii", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Find the other possible value of k when x + 3 is a factor of f(x).", "answerGuide": "Set f(−3) = 0, solve the resulting quadratic in k, and state the root other than 5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q04bi", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "When k = 5, write f(x) as a product of two algebraic factors.", "answerGuide": "Expand the cubic, divide by the known factor x + 3, and state the resulting factorisation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q04bii", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Show that f(x) = 0 has only one real solution when k = 5.", "answerGuide": "Use the discriminant of the quadratic factor to show it has no real roots; the linear factor supplies the sole real root.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q05i", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Express x in terms of a using the given logarithmic equation.", "answerGuide": "Combine the logarithms, remove the logarithms using base a, and rearrange to the requested expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=16", "qpPages": [ 5, 5 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q05ii", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the quadratic equation in log₅ y.", "answerGuide": "Factor the quadratic in log₅ y, then convert both logarithmic values back to y.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=16", "qpPages": [ 5, 5 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q06a", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of the centre of the circle.", "answerGuide": "Complete the square in x and y and read the centre from the standard circle form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q06b", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the radius when k = −32.", "answerGuide": "Complete the square, substitute k = −32, and take the positive square root of the radius squared.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q06c", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the range of k for which the circle lies completely within the first quadrant.", "answerGuide": "Use the centre (3, 7): the radius must be less than 3 and positive. Convert those two conditions on √(58 − k) into inequalities for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q07i", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 7 sin 2θ = 5 cos 2θ for 0 ≤ θ < 180°, giving answers to one decimal place.", "answerGuide": "Use tan 2θ = 5/7, find every angle in the doubled interval, then halve the values and apply the stated range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q07ii", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 24 tan x = 5 cos x for 0 ≤ x < 2π, giving answers to three decimal places.", "answerGuide": "Replace tan x with sin x/cos x, use cos²x = 1 − sin²x to form a quadratic in sin x, then select all roots in range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q08a", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the exact area of the finite region bounded by y = (x + 2)²(4 − x) and the x-axis.", "answerGuide": "Expand the cubic, identify the intercepts that bound the finite region, and integrate between them.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=22", "qpPages": [ 6, 6 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q08b", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Use the area from part (a) to find the area bounded by the stated transformed curve and the x-axis.", "answerGuide": "Identify the horizontal and vertical scale changes from the original curve and apply their product to its area.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=22", "qpPages": [ 6, 6 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q09i", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the maximum speed in second gear when the first and fifth gear speeds are 22 and 130 km h⁻¹.", "answerGuide": "Use the geometric progression to find the common ratio from the first and fifth terms, then multiply the first speed by that ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=24", "qpPages": [ 7, 7 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q09iia", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(ii)(a)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the maximum value of Sₙ for the arithmetic sequence beginning 208, 207.2, …", "answerGuide": "Use the nth-term rule to locate the terms around zero, then evaluate the corresponding arithmetic-series sum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=24", "qpPages": [ 7, 7 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q09iib", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(ii)(b)", "marks": 1, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the smallest N for which Sₙ is negative.", "answerGuide": "Use the arithmetic-series sum and identify the first integer after the positive maximum for which the sum is below zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=24", "qpPages": [ 7, 7 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q10a", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Show the given perimeter expression for the stage design with area 30 m².", "answerGuide": "Form the area equation from the rectangle and two sectors, eliminate θ, then substitute into the perimeter expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=26", "qpPages": [ 8, 8 ], "msPages": [ 26, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q10b", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Use calculus to find the minimum perimeter in the required exact form.", "answerGuide": "Differentiate the perimeter function, solve dP/dr = 0 for the positive radius, and substitute it into P.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=26", "qpPages": [ 8, 8 ], "msPages": [ 26, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA12-01A-q10c", "paperId": "ial-2025-10-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Justify that the perimeter value from part (b) is a minimum.", "answerGuide": "Show the second derivative is positive at the stationary point or establish that the derivative changes from negative to positive.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2A_MS.pdf#page=26", "qpPages": [ 8, 8 ], "msPages": [ 26, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q01", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find a particle’s speed after a specified vector impulse changes its momentum.", "answerGuide": "Use impulse = change in momentum to find the new velocity vector, then calculate its magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q02a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the mass ratio that makes the centre of mass lie on the x-axis.", "answerGuide": "Set the total moment about the x-axis to zero and solve for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q02b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the x-coordinate of the centre of mass after using the value of k.", "answerGuide": "Take moments about the y-axis and divide by the total mass to obtain c.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q03a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the velocity vector when the particle moves parallel to the j direction.", "answerGuide": "Differentiate the position vector, set the i-component of velocity to zero, then substitute the resulting time into both components.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=10", "qpPages": [ 2, 2 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q03b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration magnitude at t = 4.", "answerGuide": "Differentiate the velocity vector and evaluate its magnitude at t = 4.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=10", "qpPages": [ 2, 2 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q04a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use the centre-of-mass distance of a two-rhombus lamina to show the stated exact value of cos θ.", "answerGuide": "Use the symmetry of the two identical rhombuses to locate their centres of mass, then form the weighted-distance relation for the combined lamina.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q04b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the attached particle’s weight as a fraction of the lamina’s weight when it hangs in equilibrium.", "answerGuide": "Take moments about the suspension point B, using the horizontal centre-of-mass distance and the attached weight at A.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q05a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension for a uniform rod in limiting equilibrium against a rough wall.", "answerGuide": "Take moments about the wall contact A, including the rod’s weight and the string’s direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q05b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction between the rod and the wall.", "answerGuide": "Resolve forces horizontally and vertically to obtain the normal reaction and friction, then use the limiting condition F = μR.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q05c", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the direction of the resultant force exerted by the wall on the rod.", "answerGuide": "Use the friction and normal-reaction components to calculate the resultant’s angle and state its direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q06a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the particle’s velocity as a function of time from its acceleration and one velocity condition.", "answerGuide": "Integrate the acceleration and substitute the given velocity at t = 3 to determine the constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q06b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the distance between the particle’s first two instantaneous rest points.", "answerGuide": "Solve v = 0 for the two times, integrate velocity between them, and take the positive distance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q07a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the vertical separation of two points on a projectile’s path from its speeds at those points.", "answerGuide": "Apply conservation of mechanical energy between A and B to determine the change in height.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q07b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the launch angle when the projectile takes 1.5 seconds to reach B.", "answerGuide": "Use the vertical displacement equation with the height from part (a), then solve for α.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q07c", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the angle below the horizontal of the projectile’s velocity at B.", "answerGuide": "Use the horizontal component to relate the launch and arrival angles, then use the vertical component or the vertical motion equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q07d", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total time for which the projectile’s speed is below 8 m s⁻¹.", "answerGuide": "Find the relevant times at the speed threshold and combine them with the time at maximum height to obtain the interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q08a", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 7, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the two particles’ speeds immediately after their direct collision.", "answerGuide": "Use conservation of momentum and Newton’s law of restitution to solve for the post-collision velocities, then report speeds.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WME02-01A-q08b", "paperId": "ial-2025-10-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the allowable interval of e from a reversal condition and a no-collision condition.", "answerGuide": "Use the post-collision velocities to form the strict inequality for A to reverse direction and the speed condition preventing B from catching C.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2A_MS.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2020-s-11-q01", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the first term and common difference of an arithmetic progression from the sum of its first nine terms and its next four terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form equations for S9 and the next four terms (or S13−S9), then solve simultaneously. The first term is 7 and the common difference is 1.5.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q02", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the positive k when the coefficient of 1/x in (kx + 1/x)^5 + (1 − 2/x)^8 is 74.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The 1/x coefficients are 10k² and −16. Set 10k²−16=74, giving k=3.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q03a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "exponentials", "sequences" ], "primaryTopicId": "exponentials", "summary": "Model the necklace price n years after 2000 and find its 2008 price when it rises 5% each year from $36,000.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use 36000(1.05)^n. For 2008, n=8, giving about $53,200.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q03b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Find the total revenue for one necklace sold each year from 2000 through 2009.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sum ten geometric terms with first term $36,000 and ratio 1.05. The total is about $453,000.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q04a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "State the range of f(x)=3/2 cos(2x)+1/2 for 0≤x≤π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Since −1≤cos(2x)≤1, the range is −1≤f(x)≤2.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q04b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "For g(x)=f(x)+k with k>0 and the x-axis tangent to y=g(x), find k and describe the transformation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The minimum of f is −1, so k=1. Translate y=f(x) vertically upward by 1 unit.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q04c", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Give the equation of the reflection of y=f(x) in the x-axis in the form y=a cos(2x)+b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Negate f(x): y=−3/2 cos(2x)−1/2.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q05a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "A line y=mx+c is tangent to xy=16. Express m in terms of c.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute y=mx+c to form mx²+cx−16=0. Set the discriminant c²+64m to zero; m=−c²/64.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q05b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "With m=−4, find c-values for which the line intersects xy=16 at two distinct points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitution gives −4x²+cx−16=0. Require c²−256>0, so c<−16 or c>16.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q06a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Given f(x)=x/2−a and g(x)=3x+b, use g(g(2))=10 and f⁻¹(2)=14 to find a and b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "g(g(2))=18+4b=10 gives b=−2. From f(14)=2, 7−a=2, so a=5.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q06b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Using the values of a and b, find g(f(x)) in the form cx+d.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute f(x)=x/2−5 into g(x)=3x−2. Then g(f(x))=3x/2−17.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q07a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove (1+sinθ)/cosθ + cosθ/(1+sinθ) is identically 2/cosθ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine over cosθ(1+sinθ). Use sin²θ+cos²θ=1 to reduce the numerator to 2(1+sinθ), then cancel.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q07b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Using part (a), solve the equation with right side 3/sinθ for 0≤θ≤2π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reduce to 2/cosθ=3/sinθ, so tanθ=3/2. The solutions are approximately 0.983 and 4.12 radians.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q08", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of shaded region BXC in a radius-6 cm semicircle when arc AB is 15 cm and BX⊥AC.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "AOB=15/6=2.5 rad and BOC=π−2.5. Find arc BC=6(π−2.5), BX=6sin(π−2.5), and XC=6−6cos(π−2.5). Their sum is 8.63 cm.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q09a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "For y=(3−2x)^3+24x, find dy/dx and d²y/dx².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "dy/dx=−6(3−2x)²+24 and d²y/dx²=24(3−2x).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q09b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates of the stationary points of y=(3−2x)^3+24x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set dy/dx=0 to get x=1/2 or 5/2. Substitution gives (1/2,20) and (5/2,52).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q09c", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Determine the nature of both stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At x=1/2 the second derivative is 48, so it is a minimum. At x=5/2 it is −48, so it is a maximum.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q10a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle with diameter endpoints A(−1,−2) and B(7,4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The midpoint is (3,1) and radius is 5. The equation is (x−3)²+(y−1)²=25.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q10b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent equation at B to the circle with diameter AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The radius from (3,1) to (7,4) has slope 3/4, so the tangent slope is −4/3. Through B: 4x+3y=40.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q10c", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Reflect circle C in its tangent T at B and find the reflected circle equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The centre (3,1) reflects across T to (11,7); the radius stays 5. The equation is (x−11)²+(y−7)²=25.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q11a", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find intersections A and B of y=8/(x+2) with 2y+x=8, and find C where the curve tangent is parallel to AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The intersections are A=(0,4), B=(6,1). Since AB has slope −1/2, solve dy/dx=−8/(x+2)²=−1/2 to obtain C=(2,2).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-11-q11b", "paperId": "cie-2020-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the region between y=8/(x+2) and 2y+x=8 about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_11.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_11.pdf#page=13", "qpPages": [ 17, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use washers from x=0 to 6: π∫[(4−x/2)²−(8/(x+2))²]dx=18π cubic units.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q01a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the coefficient of x² in (x−2/x)^6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose two factors of −2/x: C(6,2)(−2)²=60.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q01b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the coefficient of x² in (2+3x²)(x−2/x)^6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the x² coefficient and constant term of the binomial: 2(60)+3(−160)=−360.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q02a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Express 3cosθ=8tanθ as a quadratic equation in sinθ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply through by cosθ and use cos²θ=1−sin²θ. This gives 3sin²θ+8sinθ−3=0.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q02b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Hence find the acute angle satisfying 3cosθ=8tanθ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor as (3sinθ−1)(sinθ+3)=0. The acute solution has sinθ=1/3, so θ≈19.5°.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q03a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the radius of a spherical balloon after 30 seconds if its volume grows at 600 cm³/s from empty.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At 30 seconds V=18,000 cm³. Solve 4πr³/3=18,000 to get r≈16.3 cm.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q03b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the balloon radius growth rate after 30 seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate V=4πr³/3: dV/dt=4πr²dr/dt. With dV/dt=600 and r≈16.3, dr/dt≈0.181 cm/s.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q04", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "The nth term of an arithmetic progression is (3n−15)/2. Find n when the sum of the first n terms is 84.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The first term is −6 and common difference 1.5. Substitute into Sₙ=n/2[2a+(n−1)d]=84 and solve; n=16.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q05a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "For f(x)=a−2x, express f(f(x)) and f⁻¹(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compose to get f(f(x))=4x−a. Solving y=a−2x for x gives f⁻¹(x)=(a−x)/2.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q05b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Given f(f(x))=f⁻¹(x), find x in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set 4x−a=(a−x)/2 and solve: x=a/3.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q06a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "For y=2x²+kx+k−1, find k if y=2x+3 is tangent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equating gives 2x²+(k−2)x+(k−4)=0. Tangency requires zero discriminant: (k−2)²=8(k−4), so k=6.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q06b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "When k=2, write the curve in vertex form and state its vertex.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square: y=2(x+1/2)²+1/2. The vertex is (−1/2,1/2).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q07a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Show the exact length BC in terms of r for the sector with radii 2r and r and angle π/6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Cosine rule gives BC²=r²+(2r)²−4r²cos(π/6)=r²(5−2√3), so BC=r√(5−2√3).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q07b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact perimeter of the shaded sector region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add arc AB=πr/3, AC=r, and BC=r√(5−2√3).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q07c", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact area of the shaded sector region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract triangle OBC area r²/2 from sector area πr²/3. The result is r²(π/3−1/2).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q08", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume when the region bounded by y=6/x, y=2 and x=1 is rotated about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For 2≤y≤6, washer radii are 6/y and 1. Thus V=π∫[(6/y)²−1]dy=8π.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q08b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "For y=6/x, show the tangent parallel to y+2x=0 occurs at a point on y=2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set dy/dx=−6/x² to −2, so x²=3. Since y=6/x, y=2x at that point.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q09a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "State the ranges of f(x)=2−3sin(2x) and g(x)=−2f(x) on [0,π].", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "As sin(2x) ranges from −1 to 1, f ranges from −1 to 5 and g from −10 to 2.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q09b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch y=g(x) on the supplied axes when g(x)=−2f(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch y=−2f(x): reflect across the x-axis and stretch vertically by factor 2; its range is −10 to 2.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q09c", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Describe transformations mapping y=f(x) to h(x)=g(x+π).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect in the x-axis, stretch vertically by factor 2, then translate left by π.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q10a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "For y=54x−(2x−7)³, find dy/dx and d²y/dx².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "dy/dx=54−6(2x−7)² and d²y/dx²=−24(2x−7).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q10b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point coordinates of y=54x−(2x−7)³.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Setting dy/dx=0 gives x=2 or 5; the points are (2,135) and (5,243).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q10c", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Determine the nature of each stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The second derivative is positive at x=2, so (2,135) is a minimum; it is negative at x=5, so (5,243) is a maximum.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q11a", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the centre C and radius of x²+y²−8x+4y−5=0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete squares: (x−4)²+(y+2)²=25. Centre (4,−2), radius 5.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q11b", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Show the tangent to the circle at P(1,2) is 4y=3x+5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "CP has slope −4/3, so the perpendicular tangent slope is 3/4. Through P the equation is 4y=3x+5.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q11c", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find Q on the circle when PQ is horizontal and P=(1,2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=14", "qpPages": [ 15, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect P across the vertical line through centre x=4: Q=(7,2).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-12-q11d", "paperId": "cie-2020-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(d)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection T of the tangents at P and Q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_12.pdf#page=15", "qpPages": [ 15, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The tangent at Q has equation 4y+3x=29. Solve with 4y=3x+5 to get T=(4,17/4).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q01", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find m-values for which y=mx+1 meets y=3x²+2x+4 at two distinct points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate and require a positive discriminant. This gives (m+4)(m−8)>0, hence m<−4 or m>8.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q02", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Find the curve equation when dy/dx=3√x−3/√x and it passes through (4,7).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate to y=2x^(3/2)−6√x+c. Using (4,7) gives c=3, so y=2x^(3/2)−6√x+3.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q03a", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the transformed graph equation for the first broken-line graph in terms of f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflection of y=f(x) in the y-axis: y=f(−x).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q03b", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the transformed graph equation for the second broken-line graph in terms of f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Vertical stretch by factor 2: y=2f(x).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q03c", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the transformed graph equation for the third broken-line graph in terms of f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect in the x-axis and translate 4 units right: y=−f(x−4).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q04a", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand (1+a)^5 through the term in a³.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial coefficients through power a³: 1+5a+10a²+10a³.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q04b", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Hence expand [1+(x+x²)]^5 through the term in x³.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute a=x+x² and retain powers through x³: 1+5x+15x²+30x³.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q05", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the cord length around a radius-5 cm pulley and a pin 13 cm below its centre, using the two tangents and major arc.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Each tangent is √(13²−5²)=12 cm. The tangent angle gives minor central angle about 2.352 rad; use the remaining major arc on radius 5 and add both tangents. Total length 43.7 cm.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q06a", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "For y=√(5x−1) with dx/dt=2, find dy/dt when x=1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "dy/dt=(dy/dx)(dx/dt), with dy/dx=5/[2√(5x−1)]. At x=1 the rate is 5/2 units per minute.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q06b", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "For y=√(5x−1) with dx/dt=2, find x when dy/dt=5/8 units per minute.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use dy/dt=5/√(5x−1). Setting this to 5/8 gives √(5x−1)=8 and x=13.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q07a", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove tanθ/(1+cosθ)+tanθ/(1−cosθ) is identically 2/(sinθ cosθ).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine fractions to get 2tanθ/sin²θ. Replace tanθ with sinθ/cosθ to obtain 2/(sinθ cosθ).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q07b", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Using the identity, solve the equation with right side 6tanθ for 0°<θ<180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set 2/(sinθ cosθ)=6tanθ, so sin²θ=1/3. The solutions are 54.7° and 125.3°.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q08a", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the sum to infinity of the GP with first term sin²θ and second term sin²θ cos²θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The common ratio is cos²θ. S∞=sin²θ/(1−cos²θ)=1.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q08b", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the common difference of the AP whose first two terms are sin²θ and sin²θ cos²θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=10", "qpPages": [ 13, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "d=sin²θcos²θ−sin²θ=−sin⁴θ.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q08bii", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "For that AP, find the sum of its first 16 terms when θ=π/3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Here a=3/4 and d=−9/16. S16=8[2a+15d]=−111/2 (−55.5).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q09a", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Write f(x)=x²−4x+3 in the form (x−a)²+b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square: f(x)=(x−2)²−1.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q09b", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the smallest c for which f(x)=x²−4x+3, restricted to x>c, is one-to-one.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The turning point is x=2. The minimum c giving a one-to-one restriction on x>c is 2.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q09c", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "For c=5, find f⁻¹(x) and its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=11", "qpPages": [ 15, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the positive root on x>5: f⁻¹(x)=2+√(x+1), with domain x>8.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; Cambridge published no examiner report, so no difficulty ranking is assigned." }, { "id": "cie-2020-s-13-q09d", "paperId": "cie-2020-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "For c=5, find g(f(x)) and its range, where g(x)=1/(x+1).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_13.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_13.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "gf(x)=1/(x−2)². With x>5, the range is 0 0 and b > 0 to derive an impossible condition, and reject the assumption.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q02", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": null, "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the exact exponential-logarithmic integral by parts, expressing the result in the requested form.", "answerGuide": "Choose the logarithmic factor and exponential factor for integration by parts, evaluate the boundary terms, and simplify to integers in the stated expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q03", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": null, "marks": 7, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate the given curve relation and find the exact gradient at (2, 3).", "answerGuide": "Differentiate both sides with respect to x, collect the dy/dx terms, solve for dy/dx, then substitute x = 2 and y = 3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=11", "qpPages": [ 2, 2 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q04a", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use a binomial expansion to derive the stated approximation formula for the specified range of x.", "answerGuide": "Expand the fractional power in ascending powers, retain the required terms, and show that the omitted terms satisfy the stated error bound.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q04b", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Substitute x = 1/20 into the approximation formula to estimate 10 as a simplified fraction.", "answerGuide": "Evaluate the retained series terms at x = 1/20 and simplify the resulting rational number.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q05", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "differentiation", "geometry" ], "primaryTopicId": "differentiation", "summary": "Find the volume growth rate of an expanding cylindrical rod at radius 2 cm.", "answerGuide": "Write the cross-sectional area and volume in terms of x, differentiate each with respect to time, use the given area rate to find dx/dt, then evaluate dV/dt at x = 2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=15", "qpPages": [ 3, 3 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q06a", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Show that the volume of the rotated parametric region has the stated integral form and identify its constants.", "answerGuide": "Use V = π∫y² dx, substitute x = 3 sin t and y = 2 sin 2t, convert dx/dt, and use the parameter limits for the shaded region.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=17", "qpPages": [ 4, 4 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q06b", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the volume exactly using u = sin t or another valid substitution.", "answerGuide": "Substitute u = sin t in the integral from part (a), change the limits, integrate the resulting polynomial, and express the answer in the requested exact form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=18", "qpPages": [ 4, 4 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q07a", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Express the given rational function as partial fractions.", "answerGuide": "Resolve the denominator into its linear factors, equate coefficients, and solve for the constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=20", "qpPages": [ 5, 5 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q07b", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the separable differential equation with the initial condition x = 0 at t = 0.", "answerGuide": "Separate variables using the partial fractions from part (a), integrate both sides, apply the initial condition, then rearrange to the required exponential form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=20", "qpPages": [ 5, 5 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q07c", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the time when the substance mass is 1 gram for k = 0.1.", "answerGuide": "Substitute x = 1 and k = 0.1 into the solution from part (b), then solve the logarithmic equation and round to three significant figures.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=22", "qpPages": [ 5, 5 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q08a", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the coordinates of A and show that B has coordinates (15, 0).", "answerGuide": "Set the parametric y-coordinate to zero, solve for the parameter values, and substitute them into x and y.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=23", "qpPages": [ 6, 6 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q08b", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at B in integer-coefficient form.", "answerGuide": "Use dy/dx = (dy/dt)/(dx/dt) at the parameter for B, then form the line through B with that gradient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=23", "qpPages": [ 6, 6 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q08c", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the coordinates of the second intersection X of the tangent at B with the curve.", "answerGuide": "Substitute the tangent equation into the parametric curve relation, factor out the known intersection at B, and use the other parameter value to obtain X.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=24", "qpPages": [ 6, 6 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q09a", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find the constants a and b from the coordinates of A and B on the given 3D line.", "answerGuide": "Substitute each point’s coordinates into the vector equation and solve for its line parameter and unknown coordinate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=26", "qpPages": [ 7, 7 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q09b", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find vector AB.", "answerGuide": "Subtract the position vector of A from that of B.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=26", "qpPages": [ 7, 7 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q09c", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "vectors", "trigonometry" ], "primaryTopicId": "vectors", "summary": "Find the angle CAB.", "answerGuide": "Form vectors AB and AC, then use their scalar product and magnitudes in the cosine formula.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=26", "qpPages": [ 7, 7 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q09d", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find the exact area of triangle CAB.", "answerGuide": "Use half the magnitude of the cross product of AB and AC, or an equivalent area method.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=26", "qpPages": [ 7, 7 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WMA14-01A-q09e", "paperId": "ial-2025-10-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(e)", "marks": 4, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find both points D on the line for which triangle CAD has twice the area of triangle CAB.", "answerGuide": "Use the line parameter and the proportional-area condition to solve for both possible positions, then give the two coordinate triples.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4A_MS.pdf#page=27", "qpPages": [ 7, 7 ], "msPages": [ 27, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q01a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the curve and simplify the derivative.", "answerGuide": "Apply the power rule to each term, including the fractional power, and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q01b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the tangent at the given point and write it as y = mx + c.", "answerGuide": "Evaluate the derivative at x = 4, then use the point and gradient to form the tangent.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q02a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rationalise and simplify a surd expression into a + b√2.", "answerGuide": "Multiply by the conjugate, simplify the numerator and denominator, and express the result in the required surd form.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q02b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the linear equation involving x and √2, giving the answer as p + q√2.", "answerGuide": "Collect the x terms, isolate x, then use the simplified surd result from part (a).", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q03a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the perimeter condition to show the lower bound for x.", "answerGuide": "Add the six edge lengths, apply the condition that the perimeter exceeds 29 m, and solve for x.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q03b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the quadratic inequality that represents the garden-area limit.", "answerGuide": "Write the compound garden area as a quadratic in x, compare it with 72, then solve the resulting inequality and keep the interval between its roots.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson's June 2019 PEF The PEF says complete solutions were uncommon; many gave the contextual range or the next part’s combined range instead of solving the requested quadratic inequality. (pp. 3–4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2019-6-WMA11-01-q03c", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Combine the perimeter and area restrictions to state the possible range of x.", "answerGuide": "Intersect the bounds from parts (a) and (b) to give the contextual interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q4", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate the given expression and give the result in its simplest form.", "answerGuide": "Rewrite the integrand as powers of x, integrate each term, and include a constant of integration.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q05a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find all real roots of the cubic equation using algebra.", "answerGuide": "Factor out the common x term, solve the remaining quadratic, and include the zero root.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q05b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Transfer the roots of a cubic in x to a polynomial in (y−5)² to find all real y values.", "answerGuide": "Set u=(y−5)² and reuse the roots from part (a). Discard the negative value of u and take both square-root branches for the positive value, while retaining the zero-root case.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that candidates often did not connect this transformed equation to part (a), and some lost roots by omitting zero or one square-root branch. (p. 4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-6-WMA11-01-q06a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the line constant that makes a straight line meet a quadratic curve at exactly one point.", "answerGuide": "Equate the line and curve to obtain a quadratic in x. Require its discriminant to be zero, then solve for the constant, c=−49/4.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson's June 2019 PEF The PEF identifies frequent sign confusion when forming the discriminant, especially because the constant appears with the opposite sign in the intersection equation. (p. 4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-6-WMA11-01-q06b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of the single intersection point.", "answerGuide": "Solve the repeated-root intersection equation using c from part (a), then substitute to find y.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q07a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the area of the major circular sector.", "answerGuide": "Use half r²θ with radius 6 and the major angle 2π − 0.7 radians.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q07b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the obtuse angle in a triangle from a known side and a given angle in radians.", "answerGuide": "Apply the sine rule to obtain the sine of the required angle, then choose the supplementary value because the question specifies an obtuse angle; this gives approximately 2.258 radians.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that most candidates returned the acute sine-rule angle even though the question specified the obtuse solution. (p. 4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA11-01-q07c", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of a composite shape made from a major circular sector and a triangle.", "answerGuide": "Find the major arc using radius 6 and angle 2π−0.7. Use the obtuse triangle angle to find OD, then add the arc, AD, and CD=6−OD; the perimeter is about 43.1 cm.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson's June 2019 PEF The PEF identifies OD as the main obstacle: candidates often abandoned after obtaining an implausible length and some confused the arc with the chord. (p. 4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA11-01-q08a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the equation of the normal to the curve at P.", "answerGuide": "Evaluate the given derivative at x = 4, take the negative reciprocal gradient, and use P(4, 1).", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q08b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Recover f(x) from its derivative and the point P.", "answerGuide": "Integrate each term of f′(x), add a constant, and substitute P(4, 1) to determine it.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q09a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "State the minimum point of a vertically scaled sine curve over one full revolution.", "answerGuide": "A factor of 4 scales the sine range to [−4,4]; the minimum occurs at 270°, giving (270°,−4).", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson's June 2019 PEF The PEF says correct coordinate pairs were rare; candidates often gave only the y-value or supplied several unlabelled possible points. (p. 5)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA11-01-q09bi", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Sketch and label y = 1 + sin θ over 0° to 360°.", "answerGuide": "Draw a smooth sine curve shifted up by one, using its key points at 0°, 90°, 180°, 270° and 360°.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that the tangent sketch was often misplaced or reflected, with some candidates omitting its first-quadrant branch; some sine sketches were also drawn as straight lines. (p. 5)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA11-01-q09bii", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Sketch and label y = tan θ over 0° to 360°.", "answerGuide": "Draw tangent branches through 0°, 180° and 360°, with vertical asymptotes at 90° and 270°.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that the tangent sketch was often misplaced or reflected, with some candidates omitting its first-quadrant branch; some sine sketches were also drawn as straight lines. (p. 5)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA11-01-q09ci", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(i)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Count the solutions of tan θ = 1 + sin θ over six revolutions.", "answerGuide": "Use one-revolution intersections from the graphs and multiply the count by six.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q09cii", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(ii)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Count the solutions of tan θ = 1 + sin θ over 0° to 180°.", "answerGuide": "Count the intersections on the stated half-turn using the two curves.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q10a", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State the coordinates of the point where the curve touches the x-axis.", "answerGuide": "Identify the repeated factor of the curve and set it equal to zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=19", "qpPages": [ 26, 28 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q10b", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the expanded curve function.", "answerGuide": "Expand the product and apply the power rule to each term.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=19", "qpPages": [ 26, 28 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA11-01-q10c", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Show that the tangent at the specified point is horizontal and find its constant y-value.", "answerGuide": "Differentiate the factored curve, evaluate its gradient at x=5/2 to show it is zero, then evaluate the curve there to obtain the horizontal tangent y=−54.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=19", "qpPages": [ 26, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson's June 2019 PEF The PEF calls this part especially challenging and says very few completed it; many found only the point’s y-coordinate without proving that the gradient was zero. (p. 6)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-6-WMA11-01-q10d", "paperId": "ial-2019-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(d)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the possible horizontal translations that make the curve pass through the origin.", "answerGuide": "Set f(a) = 0 for the translated function at x = 0 and use the curve's x-intercepts.", "qpUrl": "https://www.alevelcopilot.com/whfile/25/01/a2/88/97/26/72/5a/b3/02/73/c0/07/8a/64/83.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/f8/35/cb/a4/02/e4/be/8c/75/58/1e/0c/5d/8a/b8/1a.pdf#page=19", "qpPages": [ 26, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q01ai", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(i)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the second term of the recurrence sequence.", "answerGuide": "Substitute a₁ = 3 into the recurrence aₙ₊₁ = 4 − aₙ.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q01aii", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(ii)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use the recurrence pattern to find the 107th term.", "answerGuide": "Generate the first few terms: the values alternate 3, 1, 3, 1, so every odd-indexed term is 3.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson's June 2019 PEF The PEF notes that many candidates assumed the recurrence generated an arithmetic sequence and missed the alternating pattern needed for this term. (p. 3)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2019-6-WMA12-01-q01b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Evaluate a 200-term sum built from a recurrence with alternating values.", "answerGuide": "The recurrence alternates 3 and 1, so the summand alternates 5 and 1. Group the 200 terms into 100 pairs, each summing to 6, giving 600.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson's June 2019 PEF The PEF says this part strongly separated candidates: the period-based sum was accessible to stronger responses, while weaker candidates left it blank or tried arithmetic/geometric sum formulas. (p. 3)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2019-6-WMA12-01-q02ai", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the centre of the circle by completing the square.", "answerGuide": "Complete the squares in x and y to identify the centre coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q02aii", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact radius of the circle.", "answerGuide": "Read the squared radius after completing the squares and simplify the surd.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q02b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent at P(5, 4) in the form y = mx + c.", "answerGuide": "Find the radius gradient from P to the centre, take its negative reciprocal, and use point-slope form.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q03i", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove a universal quadratic inequality by rewriting its difference as a square.", "answerGuide": "Move both sides to one side and simplify to (x−5)²≥0. State that a square is non-negative for every real x, which proves the result.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson's June 2019 PEF The PEF identifies proof as new content and reports that few candidates gave a complete general argument; many relied on checking sample values. (pp. 3–4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2019-6-WMA12-01-q03ii", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Give a counterexample to disprove the prime-number statement.", "answerGuide": "Choose an integer n that makes 2ⁿ + 1 composite and state that it is not prime.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q04a", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first four terms of the binomial expansion in ascending powers of x.", "answerGuide": "Apply the binomial series through x³, simplifying each coefficient.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q04b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Combine the two truncated expansions and show the result has the stated form a + bx².", "answerGuide": "Use the expansion from part (a) with the sign changed in the second series; show odd-power terms cancel.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q05a", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the maximum annual profit using calculus.", "answerGuide": "Differentiate P(x), solve P′(x) = 0 for positive x, and evaluate the profit in pounds.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q05b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use calculus to justify that the profit found is a maximum.", "answerGuide": "Evaluate the second derivative at the maximizing x and state that it is negative.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q06a", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the factor theorem to show that k = 9.", "answerGuide": "Substitute x = 3 into f(x), set f(3) = 0, and solve for k with the intermediate equation shown.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=16", "qpPages": [ 12, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q06b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Fully factorise the cubic polynomial.", "answerGuide": "Use the factor x − 3, then factorise the remaining quadratic as a product of linear factors.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=16", "qpPages": [ 12, 15 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q06c", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the cubic-in-cos θ equation for 0° ≤ θ ≤ 360°.", "answerGuide": "Use the factors from part (b), keep the valid cosine value, and find both angles in the range.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=16", "qpPages": [ 12, 15 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q07a", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use the fixed annual increase model to find the salary in the second year.", "answerGuide": "Use nine annual increments between year 1 and year 10 to find the common difference, then add it once to the year-1 salary, giving £17,900.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=18", "qpPages": [ 16, 19 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that many candidates counted ten rather than nine increments between years 1 and 10, leading to an incorrect common difference and second-year salary. (p. 5)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA12-01-q07b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find Kim's year-2 salary under the fixed percentage increase model.", "answerGuide": "Use nine geometric growth intervals to find the ratio, then multiply the year-1 salary by it.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=18", "qpPages": [ 16, 19 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q07c", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Compare the ten-year salary totals under arithmetic and geometric growth models.", "answerGuide": "Find each model’s sum for ten years and subtract the totals. Keep the geometric ratio unrounded through the sum calculation; the difference is £7,480.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=18", "qpPages": [ 16, 19 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson's June 2019 PEF The PEF says fully correct answers were rare; rounding the common ratio or using the wrong number of terms frequently lost the final accuracy. (p. 5)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA12-01-q08i", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation exactly using logarithms and express the result in the requested form.", "answerGuide": "Rewrite 8 as 2³, take log base 2, and isolate x. Use log₂6=1+log₂3 to express x as −1/3+(1/6)log₂3.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=20", "qpPages": [ 20, 23 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that many attempts were incomplete and that one mark out of four was the most common score, often because the log of 6 was not handled fully. (p. 5)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-6-WMA12-01-q08ii", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation for y and retain only valid solutions.", "answerGuide": "Apply log laws, convert to an equation in y, solve the quadratic, and check the logarithm domains.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=20", "qpPages": [ 20, 23 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q09a", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use trigonometric identities to rewrite an equation as a quadratic in cos θ.", "answerGuide": "Replace tan θ with sin θ/cos θ, clear the denominator, then use sin²θ=1−cos²θ and rearrange to 5cos²θ−cosθ−4=0.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=22", "qpPages": [ 24, 27 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson's June 2019 PEF The PEF explicitly describes this part as highly discriminating; candidates who did not use the relevant identities often left it or failed to reach the required form. (p. 6)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-6-WMA12-01-q09b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the related equation over the stated interval for x.", "answerGuide": "Apply the quadratic in cos 2x, find the valid cosine values, and return to x within the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=22", "qpPages": [ 24, 27 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q10a", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the positive x-values for which the given function is increasing.", "answerGuide": "Differentiate f(x)=36/x²+2x−13, set f′(x)>0, and solve for x>0 to obtain x>∛36.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=24", "qpPages": [ 28, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Pearson's June 2019 PEF The PEF notes that some candidates integrated or used trial and error, and that only a minority gave the required exact interval. (p. 6)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-6-WMA12-01-q10b", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Show the stated value of the definite integral using calculus.", "answerGuide": "Integrate each term and substitute the limits 2 and 9 with correct brackets.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=24", "qpPages": [ 28, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-6-WMA12-01-q10ci", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)(i)", "marks": 1, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the equal-area information to state the value of the related integral.", "answerGuide": "Apply the signed-area relationship shown by the graph and the supplied integral value.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=24", "qpPages": [ 28, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that only a minority used the link between the given integral and the required one; a sign error in part (i) and mishandling the added constant in part (ii) were common. (p. 6)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-6-WMA12-01-q10cii", "paperId": "ial-2019-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)(ii)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the constant k that makes the modified definite integral zero.", "answerGuide": "Integrate the added constant term, apply the limits 2 and 6, and solve the resulting linear equation for k.", "qpUrl": "https://www.alevelcopilot.com/whfile/95/c0/4b/1a/ff/9f/9c/9d/78/2f/94/7c/0a/2a/b7/75.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/be/66/34/32/34/16/6c/95/93/d3/bf/70/8b/fb/ca/51.pdf#page=24", "qpPages": [ 28, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Pearson's June 2019 PEF The PEF reports that only a minority used the link between the given integral and the required one; a sign error in part (i) and mishandling the added constant in part (ii) were common. (p. 6)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20190815.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "cie-2020-s-21-q01", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve ln(x + 1) - ln x = 2 ln 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms, equate (x + 1)/x to 4, and solve to obtain x = 1/3.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q02", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "For p(x) = 6x^3 + ax^2 + 9x + b, use the factors (x - 2) and (2x + 1) to find a and b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor theorem at x = 2 and x = -1/2, then solve the resulting simultaneous equations: a = -19 and b = 10.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q03", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "For the parametric curve x = e^t - 2e^(-t), y = 3e^(2t) + 1, find the tangent at t = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find (x, y) = (-1, 4) and dy/dx = 2 at t = 0, then write the tangent y = 2x + 6.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q04a", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch y = |3x + 2a| and y = |3x - 4a| for positive a, and state where each graph meets both axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the two V-shaped graphs; their x-intercepts are -2a/3 and 4a/3, and their y-intercepts are 2a and 4a.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q04b", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the coordinates where y = |3x + 2a| and y = |3x - 4a| intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the absolute-value expressions and solve for the crossing point (a/3, 3a).", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q04c", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use the graphs to solve |3x + 2a| < |3x - 4a|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the two graph heights on either side of their crossing; the solution is x < a/3.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q05a", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "For y = x^3 cos(2x), show that the x-coordinate of the maximum M satisfies x = cube_root(1.5 x^2 cot(2x)).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product rule and set the derivative to zero; rearrange to x = 1.5 cot(2x), then write it in the stated cube-root form.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q05b", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "Use the equation from part (a) to show that the x-coordinate of M is between 0.59 and 0.60.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute 0.59 and 0.60 into the rearranged root equation and show the resulting values have opposite signs.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q05c", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "Iterate from the equation in part (a) to find the x-coordinate of M to 3 significant figures, giving iterations to 5 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate x_(n+1) = cube_root(1.5 x_n^2 cot(2x_n)) with each iterate to 5 significant figures; the final x-coordinate is 0.596 to 3 significant figures.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q06a", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove sin(2θ)(cosec θ - sec θ) = 2√2 cos(θ + π/4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use sin(2θ) = 2sinθcosθ to obtain 2cosθ - 2sinθ, then express it in cosine phase-shift form.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q06b", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve sin(2θ)(cosec θ - sec θ) = 1 for 0 < θ < π/2, to 3 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (a) and solve in the stated interval; θ = 0.424 radians.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q06c", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find an antiderivative of sin x (cosec(x/2) - sec(x/2)).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Simplify the integrand to 2cos(x/2) - 2sin(x/2), then integrate to obtain 4sin(x/2) + 4cos(x/2) + C.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q07a", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide 9x^3 - 6x^2 - 20x + 1 by 3x + 2; find the quotient and show that the remainder is 9.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Perform polynomial division; the quotient is 3x^2 - 4x - 4 and the remainder is 9.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q07b", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate ∫ from 1 to 6 of (9x^3 - 6x^2 - 20x + 1)/(3x + 2) dx, giving the answer as a + ln b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the quotient and remainder from part (a), integrate each term, apply the bounds, and simplify to 125 + ln 64.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-21-q07c", "paperId": "cie-2020-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the exact root of 9e^(9y) - 6e^(6y) - 20e^(3y) - 8 = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_21.pdf", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set u = e^(3y), factor the cubic, take the valid positive root u = 2, and obtain y = (ln 2)/3.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q01", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Given 2^y = 9^(3x), use logarithms to show y = kx and find k to 3 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms and rearrange to y = (3 ln 9 / ln 2)x; k = 9.51.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q02", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact stationary-point coordinates for y = 5x e^(x/2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate by the product rule, set the derivative to zero to find x = -2, then substitute to get y = -10/e.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q03", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "For cos(3x) + 5 sin y = 3, find dy/dx at (π/9, π/6).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, substitute the given point, and obtain dy/dx = 3/5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q04", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Given y = A x^(-2p) and two points on the ln y against ln x graph, find A and p.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use ln y = ln A - 2p ln x; the line gradient is -2.6, so p = 1.3. Substitution gives ln A = 1.252 and A = 3.5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q05a", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch y = |2x - 3| and y = 3x + 5 on the same axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the absolute-value V with vertex at (3/2, 0), together with the line of gradient 3 and y-intercept 5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q05b", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve 3x + 5 < |2x - 3|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split at x = 3/2 and solve on each interval; the valid solution is x < -2/5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q06a", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "For p(x) = 6x^3 + ax^2 - 4x - 3, given x + 3 is a factor, find a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = -3 and set p(-3) = 0; a = 17.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q06b", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Using the value of a, factorise p(x) completely.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide by x + 3 and factor the remaining quadratic: p(x) = (x + 3)(3x + 1)(2x - 1).", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q06c", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Hence solve p(cosec θ) = 0 for 0° < θ < 360°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factorisation and retain the feasible cosecant root; solving sin θ = -1/3 gives 199.5° and 340.5°.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q07a", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Given ∫ from 0 to a of [4/(2x + 1) + 8x] dx = 10, show a^2 = 2.5 - 0.5 ln(2a + 1).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate to get 2ln(2a + 1) + 4a^2, apply the bounds, and rearrange to the stated relation.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q07b", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Using the relation from part (a), show 1 < a < 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the difference between the two sides at a = 1 and a = 2; the opposite signs establish a root between them.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q07c", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration based on part (a) to find a to 4 significant figures, with each iteration to 6 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate using the stated relation; the result is a = 1.358.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q08a", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Show 3 sin(2θ) cot θ = 6 cos^2 θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use sin(2θ) = 2sinθcosθ and cotθ = cosθ/sinθ.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q08b", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 3 sin(2θ) cot θ = 5 for 0 < θ < π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (a), solve cos^2θ = 5/6, and select both roots in the interval: θ = 0.421 and 2.72 radians.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-22-q08c", "paperId": "cie-2020-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate exactly ∫ from π/4 to π/2 of 3 sin x cot(x/2) dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_22.pdf", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as 3 + 3cos x, integrate, and apply the bounds to obtain 3π/4 + 3 - 3√2/2.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q01", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Given 2^y = 9^(3x), use logarithms to show y = kx and find k to 3 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms and rearrange to y = (3 ln 9 / ln 2)x; k = 9.51.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q02", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact stationary-point coordinates for y = 5x e^(x/2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate by the product rule, set the derivative to zero to find x = -2, then substitute to get y = -10/e.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q03", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "For cos(3x) + 5 sin y = 3, find dy/dx at (π/9, π/6).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, substitute the given point, and obtain dy/dx = 3/5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q04", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Given y = A x^(-2p) and two points on the ln y against ln x graph, find A and p.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use ln y = ln A - 2p ln x; the line gradient is -2.6, so p = 1.3. Substitution gives ln A = 1.252 and A = 3.5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q05a", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch y = |2x - 3| and y = 3x + 5 on the same axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the absolute-value V with vertex at (3/2, 0), together with the line of gradient 3 and y-intercept 5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q05b", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve 3x + 5 < |2x - 3|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split at x = 3/2 and solve on each interval; the valid solution is x < -2/5.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q06a", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "For p(x) = 6x^3 + ax^2 - 4x - 3, given x + 3 is a factor, find a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = -3 and set p(-3) = 0; a = 17.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q06b", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Using the value of a, factorise p(x) completely.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide by x + 3 and factor the remaining quadratic: p(x) = (x + 3)(3x + 1)(2x - 1).", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q06c", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Hence solve p(cosec θ) = 0 for 0° < θ < 360°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factorisation and retain the feasible cosecant root; solving sin θ = -1/3 gives 199.5° and 340.5°.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q07a", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Given ∫ from 0 to a of [4/(2x + 1) + 8x] dx = 10, show a^2 = 2.5 - 0.5 ln(2a + 1).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate to get 2ln(2a + 1) + 4a^2, apply the bounds, and rearrange to the stated relation.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q07b", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Using the relation from part (a), show 1 < a < 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the difference between the two sides at a = 1 and a = 2; the opposite signs establish a root between them.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q07c", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration based on part (a) to find a to 4 significant figures, with each iteration to 6 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate using the stated relation; the result is a = 1.358.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q08a", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Show 3 sin(2θ) cot θ = 6 cos^2 θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use sin(2θ) = 2sinθcosθ and cotθ = cosθ/sinθ.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q08b", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 3 sin(2θ) cot θ = 5 for 0 < θ < π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (a), solve cos^2θ = 5/6, and select both roots in the interval: θ = 0.421 and 2.72 radians.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-23-q08c", "paperId": "cie-2020-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate exactly ∫ from π/4 to π/2 of 3 sin x cot(x/2) dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_23.pdf", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as 3 + 3cos x, integrate, and apply the bounds to obtain 3π/4 + 3 - 3√2/2.", "administrationNote": "May/June 2020 examinations were cancelled. This archived paper was not administered and is indexed as unranked practice; no difficulty ranking is assigned." }, { "id": "ial-2025-10-WST01-01A-q01a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate Syy for the sample of patient ages and heart-blood volumes.", "answerGuide": "Use the supplied totals to calculate Σy² − (Σy)²/n.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q01b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation coefficient for the sample.", "answerGuide": "Substitute the supplied Sxx, Syy and Sxy into the correlation formula and round appropriately.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q01c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the direction of the correlation between age and blood volume per heartbeat.", "answerGuide": "State the negative association in context: higher age is associated with lower pumped volume.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q01d", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State whether the correlation supports using a linear regression model, with a reason.", "answerGuide": "Use the magnitude of the correlation coefficient from part (b) to justify whether a linear model is supported.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q01e", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(e)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the regression line of y on x in the form y = a + bx.", "answerGuide": "Calculate the regression gradient from Sxy/Sxx, then find the intercept using the sample means.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q01fi", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(f)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the regression line to estimate the heart volume for the 40-year-old patient.", "answerGuide": "Substitute x = 40 into the regression equation and round the estimate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q01fii", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(f)(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Comment on the reliability of the 40-year-old patient’s estimate.", "answerGuide": "Identify that age 40 lies within the sampled age range, so the estimate is interpolation and is reasonably reliable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q02a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Write down the final cumulative probability d.", "answerGuide": "Use the fact that the cumulative distribution function reaches 1 at the largest value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q02b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the unknown probabilities a, b and c in the table.", "answerGuide": "Use the cumulative probabilities as running sums and use the fact that the probabilities total 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q02c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find P(X > 4).", "answerGuide": "Use the probability at x = 5 or the complement of F(4).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q02d", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that two independent observations are both odd.", "answerGuide": "Add the probabilities of the odd values for one observation and square the result.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q02e", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(e)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Given both observations are odd, find the probability that their sum is 6.", "answerGuide": "Use conditional probability: add the ordered odd pairs that sum to 6, then divide by the probability that both observations are odd.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q03a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Write down the time exceeded by 75% of the Westyou runners.", "answerGuide": "Read the lower quartile from the Westyou box plot.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q03b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis", "data-representation" ], "primaryTopicId": "data-analysis", "summary": "Find the median and interquartile range for Eastyou.", "answerGuide": "Read the median, lower quartile and upper quartile from the ordered stem-and-leaf data, then subtract Q1 from Q3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q03c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Draw a box plot for the Eastyou race times, identifying outliers.", "answerGuide": "Calculate the 1.5 × IQR fences, then plot the median, quartiles, whiskers and any outliers on a suitable scale.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q03d", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State the skewness of each club’s race-time distribution and give reasons.", "answerGuide": "Compare the quartile spacings or the median’s position within each box to justify the direction or absence of skew.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q04a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Estimate the proportion of students who jump less than 2.5 m under the normal model.", "answerGuide": "Standardise 2.5 m using the given mean and standard deviation, then read the lower-tail probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q04b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the minimum first-jump distance needed to qualify the top 40%.", "answerGuide": "Set the upper-tail probability to 0.4, find the corresponding standard normal value, and transform back to metres.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q04c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the median first-jump distance among the students who qualify.", "answerGuide": "The conditional median leaves half of the top 40% above it, so find the distance with upper-tail probability 0.2 and standardise back.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q04d", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a randomly selected student earns a certificate on the second jump.", "answerGuide": "Multiply the probability of qualifying after the first jump by the independent probability that the second jump exceeds 4.1 m.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q05a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that both draws from the first bag are blue.", "answerGuide": "Multiply the two independent blue-bead probabilities because the first bead is replaced.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q05b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly one of the two draws from the first bag is red.", "answerGuide": "Add the two mutually exclusive orders: red then blue, and blue then red.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q05c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of drawing two yellow beads and one green bead in three draws without replacement.", "answerGuide": "Count the three possible positions for the green bead and multiply the corresponding sequential probabilities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q05d", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of getting at least one green bead in four draws without replacement.", "answerGuide": "Use the complement: subtract the probability that all four selected beads are yellow from 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q06a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(X) in terms of a and b.", "answerGuide": "Multiply each value of X by its probability and add the terms.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q06b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Use E(X) = 0 and Var(X) = E(X²) to find a and b.", "answerGuide": "Apply the mean-zero condition and the total-probability condition to form two equations, then solve them simultaneously; the mark scheme scores the two displayed subparts together.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q06c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Var(1 − 3X).", "answerGuide": "Use Var(a + bX) = b² Var(X), first finding Var(X) from E(X²) and the distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q06di", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(d)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Given Y = 1 − X, find P(Y < 0).", "answerGuide": "Convert Y < 0 to X > 1 and add the probabilities of the qualifying X values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q06dii", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(d)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the largest k for which P(Y < k) = 0.2.", "answerGuide": "Order the possible Y values with their probabilities and identify the largest threshold whose cumulative probability remains 0.2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q07a", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the mean and standard deviation of the ten rabbit weights.", "answerGuide": "Use the supplied totals to calculate the mean and apply the standard-deviation formula for the full data set.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q07b", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe the skewness of the rabbit-weight data, given the median.", "answerGuide": "Compare the mean from part (a) with the stated median; a mean above the median indicates positive skew.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q07c", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State how adding rabbits of 776 g and 896 g affects the mean, with a reason.", "answerGuide": "Show that the two added weights average to the original mean, so the combined mean is unchanged.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST01-01A-q07d", "paperId": "ial-2025-10-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "State how adding the two rabbits affects the standard deviation, using the change in squared deviations.", "answerGuide": "Compare the added squared deviations about the old mean with the original total and account for the larger sample size; the spread decreases.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1A_MS.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q01a", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Calculate the upper-tail probability P(X > 4) from the piecewise cumulative distribution function.", "answerGuide": "Use 1 − F(4), substituting the correct CDF branch at x = 4.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q01b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a such that the probability that X lies between 3 and a is 0.642.", "answerGuide": "Write F(a) − F(3) = 0.642, evaluate the CDF at the required breakpoints, then solve using the correct branch for a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q01c", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Differentiate the piecewise CDF to give the probability density function on its full support.", "answerGuide": "Differentiate each non-constant CDF segment and state the correct interval for each density expression and zero elsewhere.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q02a", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Suggest a suitable sampling frame for the restaurant customer survey.", "answerGuide": "Use a complete list or register of the restaurant’s eligible customers over the survey period.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q02b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the sampling units in the survey.", "answerGuide": "The sampling unit is one customer who eats at the restaurant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q02c", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give one advantage and one disadvantage of taking a census rather than a sample.", "answerGuide": "Give one valid benefit, such as complete coverage, and one cost, such as greater time or expense.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q02d", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(d)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at the 5% level whether more than 30% of customers want greater menu choice.", "answerGuide": "Define the binomial count, state H₀: p = 0.3 and H₁: p > 0.3, calculate the upper-tail p-value or critical region, and give a contextual conclusion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q03ai", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Write an exact binomial expression for the probability that at most one person in the sample is left-handed.", "answerGuide": "Use Y ~ B(60, 0.1) and add P(Y = 0) and P(Y = 1) in binomial form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q03aii", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Evaluate the exact probability that at most one sampled person is left-handed.", "answerGuide": "Evaluate the binomial expression from part (a)(i) and give the answer to 3 significant figures.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q03b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Estimate the probability that at most one person is left-handed using a Poisson approximation.", "answerGuide": "Use a Poisson distribution with mean np = 6 and calculate P(Y ≤ 1).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q03c", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Estimate the probability that at most one person is left-handed using a normal approximation.", "answerGuide": "Use the approximating normal distribution with mean 6 and variance 5.4, apply the continuity correction at 1.5, standardise and evaluate the lower-tail probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q03d", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(d)", "marks": 1, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Explain why the Poisson approximation is more suitable than the normal approximation here.", "answerGuide": "State that p is small and n is large, or compare the estimates with the exact probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q04a", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "graphs" ], "primaryTopicId": "distributions", "summary": "Sketch the proposed density function and decide whether it is a valid probability density function.", "answerGuide": "Sketch the stated piecewise curve and note that it takes negative values over part of its domain, so it cannot be a density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q04b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "differentiation" ], "primaryTopicId": "distributions", "summary": "Use calculus to find the mode of the given density for Y.", "answerGuide": "Differentiate the density, set the derivative to zero, solve for the stationary point and identify the maximum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q04c", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use algebraic integration to find the normalising constant k.", "answerGuide": "Integrate the density over its support and set the total probability equal to 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q04d", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the median of Y to 3 significant figures.", "answerGuide": "Set the cumulative probability from the lower support endpoint to m equal to 0.5, integrate, solve for m and select the root in the support.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q04e", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Describe the skewness of Y and give a reason.", "answerGuide": "Compare the median with the mode found earlier; their near equality supports an approximately unskewed distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q05ai", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that exactly seven cars arrive in a 10-minute period.", "answerGuide": "The 10-minute Poisson mean is 6; evaluate P(X = 7).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q05aii", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that more than seven cars arrive in a 10-minute period.", "answerGuide": "Use the complement 1 − P(X ≤ 7) for a Poisson distribution with mean 6.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q05b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 9, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the duration n when a normal approximation gives probability 0.2266 of more than 40 arrivals.", "answerGuide": "Let the Poisson mean be λ = 0.6n and approximate with N(λ, λ). Apply the continuity correction at 40.5, use the corresponding z-value, solve for λ, then find n.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q06a", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "List all possible unordered samples of two coins from the bag.", "answerGuide": "List the six pairs drawn from 1p, 5p and 10p coins, allowing repeated types and treating mixed pairs as unordered.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q06b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Use the distribution of the highest coin value to find q, r and s.", "answerGuide": "Equate the probabilities for maximum values 1p and 5p to the given distribution, solve for q and r, then use the 10p equation or q + r + s = 1 to find s.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q07a", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Express E(3 − 2X) in terms of a and b for X uniform on [a, b].", "answerGuide": "Use linearity of expectation and E(X) = (a + b)/2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q07b", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(X > b/3 + 2a/3) for X uniform on [a, b].", "answerGuide": "Subtract the threshold b/3 + 2a/3 from b and divide by the full interval length b − a to get the upper-tail probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q07c", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Given E(X) = 0, express E(3X²) in terms of b only.", "answerGuide": "Use E(X) = (a + b)/2 = 0 to obtain a = −b, then integrate 3x² over the uniform interval and divide by its length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-10-WST02-01A-q07d", "paperId": "ial-2025-10-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Given E(X) = 0 and the range is 18, find Var(X).", "answerGuide": "Use the zero mean and range to identify the interval endpoints, then apply Var(X) = (b − a)²/12.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2020-s-31-q01", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential inequality, giving the exact set of x-values.", "answerGuide": "Take logarithms to obtain a linear inequality in x, then solve and express the boundary using logarithms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q02a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand (2 − 3x)⁻² through the x² term.", "answerGuide": "Factor out 2⁻² and use the binomial expansion for (1 − 3x/2)⁻², retaining terms through x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q02b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "State the interval of x-values for which the expansion in part (a) is valid.", "answerGuide": "Apply the binomial-series condition |3x/2| < 1 and simplify the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q03", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Convert the tangent equation into a quadratic in tan θ and solve it over 0° ≤ θ ≤ 180°.", "answerGuide": "Use the tangent addition and subtraction formulae, substitute tan 60° = √3, form the quadratic in tan θ, then list both angles in the required interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q04a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the stationary point of the exponential-trigonometric curve.", "answerGuide": "Differentiate using the product rule, set dy/dx = 0, reduce to a trigonometric equation and solve in the given interval to 2 decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q04b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Determine whether the stationary point is a maximum or a minimum.", "answerGuide": "Use a sign change in the derivative around the stationary x-value or evaluate the second derivative there.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q05a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide the cubic polynomial by x² + 3 and give the quotient and remainder.", "answerGuide": "Perform polynomial long division and identify the quotient and constant remainder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q05b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the division result to evaluate the definite rational integral exactly.", "answerGuide": "Rewrite the integrand as the quotient plus a constant over x² + 3, integrate the polynomial and arctangent terms, and apply the limits 1 and 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q06a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry", "geometry" ], "primaryTopicId": "trigonometry", "summary": "Show that the angle x satisfies the stated tangent equation when the shaded area equals the circle’s area.", "answerGuide": "Express each tangent length using r tan x, write the shaded area as the tangent quadrilateral minus the minor sector, equate it to πr² and simplify.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q06b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Verify by calculation that the root lies between 1 and 1.4.", "answerGuide": "Evaluate the two sides of the equation at 1 and 1.4 and show that their difference changes sign.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q06c", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use the given iteration to find the root to 2 decimal places.", "answerGuide": "Iterate from a suitable starting value, show successive results to 4 decimal places, and demonstrate enough iterations to justify the rounded root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q07a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show that the derivative of the given quotient is negative throughout the stated interval.", "answerGuide": "Differentiate by the quotient rule, simplify using sin²x + cos²x = 1, and use the denominator’s sign on the interval to establish the inequality.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q07b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of the given trigonometric quotient exactly.", "answerGuide": "Recognise the logarithmic derivative of 1 + sin x, integrate as ln(1 + sin x), and substitute the stated limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q08a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 8, "topicIds": [ "differentiation", "exponentials", "algebra" ], "primaryTopicId": "differentiation", "summary": "Set up and solve the differential equation for the curve through the two stated points.", "answerGuide": "Write dy/dx = ky/(x√x), separate variables and integrate, then use both coordinates to determine k and the integration constant before expressing y in terms of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q08b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Describe the limiting value of y as x tends to infinity.", "answerGuide": "Use the expression from part (a) and let 1/√x tend to zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q09a", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Use a scalar product to show that angle ABC is a right angle.", "answerGuide": "Form vectors BA and BC from the given position vectors, calculate their scalar product and show it is zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q09b", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Show that triangle ABC is isosceles.", "answerGuide": "Find the lengths of AB and BC from their component vectors and show that they are equal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q09c", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find the exact perpendicular distance from O to the line through B and C.", "answerGuide": "Write a general point on BC, impose the perpendicular condition using a scalar product (or an equivalent distance method), solve for the projection parameter and find the exact distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=11", "qpPages": [ 15, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q10ai", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the given complex quotient in Cartesian form, with real and imaginary parts in terms of a.", "answerGuide": "Multiply numerator and denominator by the conjugate of the denominator and use i² = −1 to separate real and imaginary parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q10aii", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)(ii)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Find the exact real value of a that gives the stated argument of the conjugate complex number.", "answerGuide": "Use the Cartesian form from part (i), account for conjugation and the correct argument quadrant, then solve the resulting trigonometric equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q10bi", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)(i)", "marks": 4, "topicIds": [ "geometry", "coordinate-geometry" ], "primaryTopicId": "geometry", "summary": "Sketch the Argand-plane region satisfying the two modulus inequalities.", "answerGuide": "Interpret the first inequality as a perpendicular-bisector half-plane and the second as a closed disk centred at 2 + i with radius 2; shade their intersection.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=13", "qpPages": [ 17, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-31-q10bii", "paperId": "cie-2020-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 2, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Find the least possible argument of a point in the shaded region.", "answerGuide": "Identify the boundary point where a ray from the origin is tangent to the circle within the allowed half-plane, then calculate its argument.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_31.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_31.pdf#page=13", "qpPages": [ 17, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2019-s-11-q01i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the parameter that makes the constant term of the binomial expansion equal 540.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the general term and select the term whose powers of x cancel; equate its coefficient to 540 and solve for the parameter." }, { "id": "cie-2019-s-11-q01ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "With that parameter, determine the coefficient of x² in the same expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the binomial term with net power x² and substitute the value found in part (i)." }, { "id": "cie-2019-s-11-q02i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the constant in a line that touches the given curve at a point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match the line’s gradient to the curve’s derivative, solve for the contact point, then substitute into the line equation." }, { "id": "cie-2019-s-11-q02ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates where the line is tangent to the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the tangent condition from part (i) to obtain the point on the curve and verify both coordinates satisfy the curve equation." }, { "id": "cie-2019-s-11-q03", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the boundary length of a circular sector using its radius and area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the sector angle from area = ½r²θ, use arc length rθ, and add the two radii." }, { "id": "cie-2019-s-11-q04i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the unknown coordinate in the trapezium using its right angle and parallel sides.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the gradient of AB; use perpendicularity to get the gradient of BC, then substitute the coordinates of B and C." }, { "id": "cie-2019-s-11-q04ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the fourth vertex of the trapezium after finding the parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the parallel-side gradient to write the line through A, then intersect it with the horizontal line through C." }, { "id": "cie-2019-s-11-q05i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Rewrite the quadratic function in completed-square form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor out the leading coefficient and complete the square in x, preserving the constant term." }, { "id": "cie-2019-s-11-q05ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the maximum value of the quadratic function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the maximum from the completed-square form, noting that its squared term has a negative coefficient." }, { "id": "cie-2019-s-11-q05iii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the equation involving the composite of the two stated functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute f(x) into g, include the added constant, set the result to zero, and solve the resulting quadratic." }, { "id": "cie-2019-s-11-q06i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove an identity involving a squared expression in secant and tangent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite secant and tangent in sine and cosine, combine over a common denominator, and use sin²x + cos²x = 1." }, { "id": "cie-2019-s-11-q06ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the equation obtained by setting the trigonometric expression equal to one third.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the identity from part (i), reduce to an equation in sin 2x, and list all solutions in the stated interval." }, { "id": "cie-2019-s-11-q07i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Express the two position vectors needed to form the angle at M in the three-dimensional square-prism diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The x-components of both required vectors proved very challenging, although y- and z-components were often correct.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the square base and matching upper face to write each position vector in i, j and k components, then subtract endpoint position vectors carefully." }, { "id": "cie-2019-s-11-q07ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use a scalar product to find the angle between the two vectors from the 3D diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the vectors from part (i), calculate their dot product and magnitudes, then apply the inverse-cosine formula." }, { "id": "cie-2019-s-11-q08a", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the infinite sum of a geometric progression from its third and fourth terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use adjacent terms to find the common ratio, recover the first term, then apply S∞ = a/(1−r)." }, { "id": "cie-2019-s-11-q08b", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Calculate the total recycled waste over 24 months under two different monthly growth schemes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says fewer candidates attempted this part than part (a), suggesting that identifying the required finite-sum formula was more difficult.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For the fixed monthly increase, sum the arithmetic progression; for the percentage increase, sum the geometric progression over 24 terms." }, { "id": "cie-2019-s-11-q09i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(i)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the range of the cosine function after its vertical shift and stretch.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use −1 ≤ cos x ≤ 1 and transform both endpoint values under the stated function." }, { "id": "cie-2019-s-11-q09ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch the cosine-based function over the stated interval, showing the correct range and turning behaviour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Fewer sketches received full credit because the gradients at the limiting x-values were often not approximately zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Mark the endpoints and extrema from the cosine cycle, then draw a smooth symmetric curve with zero gradient at the limiting x-values." }, { "id": "cie-2019-s-11-q09iii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(iii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the largest domain endpoint for which the cosine-based function has an inverse.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the first interval from zero on which cosine is one-to-one, and state its endpoint." }, { "id": "cie-2019-s-11-q09iv", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(iv)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Find the inverse rule for the restricted cosine function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange y = 2−3cos x for x using arccos, and use the restricted domain to define the inverse range." }, { "id": "cie-2019-s-11-q10i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Recover the curve equation from its second derivative and a stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=14", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates often omitted an integration constant, reused one constant for both integrations, or used the wrong stationary gradient in the first integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate twice, use the stationary-point condition to determine the linear constant, and substitute the given point to find the remaining constant." }, { "id": "cie-2019-s-11-q10ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the other stationary x-coordinate of the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the first derivative from part (i) to zero, factor it, and select the stationary root other than x = 3." }, { "id": "cie-2019-s-11-q10iii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Classify both stationary points on the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the second derivative at each stationary x-coordinate; positive indicates a minimum and negative a maximum." }, { "id": "cie-2019-s-11-q11i", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(i)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given reciprocal-root curve and find its indefinite integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=20", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule to the power of (1+4x), then integrate the curve term by term using a substitution or power rule." }, { "id": "cie-2019-s-11-q11ii", "paperId": "cie-2019-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(ii)", "marks": 6, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the shaded area formed by the curve, its normal, and the coordinate axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_11.pdf#page=20", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_11.pdf#page=16", "qpPages": [ 20, 21 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The rigorous approach and correct integration limits were often missing, despite many candidates attempting integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the correct integration limits from the diagram and show how the area is assembled; alternatively use the triangle method where its geometry is justified." }, { "id": "cie-2019-s-12-q01", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the coefficient of x in the stated fifth-power expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial general term and identify the term whose net power of x is one." }, { "id": "cie-2019-s-12-q02", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find where the perpendicular bisector of two given points meets the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the midpoint of AB, use the negative reciprocal of AB’s gradient for the bisector, then set x = 0." }, { "id": "cie-2019-s-12-q03i", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "differentiation", "kinematics" ], "primaryTopicId": "differentiation", "summary": "Find the instantaneous y-rate when x is changing at a constant negative rate at the specified curve point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use dy/dt = (dy/dx)(dx/dt), evaluating the derivative and both rates at P." }, { "id": "cie-2019-s-12-q03ii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Recover the curve equation from its derivative and a point on the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate dy/dx, include the constant of integration, and use P to determine that constant." }, { "id": "cie-2019-s-12-q04i", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Show that the expression formed from the sine and cosine parameters has a value independent of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part in particular proved challenging; candidates who began with the trigonometric identity were generally more successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Square the given sine and cosine relations, add them, and use sin²x + cos²x = 1 to eliminate x." }, { "id": "cie-2019-s-12-q04ii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Express a in terms of b when the angle has tangent 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set (a+b)/(a−b) equal to 2 and rearrange for a." }, { "id": "cie-2019-s-12-q05", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the angle parameter from the relationship between the perimeters of two semicircle sectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=6", "qpPages": [ 6, 6 ], "msPages": [ 6, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write each sector perimeter as two radii plus arc length, use the supplementary central angles, and solve the given ratio." }, { "id": "cie-2019-s-12-q06i", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Find the minimum and maximum y-values of the curve and line over the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the cosine range for the curve; evaluate the line at both interval endpoints and order its values." }, { "id": "cie-2019-s-12-q06ii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch the curve and line together over the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot the cosine extrema and endpoints, draw the line from its endpoint values, and label axes and intersections consistently." }, { "id": "cie-2019-s-12-q06iii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Count the intersections of the two graphs on the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sketch from part (ii) to count every crossing, including any at the interval boundaries." }, { "id": "cie-2019-s-12-q07i", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find both inverse functions and identify the excluded input for the rational inverse.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Interchange x and y and solve for y in each rule; use the denominator to state where the rational inverse is undefined." }, { "id": "cie-2019-s-12-q07ii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the equation formed by composing the two functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute g(x) into f, set the composite equal to the given value, and solve while checking the domain restriction." }, { "id": "cie-2019-s-12-q08i", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 2, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the vector parameter that makes the two position vectors perpendicular.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set their dot product to zero and solve the resulting linear equation." }, { "id": "cie-2019-s-12-q08ii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find parameter values for which the two position vectors have equal lengths.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate their squared magnitudes, simplify the resulting quadratic, and solve for k." }, { "id": "cie-2019-s-12-q08iii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the unit vector from O toward C using the given relation between AC and CB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The final vector part of the question was a multi-step challenge; candidates often failed to form the vector or normalise it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use AC = 2CB to obtain the position vector OC, substitute the specified parameter value, then divide by the magnitude of OC." }, { "id": "cie-2019-s-12-q09", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9", "marks": 8, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter and point where a tangent to one curve is also tangent to a second curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 19 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this a real challenge for many candidates and notes frequent misinterpretation of which curve and point each condition described.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the first tangent gradient and equation, impose equal gradient and common-point conditions on the second curve, then solve for the parameter and contact point." }, { "id": "cie-2019-s-12-q10ai", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)(i)", "marks": 4, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "sequences", "summary": "Show the common difference of the arithmetic progression is one third of its first term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the first-ten and next-five sums with the arithmetic-series formula, equate them, and simplify for d." }, { "id": "cie-2019-s-12-q10aii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)(ii)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the difference between two specified terms to find the first term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the tenth and fourth terms in terms of a and d, substitute the result from part (i), and solve." }, { "id": "cie-2019-s-12-q10b", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the fifth term of a geometric progression from its first term and a relation between two sums.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write S∞ and S4 in terms of the common ratio, solve the stated ratio for r, then calculate the fifth term." }, { "id": "cie-2019-s-12-q11i", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(i)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the derivative and antiderivative of the given curve expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the radicals as powers of (4x+1), differentiate and integrate term by term, including constants where needed." }, { "id": "cie-2019-s-12-q11ii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(ii)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates of the minimum point of the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative from part (i) to zero, solve for x, and substitute into the original function." }, { "id": "cie-2019-s-12-q11iii", "paperId": "cie-2019-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(iii)", "marks": 3, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Calculate the shaded area between the curve, the y-axis, and a horizontal line through its minimum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_12.pdf#page=20", "qpPages": [ 16, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A significant number did not attempt this final part, and others used incorrect limits or did not show both substitutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the antiderivative from part (i), determine the relevant x-intercept or boundary, substitute both limits, and subtract the line-bound area as required." }, { "id": "cie-2019-s-13-q01i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Express the quadratic function as a perfect square plus a constant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square in x and collect the constant term." }, { "id": "cie-2019-s-13-q01ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the strict inequality for which the quadratic function is below nine.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the completed-square expression from part (i), isolate the square, and express both exact bounds." }, { "id": "cie-2019-s-13-q02i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Complete the final three terms of the given binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Continue the binomial coefficients and powers through the remaining terms, preserving alternating signs and powers of x." }, { "id": "cie-2019-s-13-q02ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the x coefficient after multiplying the expansion by the stated quadratic factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine only terms whose powers add to one and sum their coefficients." }, { "id": "cie-2019-s-13-q03i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find a radial length from the right-triangle angle and given opposite side.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sine ratio with the angle in radians to determine the hypotenuse or required radius." }, { "id": "cie-2019-s-13-q03ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the shaded area in the tangent-and-sector diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the right-triangle and sector geometry to form the relevant sector and triangle areas, then subtract the unshaded portion." }, { "id": "cie-2019-s-13-q04i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Choose the largest lower and smallest upper domain bounds that allow the composite gf to be formed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates found this domain-composition condition challenging and confused the output interval with the input interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Map the full range of f on its stated domain and require that range to lie inside the domain of g." }, { "id": "cie-2019-s-13-q04ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Write the rule for the composite function gf.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the expression for f into g and simplify the rational expression." }, { "id": "cie-2019-s-13-q04iii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the composite function gf.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set y equal to the composite rule, interchange x and y, and solve for y while respecting the given domain." }, { "id": "cie-2019-s-13-q05i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Write an expression for the total weight loss after x weeks under the arithmetic model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first term and common difference in the arithmetic-series sum formula." }, { "id": "cie-2019-s-13-q05ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Determine the week in which the first boxer reaches the target total weight loss.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates reported a non-integer value for n and failed to recognise that the week number must be an integer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the arithmetic-series total from part (i) equal to the target, solve the quadratic, and use the context to select an integer week." }, { "id": "cie-2019-s-13-q05iii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the second boxer’s 20-week weight-loss total and decide whether the target can ever be reached.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often stopped after finding the 20-week total and did not use the sum to infinity to establish that the target is never reached.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sum the first 20 terms of the geometric progression, then compare its limiting sum to the target." }, { "id": "cie-2019-s-13-q06i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Express the vector from M to F using the diagram’s unit vectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the position vector of F from the parallel-edge lengths and height, subtract the position vector of M, and simplify components." }, { "id": "cie-2019-s-13-q06ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 1, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Express the vector from F to the midpoint N using i and j.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the midpoint of FE and subtract F’s position vector; the vertical components cancel because both lie in one horizontal plane." }, { "id": "cie-2019-s-13-q06iii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the vector from M to N by combining the earlier vector results.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use MF + FN, substitute parts (i) and (ii), and collect the three vector components." }, { "id": "cie-2019-s-13-q06iv", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iv)", "marks": 4, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the angle FMN using a scalar product.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vectors MF and MN, divide their dot product by the product of their magnitudes, and take the inverse cosine." }, { "id": "cie-2019-s-13-q07i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "State the midpoint coordinates of A and B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Average the x-coordinates and the y-coordinates separately." }, { "id": "cie-2019-s-13-q07ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Form the equation relating x and y from the fixed distance CD.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the distance formula from the midpoint D to C and set its square equal to the stated value." }, { "id": "cie-2019-s-13-q07iii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the equality of distances from C to A and B to derive a line equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the two squared distances, expand and cancel common quadratic terms, then simplify to the required linear relation." }, { "id": "cie-2019-s-13-q07iv", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the possible coordinates of C satisfying both geometric conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line relation from part (iii) into the distance equation from part (ii), solve for x, then recover y." }, { "id": "cie-2019-s-13-q08", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8", "marks": 8, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the constants in a derivative model from two stationary-point coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=6", "qpPages": [ 12, 13 ], "msPages": [ 6, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative equal to zero at both x-coordinates to determine a and b, then integrate and use the known point to obtain k." }, { "id": "cie-2019-s-13-q09i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "State the range of the sine-squared function in terms of its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use 0 ≤ sin²(2x) ≤ 1 and apply the positive scale factor and vertical shift." }, { "id": "cie-2019-s-13-q09iia", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)(a)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count where the curve reaches its maximum value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve sin²(2x)=1 over the stated interval and include each admissible endpoint solution." }, { "id": "cie-2019-s-13-q09iib", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)(b)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count where the curve reaches its vertical-shift value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve sin²(2x)=0 over the stated interval and count all allowed roots." }, { "id": "cie-2019-s-13-q09iic", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)(c)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count intersections at the stated intermediate height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set sin²(2x)=1/2 and list the number of solutions in the given interval." }, { "id": "cie-2019-s-13-q09iii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(iii)", "marks": 5, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Solve the specified trigonometric equation for the given values of the curve parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says Question 9 was challenging, with many omissions and incomplete solution sets; it stresses finding all four solutions over the stated range.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the parameter values into the function, isolate the sine-squared term, solve over the stated x-interval, and list all solutions." }, { "id": "cie-2019-s-13-q10i", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(i)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at the specified point on the square-root curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=14", "qpPages": [ 16, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the chain rule, evaluate the gradient at x = 4, find the point’s y-coordinate, and use point-slope form." }, { "id": "cie-2019-s-13-q10ii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 5, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the shaded area between the curve and its tangent using the diagram’s boundaries.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=15", "qpPages": [ 16, 18 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report stresses that calculator-only answers were not accepted and that the integration and both limit substitutions had to be shown.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve over the specified interval, evaluate both limits explicitly, and subtract or add the tangent-line area with the correct orientation." }, { "id": "cie-2019-s-13-q10iii", "paperId": "cie-2019-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "differentiation", "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the point where the y-coordinate changes at half the x-coordinate’s rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_13.pdf#page=16", "qpPages": [ 16, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set dy/dt = (1/2)dx/dt, cancel the nonzero dx/dt, and solve dy/dx = 1/2 on the curve." }, { "id": "cie-2019-s-21-q01", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show the logarithmic difference simplifies to the logarithm of x plus two.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms into one quotient, factor both polynomials, cancel common factors, and apply the log product rule." }, { "id": "cie-2019-s-21-q02i", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a strict inequality comparing two absolute-value expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find where each linear expression changes sign, solve on the resulting intervals, and retain the values satisfying the strict inequality." }, { "id": "cie-2019-s-21-q02ii", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the inequality result to find the integer n that satisfies the condition involving the geometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few linked the parts correctly, and many left n as 12.6 instead of giving the required integer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the expression from part (i) into the derived bound, solve for n, then apply the strict inequality and integer requirement." }, { "id": "cie-2019-s-21-q03", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 7, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation to an implicitly defined curve at the stated point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, substitute the point to obtain the tangent gradient, take its negative reciprocal, then use point-slope form." }, { "id": "cie-2019-s-21-q04a", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate the square of a tangent function with a scaled angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use tan²u = sec²u−1, substitute u = 3x, and integrate both terms." }, { "id": "cie-2019-s-21-q04b", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Integrate the exact exponential expression after rewriting each term over the common exponential factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number omitted division by e^x and could not proceed to an integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Divide every numerator term by e^x, integrate the resulting exponentials separately, and keep the final result exact." }, { "id": "cie-2019-s-21-q05i", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine two coefficients from a known factor and a specified remainder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p(2)=0 for the factor condition and p(−1)=27 for the remainder theorem, then solve the simultaneous equations." }, { "id": "cie-2019-s-21-q05ii", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor the cubic completely using the factor identified in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the cubic by x−2 and factor the remaining quadratic if possible over the integers." }, { "id": "cie-2019-s-21-q06i", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate a rational curve and find its gradient at the x-axis crossing.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quotient rule, solve the numerator for the x-intercept, and substitute that x-value into the derivative." }, { "id": "cie-2019-s-21-q06ii", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Rearrange the derivative condition into the given equation involving the curve parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates were able to rearrange the derivative correctly to obtain the required equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the derivative from part (i) to zero, retain the quotient-rule structure, and rearrange without cancelling terms that are not common factors." }, { "id": "cie-2019-s-21-q06iii", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "numerical-methods", "coordinate-geometry" ], "primaryTopicId": "numerical-methods", "summary": "Use the given iteration to find the maximum point's x-coordinate to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate the stated recurrence from its prescribed starting value, recording each iterate to five significant figures, then round the converged x-coordinate to three significant figures." }, { "id": "cie-2019-s-21-q07i", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove an identity linking a double-angle cosecant with cosecant and cotangent terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write sin 2θ as 2sinθcosθ and use cotθ = cosθ/sinθ to simplify both sides." }, { "id": "cie-2019-s-21-q07ii", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Use the identity proved in part (i) to verify the specified exact value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates earned credit; most used a calculator instead of applying the result as instructed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rearrange the identity, substitute the stated angle, evaluate the exact cosecant values, and show the required equality." }, { "id": "cie-2019-s-21-q07iii", "paperId": "cie-2019-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the linked trigonometric equation and report every solution in the required interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few made meaningful progress without reusing the earlier identity, and negative-angle solutions were often omitted.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (i) to form a quadratic in cosec²φ, solve it, and convert all positive and negative roots into angles in range." }, { "id": "cie-2019-s-22-q01", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express one polynomial coefficient in terms of two constants using a known factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = −1 into p(x), set the result to zero, and rearrange for m." }, { "id": "cie-2019-s-22-q02i", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an equation equating two absolute values of linear expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the expressions with both possible signs, solve the two linear equations, and check each solution." }, { "id": "cie-2019-s-22-q02ii", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the root information from part (i) to solve the related equation in e raised to a multiple of y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few recognised the “Hence” link, and many included a spurious solution from the negative branch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Replace x in the earlier root result with the exponential expression and retain only solutions consistent with the original equation." }, { "id": "cie-2019-s-22-q03", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of the stationary point of a logarithmic quotient curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate by the quotient rule, set the numerator to zero to find x, then substitute into y." }, { "id": "cie-2019-s-22-q04a", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact trigonometric definite integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use standard sine and cosine antiderivatives, apply the stated bounds, and simplify using exact angles." }, { "id": "cie-2019-s-22-q04b", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the stated definite integral using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few complete correct solutions were seen; many candidates did not know or misapplied the trapezium rule.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Evaluate the integrand at the two endpoints and midpoint, apply the width-weighted trapezium formula, and keep enough precision before rounding." }, { "id": "cie-2019-s-22-q05i", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder for a cubic divided by a linear expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division and verify by multiplying the quotient by the divisor and adding the remainder." }, { "id": "cie-2019-s-22-q05ii", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the quotient from part (i) to evaluate the resulting definite integral in the requested exact form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few used the “Hence” result or rewrote the integrand into a form that could be integrated.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as a polynomial plus a constant-over-linear term, integrate each part, apply the limits, and express the logarithm exactly." }, { "id": "cie-2019-s-22-q06i", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Rearrange the parametric condition at P into the stated fixed-point equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use y = 1 in the parametric y-equation, isolate the exponential term, and take the required square root." }, { "id": "cie-2019-s-22-q06ii", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Use the specified recurrence to approximate the parameter t at point P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Starting from t₁ = 0.7, repeatedly substitute into the given iteration, retain five significant figures at each step, and round the converged value to three significant figures." }, { "id": "cie-2019-s-22-q06iii", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the parametric curve at P using the computed parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate dy/dt and dx/dt, form dy/dx as their ratio, and substitute the converged parameter value." }, { "id": "cie-2019-s-22-q07ai", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite a sum of sine and cosine terms as one shifted sine function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use R = √(a²+b²), then match coefficients to determine the phase angle in the required range." }, { "id": "cie-2019-s-22-q07aii", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the least positive angle satisfying the amplitude-form equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the shifted sine equal to 5/R, find the first admissible angle, and subtract the phase shift." }, { "id": "cie-2019-s-22-q07b", "paperId": "cie-2019-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation using the amplitude form obtained in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_22.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates who used sine and cosine separately could not progress to a solvable equation; the tangent-form approach was required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the R-form into the equation, solve for the angle in the stated range, and convert radians or degrees only as requested." }, { "id": "cie-2019-s-23-q01", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express one polynomial coefficient in terms of two constants using a known factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = −1 into p(x), set the result to zero, and rearrange for m." }, { "id": "cie-2019-s-23-q02i", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an equation equating two absolute values of linear expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the expressions with both possible signs, solve the two linear equations, and check each solution." }, { "id": "cie-2019-s-23-q02ii", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the root information from part (i) to solve the related equation in e raised to a multiple of y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few recognised the “Hence” link, and many included a spurious solution from the negative branch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Replace x in the earlier root result with the exponential expression and retain only solutions consistent with the original equation." }, { "id": "cie-2019-s-23-q03", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of the stationary point of a logarithmic quotient curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate by the quotient rule, set the numerator to zero to find x, then substitute into y." }, { "id": "cie-2019-s-23-q04a", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact trigonometric definite integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use standard sine and cosine antiderivatives, apply the stated bounds, and simplify using exact angles." }, { "id": "cie-2019-s-23-q04b", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the stated definite integral using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few complete correct solutions were seen; many candidates did not know or misapplied the trapezium rule.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Evaluate the integrand at the two endpoints and midpoint, apply the width-weighted trapezium formula, and keep enough precision before rounding." }, { "id": "cie-2019-s-23-q05i", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder for a cubic divided by a linear expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division and verify by multiplying the quotient by the divisor and adding the remainder." }, { "id": "cie-2019-s-23-q05ii", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the quotient from part (i) to evaluate the resulting definite integral in the requested exact form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few used the “Hence” result or rewrote the integrand into a form that could be integrated.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as a polynomial plus a constant-over-linear term, integrate each part, apply the limits, and express the logarithm exactly." }, { "id": "cie-2019-s-23-q06i", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Rearrange the parametric condition at P into the stated fixed-point equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use y = 1 in the parametric y-equation, isolate the exponential term, and take the required square root." }, { "id": "cie-2019-s-23-q06ii", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Use the specified recurrence to approximate the parameter t at point P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Starting from t₁ = 0.7, repeatedly substitute into the given iteration, retain five significant figures at each step, and round the converged value to three significant figures." }, { "id": "cie-2019-s-23-q06iii", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the parametric curve at P using the computed parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate dy/dt and dx/dt, form dy/dx as their ratio, and substitute the converged parameter value." }, { "id": "cie-2019-s-23-q07ai", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite a sum of sine and cosine terms as one shifted sine function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use R = √(a²+b²), then match coefficients to determine the phase angle in the required range." }, { "id": "cie-2019-s-23-q07aii", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the least positive angle satisfying the amplitude-form equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the shifted sine equal to 5/R, find the first admissible angle, and subtract the phase shift." }, { "id": "cie-2019-s-23-q07b", "paperId": "cie-2019-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation using the amplitude form obtained in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates who used sine and cosine separately could not progress to a solvable equation; the tangent-form approach was required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the R-form into the equation, solve for the angle in the stated range, and convert radians or degrees only as requested." }, { "id": "cie-2019-s-31-q01", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the definite integral of an absolute-value linear function using three trapezia.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the integrand at the four equally spaced x-values, including the sign change, then apply the composite trapezium formula." }, { "id": "cie-2019-s-31-q02", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation and round the solution to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms using log laws, convert the equality to an algebraic equation, check the domain, and round." }, { "id": "cie-2019-s-31-q03", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of an implicitly defined curve at the given point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each term with respect to x using the chain and product rules, then substitute the coordinates." }, { "id": "cie-2019-s-31-q04", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a cotangent equation by reducing it to a quadratic in tangent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the angle-addition formula for cot(θ+45°), form the stated quadratic in tan θ, and retain roots in the interval." }, { "id": "cie-2019-s-31-q05i", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the reciprocal of sin squared theta.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite as cosec²θ and apply the chain rule, using the derivative of cosec θ or the standard cosecant identity." }, { "id": "cie-2019-s-31-q05ii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "differentiation", "integration", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Solve the differential equation by separating variables and integrating the trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates did not separate variables, and only stronger candidates recognised the trigonometric function needed for integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate x and y terms, integrate both sides, use the result from part (i) to handle the trigonometric factor, and apply any condition." }, { "id": "cie-2019-s-31-q06i", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the triple-angle identity for sine by expanding a sum of angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand sin(2x+x), replace sin 2x and cos 2x with double-angle forms, and simplify to the required cubic expression." }, { "id": "cie-2019-s-31-q06ii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact definite integral of sin cubed over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (i) to rewrite sin³x as a linear combination of sine terms, then integrate and apply the limits." }, { "id": "cie-2019-s-31-q07i", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Show the stated relation between the gradients of the two functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a minority reached the given result correctly; common errors were sign mistakes and multiplying the derivatives instead of applying the correct condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate both functions carefully, including the sign and chain-rule factors, then use the tangent-normal or perpendicular-gradient relationship as specified." }, { "id": "cie-2019-s-31-q07ii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the root a lies between two consecutive integers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute each endpoint into the fixed-point equation or equivalent residual and check that the root is bracketed." }, { "id": "cie-2019-s-31-q07iii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Iterate the equation for a to the requested decimal accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the fixed-point formula from part (i), show the iterations to five decimal places, and round the converged value." }, { "id": "cie-2019-s-31-q08i", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the rational function as partial fractions with repeated linear factors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up a constant term over each distinct factor and repeated-factor term, then solve by coefficient comparison or substitution." }, { "id": "cie-2019-s-31-q08ii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the required power-series expansion using the partial-fraction form from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A significant number made no attempt, and errors with constants were common in the expansions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Expand each partial-fraction term separately with the binomial theorem and retain all required powers while tracking the constants." }, { "id": "cie-2019-s-31-q09i", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(i)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the Cartesian equation of the plane through B, C and D.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A large proportion offered no response to the final two questions, including this multi-stage vectors question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form two direction vectors from B, take their vector product for a normal, and use a point on the plane to determine the constant term." }, { "id": "cie-2019-s-31-q09ii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the acute angle between plane BCD and plane OABC.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A large proportion offered no response to the final two questions, including this multi-stage vectors question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the plane normals: obtain the normal to BCD from part (i), note that OABC lies in the xy-plane, and apply the acute angle formula to the two normals." }, { "id": "cie-2019-s-31-q10i", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 5, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Write the complex number in polar form and find the modulus and argument of its fourth power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the modulus and principal argument of the original number, then apply De Moivre’s theorem to the fourth power." }, { "id": "cie-2019-s-31-q10ii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Verify the stated complex root of a real cubic and identify its other roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given root into the polynomial; use the conjugate-root property and the sum of roots to find the remaining root." }, { "id": "cie-2019-s-31-q10iii", "paperId": "cie-2019-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the Argand region satisfying a distance condition and a lower bound on the imaginary part.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_31.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the circle centred at u with radius two and the horizontal boundary Im(z)=2, then shade their intersection." }, { "id": "cie-2019-s-32-q01", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the x-cubed coefficient in a product involving a fractional binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=4", "qpPages": [ 2, 2 ], "msPages": [ 4, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand the power series through the terms needed, multiply by the linear factor, and collect all contributions to x³." }, { "id": "cie-2019-s-32-q02", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation with powers of three and round to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=4", "qpPages": [ 3, 3 ], "msPages": [ 4, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express 9 as 3², equate the exponents, solve for x, and report the requested precision." }, { "id": "cie-2019-s-32-q03", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the double-angle cotangent equation over the given degree interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=4", "qpPages": [ 4, 4 ], "msPages": [ 4, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use cot 2θ = (1−tan²θ)/(2tanθ), reduce to an algebraic equation in tan θ, and list valid angles." }, { "id": "cie-2019-s-32-q04", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact point where a logarithmic curve has tangent gradient one quarter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=4", "qpPages": [ 5, 5 ], "msPages": [ 4, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x/(1+ln x), set the derivative equal to 1/4, solve for x, and substitute to find y." }, { "id": "cie-2019-s-32-q05i", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State the other root paired with a given non-real root of a real cubic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the conjugate-root property for polynomials with real coefficients." }, { "id": "cie-2019-s-32-q05ii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine the remaining root of the cubic after using the known complex-conjugate pair.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies the complex-number question among the most challenging and notes that unsupported calculator answers received no credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the quadratic factor from the conjugate roots, divide the cubic by it or compare coefficients, and solve the remaining linear factor with full working." }, { "id": "cie-2019-s-32-q06i", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Relate the chord length to the radius and angle in the isosceles-circle diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates offered no solution or wrote the relation without justification.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the isosceles triangle into two right triangles and use the half-chord-to-radius cosine ratio, justifying the relation." }, { "id": "cie-2019-s-32-q06ii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the required area relationship for the sector and semicircle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Common errors involved using x instead of 2x in the sector area and losing factors in the algebra.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the sector area to half the semicircle area, use the sector angle correctly, and preserve all factors when simplifying." }, { "id": "cie-2019-s-32-q06iii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Check that the angle solving the fixed-point equation lies between one and one point five radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the right-hand side or residual at both endpoints and verify that the solution is bracketed." }, { "id": "cie-2019-s-32-q06iv", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iv)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration to find the angle to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the rearranged cosine formula repeatedly, show values to five decimal places, and round the stable result." }, { "id": "cie-2019-s-32-q07i", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 7, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve a separable differential equation with an initial condition for y in terms of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate e^(−y)dy from xe^x dx, integrate both sides, apply y(0)=0, and take logarithms." }, { "id": "cie-2019-s-32-q07ii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 1, "topicIds": [ "functions", "exponentials" ], "primaryTopicId": "functions", "summary": "Explain the domain restriction on x in the solution to the differential equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the logarithm’s positive-argument condition in the expression for y to show x must be less than one." }, { "id": "cie-2019-s-32-q08i", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express a rational function with a repeated denominator factor as partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use one term for each power of each distinct linear factor, then determine coefficients by substitution or comparison." }, { "id": "cie-2019-s-32-q08ii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the partial fractions from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the logarithmic terms separately, apply the bounds zero and one, and simplify to the stated exact form." }, { "id": "cie-2019-s-32-q09i", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that two lines in three dimensions do not intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write a general point on each line, equate the three coordinates, and show the simultaneous equations are inconsistent." }, { "id": "cie-2019-s-32-q09ii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the plane perpendicular to AB through its midpoint, then find where that plane meets line l.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report names this plane-equation part among the most challenging questions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use AB as a normal to the plane and the midpoint of AB as a point on it. Substitute the line l into the plane equation to find its intersection parameter, then give the position vector of P." }, { "id": "cie-2019-s-32-q10i", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove a product-to-sum identity for sine and cosine using angle expansions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand sin(3x+x) and sin(3x−x), add the results, and rearrange." }, { "id": "cie-2019-s-32-q10ii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral required for the area using the result from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Only stronger candidates reached the final answer; many used an incorrect upper limit or did not show the required working.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the product using the identity derived earlier, determine the correct upper limit from the curve, and show the exact integration and substitutions." }, { "id": "cie-2019-s-32-q10iii", "paperId": "cie-2019-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the minimum point of the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_32.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the sum of sine terms from part (i), solve dy/dx = 0 in the interval, and select the minimum." }, { "id": "cie-2019-s-33-q01", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an equation with exponential bases five and seven using logarithms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides, collect the x-terms, and round the result to three decimal places." }, { "id": "cie-2019-s-33-q02", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral by applying integration by parts twice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The question is identified as one of the paper’s more challenging questions; scattered working commonly caused sign and arithmetic errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Complete each integration by parts in an organised indefinite form, combine the terms, then substitute both limits into every term." }, { "id": "cie-2019-s-33-q03i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Show that the defined quotient of double-angle expressions simplifies to tan theta.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use sin 2θ = 2sinθcosθ and cos 2θ = 1−2sin²θ or 2cos²θ−1, then factor and cancel." }, { "id": "cie-2019-s-33-q03ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral of the function simplified in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Replace f(θ) with tan θ, integrate as −ln|cos θ|, and apply the stated radian limits." }, { "id": "cie-2019-s-33-q04i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Show the derivative of an exponential quotient is negative for every positive x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quotient rule, simplify the numerator to a negative expression, and note the denominator is positive." }, { "id": "cie-2019-s-33-q04ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the derivative result to obtain and solve the required quadratic equation in the exponential variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few candidates obtained the equation correctly, and most failed to reject the out-of-domain root or give an exact answer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Start with the correct derivative from part (i), rearrange the relation in e^a, solve the quadratic, and reject any root outside the stated domain." }, { "id": "cie-2019-s-33-q05", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve a separable differential equation for y squared from the given initial value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=6", "qpPages": [ 8, 9 ], "msPages": [ 6, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to y/(y²+5) dy = dx/(x+1), integrate both sides, apply y(0)=2, and isolate y²." }, { "id": "cie-2019-s-33-q06i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Use the factor theorem to identify the required integer root of the quartic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the quartic-root question as challenging; many guessed a root and showed little supporting work.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Test the integer candidates by substitution, show the polynomial value at the selected root, and use the factor theorem explicitly." }, { "id": "cie-2019-s-33-q06ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "Rearrange the quartic equation to the fixed-point equation for its other root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the known root to factor the quartic, then isolate the remaining root a in the required form." }, { "id": "cie-2019-s-33-q06iii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Iterate the fixed-point equation to find the smaller root to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Start from a suitable value in the specified interval, show iterates to five decimal places, and round the stable value." }, { "id": "cie-2019-s-33-q07i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate a product of a shifted sine and cosine function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the product rule and chain rule, preserving the angular shift in both terms." }, { "id": "cie-2019-s-33-q07ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Use the stationary-point condition to form the trigonometric equation for the parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates who expanded before differentiating often could not recover the required stationary-point equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the original product directly, substitute the stationary condition, and rearrange using the cosine addition identity." }, { "id": "cie-2019-s-33-q07iii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact x-coordinates of the stationary points in the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the cosine equation from part (ii), include both allowed roots, and convert back from the shifted angle." }, { "id": "cie-2019-s-33-q08i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Simplify a complex fraction to exact Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=10", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply numerator and denominator by the conjugate of the denominator, then collect real and imaginary parts." }, { "id": "cie-2019-s-33-q08ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Find the modulus and principal argument of the complex number.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Cartesian components to calculate the modulus and locate the argument in the correct quadrant." }, { "id": "cie-2019-s-33-q08iii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(iii)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Sketch the Argand region satisfying the stated modulus and perpendicular-bisector conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report lists the Argand-diagram part as one of the more challenging questions and notes missing scale, centre, radius and coordinates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot the complex number with equal axis scales, draw the circle centred at the origin with its radius marked, and shade the intersection region." }, { "id": "cie-2019-s-33-q09i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express a rational function with a repeated linear factor as partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use terms for the distinct linear factors and each power of the repeated factor, then solve for their coefficients." }, { "id": "cie-2019-s-33-q09ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the partial fractions to expand the rational function through the x³ term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the geometric-series expansion to each partial-fraction term around x = 0, multiply by its coefficient, combine like powers, and stop at x³." }, { "id": "cie-2019-s-33-q10i", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(i)", "marks": 5, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the perpendicular distance from a point to a line in three dimensions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the vector from a point on the line to P, take its cross product with the line direction, and divide by the direction magnitude." }, { "id": "cie-2019-s-33-q10ii", "paperId": "cie-2019-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 6, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the two unknown coefficients of a plane containing a given line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_33.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute a general point on the line into the plane equation and equate both the constant term and parameter coefficient." }, { "id": "cie-2019-s-41-q01", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics", "proof" ], "primaryTopicId": "mechanics", "summary": "Verify that the four coplanar forces balance in both perpendicular directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components using the supplied tangent ratios, then show that both component sums are zero." }, { "id": "cie-2019-s-41-q02i", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the vertically projected particle reaches its highest point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=5", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical motion with initial speed 25 m s⁻¹ and set the final vertical velocity to zero to find the time." }, { "id": "cie-2019-s-41-q02ii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find how long the particle remains above 23 m above the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the height above ground as the initial height plus the vertical displacement, solve the boundary equation at 23 m, and take the time between its two roots." }, { "id": "cie-2019-s-41-q02iii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(iii)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the height threshold that the particle exceeds for exactly one second.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the two crossing times of the height h one second apart in the vertical displacement equation, or use the symmetry of the trajectory about its maximum, then solve for h." }, { "id": "cie-2019-s-41-q03i", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show the engine power needed to maintain the lorry's given uphill speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At constant speed balance the engine force against the resistance and the component of weight down the slope, then multiply the driving force by 5 m s⁻¹." }, { "id": "cie-2019-s-41-q03ii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "energy-momentum", "algebra" ], "primaryTopicId": "energy-momentum", "summary": "Determine the resistance constant from the lorry's speed-dependent resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the known resistance and speed into R = kv² and solve for k." }, { "id": "cie-2019-s-41-q03iii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(iii)", "marks": 3, "topicIds": [ "energy-momentum", "algebra" ], "primaryTopicId": "energy-momentum", "summary": "Find the lorry's steady speed on a level road at the stated engine power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "On level ground the engine power balances the resistance power k v³; substitute the power and k from the earlier parts and solve for v." }, { "id": "cie-2019-s-41-q04i", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction coefficient when the particle is just about to move up the rough plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=7", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the plane. At limiting equilibrium use friction = μR and set the applied force equal to the downslope weight component plus limiting friction." }, { "id": "cie-2019-s-41-q04ii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle's acceleration after the applied force is removed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine the impending direction from the downslope weight component, set friction opposite that motion, then apply Newton's second law parallel to the plane." }, { "id": "cie-2019-s-41-q04iii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Calculate the work done against friction during the first two seconds after release.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the acceleration from part (ii) to find the distance moved in two seconds, calculate the friction force from μR, and multiply force by distance." }, { "id": "cie-2019-s-41-q05i", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the minimum value of the particle's quadratic velocity on the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square or differentiate v(t) and locate its stationary point, then check that this time lies in the interval." }, { "id": "cie-2019-s-41-q05ii", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 7, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled from the quadratic velocity function over the stated time interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Most candidates integrated from 0 to 8 without accounting for the sign change, giving displacement rather than distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the zero-velocity times, split the interval into positive and negative velocity regions, integrate on each, and reverse the negative contribution." }, { "id": "cie-2019-s-41-q06i", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the plane angle that keeps the two-particle system in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=10", "qpPages": [ 12, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each particle resolve forces along the string and the plane. Equate the common tension to the hanging weight and the component of A's weight down the smooth plane." }, { "id": "cie-2019-s-41-q06iia", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)(a)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension and acceleration when the plane angle is 20°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=10", "qpPages": [ 12, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take B's downward direction and A's up-plane direction as positive, write one Newton's second-law equation for each mass, and solve the simultaneous equations." }, { "id": "cie-2019-s-41-q06iib", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find A's speed when B has descended 0.5 m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=10", "qpPages": [ 12, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the constant acceleration from part (a), starting from rest, in v² = u² + 2as over the 0.5 m displacement." }, { "id": "cie-2019-s-41-q06iic", "paperId": "cie-2019-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)(c)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use energy, after one particle reaches the ground, to find the total further distance moved by the other particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_41.pdf#page=11", "qpPages": [ 12, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as the most difficult part and describes common errors involving tension and the stationary particle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "After the string goes slack, equate the moving particle’s kinetic energy to its gain in gravitational potential energy and add the distance already travelled." }, { "id": "cie-2019-s-42-q01", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 6, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Determine the unknown force magnitude and direction angle for a system in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve all four forces along horizontal and vertical axes, set each component sum to zero, and solve the two equations for P and the angle." }, { "id": "cie-2019-s-42-q02", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the initial speed and constant acceleration from two distance-time conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply s = ut + ½at² at 5 s and 8 s, then solve the resulting simultaneous equations for u and a." }, { "id": "cie-2019-s-42-q03", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the work done by a force moving a particle up a rough inclined plane at constant speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve weight and friction along the plane to determine the applied force, then multiply it by the 2.5 m distance." }, { "id": "cie-2019-s-42-q04", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the car's speed after travelling 100 m uphill and downhill under the same engine force and resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=6", "qpPages": [ 6, 7 ], "msPages": [ 6, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each direction, combine the engine force, resistance and signed component of weight to get the net force. Use work–energy over 100 m from rest to find the final speed in each case." }, { "id": "cie-2019-s-42-q05i", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and string tension while the two hanging particles move before A reaches the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take A downward and B upward as positive, apply Newton's second law to both particles, and solve for the common acceleration and tension." }, { "id": "cie-2019-s-42-q05ii", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find B's maximum height above the ground after A reaches the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "First use the acceleration and 0.5 m travel distance to find B's speed as A lands. Then treat B's upward motion under gravity alone and add its further rise to the initial 0.5 m." }, { "id": "cie-2019-s-42-q06", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "energy-momentum", "algebra" ], "primaryTopicId": "energy-momentum", "summary": "Determine the constants in the car's linear resistance model from two steady-speed power conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=6", "qpPages": [ 10, 11 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At each steady speed use engine force = power/speed. On level ground this equals Av + B; on the slope also account for the downslope weight component. Solve the two linear equations." }, { "id": "cie-2019-s-42-q07i", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "kinematics", "functions" ], "primaryTopicId": "kinematics", "summary": "Sketch both particles' velocity-time graphs over the five-second interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot P's quadratic velocity with its zeros at the start and at 3 s and Q's straight-line velocity from 10 to zero at 5 s; mark axes, intercepts and the correct curvature." }, { "id": "cie-2019-s-42-q07ii", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Verify that the particles meet again after five seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each velocity from 0 to 5 to find its displacement from A, then show the two displacements are equal." }, { "id": "cie-2019-s-42-q07iii", "paperId": "cie-2019-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the maximum distance between two particles from their velocity functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_42.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This was the most difficult question on the paper for almost all candidates; very few earned full marks.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the velocities to locate the maximum separation, then substitute that time into the displacement difference and report the distance." }, { "id": "cie-2019-s-43-q01", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "kinematics", "functions" ], "primaryTopicId": "kinematics", "summary": "Sketch the bus's velocity-time graph and find the distance between the stops.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the speed after five seconds of acceleration, draw the constant-speed section and final linear deceleration to zero, then add the areas of the three regions under the graph." }, { "id": "cie-2019-s-43-q02i", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 4, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the x- and y-components of the resultant of three coplanar forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force along the labelled axes using its direction, keeping the signs consistent, and add the three components on each axis." }, { "id": "cie-2019-s-43-q02ii", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the direction of the resultant force from its components.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the component values from part (i) in a tangent ratio, then choose the angle's quadrant from the signs of the x- and y-components." }, { "id": "cie-2019-s-43-q03i", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the car's speed when its engine supplies 30 kW and the car accelerates uphill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv to express the driving force, then apply Newton's second law with resistance and the uphill weight component included; solve for v." }, { "id": "cie-2019-s-43-q03ii", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the maximum engine power at the car's greatest steady uphill speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At constant speed balance the driving force with the resistance and the component of weight down the slope, then multiply the required driving force by 40 m s⁻¹." }, { "id": "cie-2019-s-43-q04i", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and string tension before the two particles reach the same height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton's second law to the two masses in opposite directions. The acceleration is set by the difference of their weights divided by their combined mass; substitute it to find tension." }, { "id": "cie-2019-s-43-q04ii", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the difference in the particles’ times to reach the stated position after the string breaks.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says part (ii) was challenging for many and lists errors in the post-break stages and time comparison.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the shared break height and speed, model both subsequent motions with the correct initial velocities and acceleration signs, and compare arrival times." }, { "id": "cie-2019-s-43-q05i", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find how far the particle travels up a smooth plane before stopping.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy: initial kinetic energy becomes gravitational potential energy, with vertical rise equal to the distance along the plane times sin 30°." }, { "id": "cie-2019-s-43-q05ii", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 8, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed after the particle travels up and then back down the rough plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates misinterpreted the two-stage motion and treated the return journey as though it covered the same distance as the ascent.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Model the two-stage journey separately, reverse the friction direction on descent, and combine the energy or force equations with correct signs." }, { "id": "cie-2019-s-43-q06i", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the displacement function and constants from the acceleration and two displacement conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration once to obtain velocity and again to obtain displacement, retaining a constant at each step. Substitute both given (t, s) pairs to solve for p and q." }, { "id": "cie-2019-s-43-q06ii", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the times when the particle is instantaneously at rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the displacement function to get velocity, set it equal to zero, and solve the resulting quadratic, retaining only valid times." }, { "id": "cie-2019-s-43-q06iii", "paperId": "cie-2019-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find total distance from the velocity function when the particle changes direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_43.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates frequently confused distance with displacement and integrated the signed velocity across the whole interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the zeros of velocity to split the time interval, integrate the speed magnitude on each section, and add the distances." }, { "id": "cie-2019-s-51-q01i", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the string tension has the stated value for the particle moving in a horizontal circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the tension vertically to balance the particle's weight, using the right triangle formed by the string and its vertical offset." }, { "id": "cie-2019-s-51-q01ii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle's speed in the horizontal circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the horizontal component of tension as the centripetal force, with the circle radius obtained from the string-length triangle." }, { "id": "cie-2019-s-51-q02", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the launch speed and angle from the particle's speed and direction four seconds later.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the velocity at four seconds into horizontal and vertical components. The horizontal component is unchanged; reverse the vertical motion equation to find the initial vertical component, then recover the launch speed and angle." }, { "id": "cie-2019-s-51-q03", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "data-analysis", "mechanics" ], "primaryTopicId": "data-analysis", "summary": "Find the centre of mass of a cylinder after a hemisphere is removed from one end.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat the removed hemisphere as negative mass. Take moments about A using the cylinder and hemisphere volumes and their centre positions, then divide the net moment by the remaining volume." }, { "id": "cie-2019-s-51-q04i", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the trajectory equation for a projectile launched at 30° with speed 25 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the initial velocity into horizontal and vertical components, write x(t) and y(t) under constant gravity, eliminate t, and state y as a function of x." }, { "id": "cie-2019-s-51-q04ii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate where the parametric curve has the specified tangent gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was more challenging for many candidates; the report describes the required implicit differentiation and tangent-gradient substitution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t, form dy/dx, set it equal to the stated tangent value, and solve for x." }, { "id": "cie-2019-s-51-q05i", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the initial extension of an elastic string when the particle released from rest reaches O and stops there.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of energy between the release point and O, accounting for the change in gravitational potential energy and the elastic energy stored at the start." }, { "id": "cie-2019-s-51-q05ii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed of the oscillating particle using the energy equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The part was identified as more challenging; candidates needed to recognise the equilibrium position and include all three energy terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise that maximum speed occurs at equilibrium and set up the three-term energy balance between the given position and equilibrium." }, { "id": "cie-2019-s-51-q06i", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "data-analysis", "coordinate-geometry" ], "primaryTopicId": "data-analysis", "summary": "State the centre-of-mass distances from the two perpendicular sides of the uniform triangular lamina.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the centroid location of a uniform right-angled triangle: each coordinate is one third of the corresponding perpendicular side measured from the right-angle vertex." }, { "id": "cie-2019-s-51-q06ii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle between side AB and the horizontal when the lamina hangs freely from B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass relative to B using the centroid distances, then use the right triangle formed by its horizontal and vertical offsets to obtain the orientation angle." }, { "id": "cie-2019-s-51-q06iii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Calculate the lamina's weight when the applied force holds AB vertical.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the suspension point B. Find the perpendicular moment arm of the 12 N force along AC, equate its moment to the weight's moment through the centroid, and solve for the weight." }, { "id": "cie-2019-s-51-q07i", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the differential relation between velocity and displacement before the elastic string becomes taut.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use acceleration = v dv/dx and apply Newton's second law while the string is slack, including gravity and the displacement-dependent resisting force with the stated sign convention." }, { "id": "cie-2019-s-51-q07ii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the upward velocity when the elastic string first becomes taut.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity-displacement equation from the launch point to the natural-length position, use the initial speed, and take the positive root for upward motion." }, { "id": "cie-2019-s-51-q07iii", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find an expression for the particle's upward acceleration after the elastic string becomes taut.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton's second law with upward positive, including gravity, the elastic tension based on extension beyond the natural length, and the opposing resistance." }, { "id": "cie-2019-s-51-q07iv", "paperId": "cie-2019-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iv)", "marks": 4, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Use the integrated force equation with correct limits to reach the final motion conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_51.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was challenging for most; very few used the correct limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the equation from part (iii), substitute the physical endpoints for the displacement, and interpret the resulting change in kinetic energy or speed." }, { "id": "cie-2019-s-52-q01i", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "State the coefficient of t² in the projectile's vertical displacement equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the given displacement expression with s = ut + ½at² for vertical acceleration −g, so the coefficient is g/2." }, { "id": "cie-2019-s-52-q01ii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Show that the projectile's initial speed is 28 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the initial vertical velocity from the coefficient of t, then divide by sin 30° to recover the launch speed." }, { "id": "cie-2019-s-52-q01iii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(iii)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the horizontal displacement after three seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the constant horizontal velocity, found from the launch speed and cos 30°, multiplied by three seconds." }, { "id": "cie-2019-s-52-q02", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "data-analysis", "coordinate-geometry" ], "primaryTopicId": "data-analysis", "summary": "Find the centre of mass of a square lamina with a smaller square removed from one corner.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the large square as positive area and the removed square as negative area. Take area-weighted moments of their centroids about A and divide by the remaining area." }, { "id": "cie-2019-s-52-q03i", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension for the particle moving on the smooth horizontal surface in a horizontal circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the circle radius from the string geometry and use the horizontal component of tension as m r ω², with the given angular speed." }, { "id": "cie-2019-s-52-q03ii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the normal force exerted by the surface on the particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical equilibrium: the surface reaction and the upward vertical component of the string tension balance the particle's weight." }, { "id": "cie-2019-s-52-q04i", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the velocity-displacement equation for the particle moving down while the elastic string stretches.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take downward displacement as positive, use a = v dv/dx, and apply Newton's second law with gravity, elastic tension and the opposing quadratic resistance." }, { "id": "cie-2019-s-52-q04ii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the kinetic and elastic potential energies when the particle has its greatest downward speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set acceleration to zero in the velocity-displacement relation to locate the speed maximum. Substitute that extension into the integrated energy relation and the elastic-energy formula." }, { "id": "cie-2019-s-52-q05i", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the natural length and modulus of the elastic string from two tension-extension conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use T = λe/a for each stated length, forming the extension as length minus natural length. Solve the two equations simultaneously." }, { "id": "cie-2019-s-52-q05ii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle's speed while moving in a horizontal circle on the stretched string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At steady circular speed the string tension supplies the centripetal force. Substitute the radius and the modulus and natural length from part (i), then solve for speed." }, { "id": "cie-2019-s-52-q06i", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Use horizontal and vertical velocity components to determine the launch angle from the resultant speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part proved challenging; candidates needed both velocity components and the correct resultant-speed equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find both velocity components at the stated time, set the sum of their squares equal to the given speed squared, and solve for the angle." }, { "id": "cie-2019-s-52-q06ii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the vertical displacement and explain its position relative to the projection level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few explained that the negative displacement meant the particle was below its projection level.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply s = ut + ½at² with the vertical sign convention, then interpret the sign of the displacement in context." }, { "id": "cie-2019-s-52-q07i", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Show the given radius of the semicircle from the wire object's centre-of-mass offset.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=11", "qpPages": [ 12, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the semicircular arc's centre offset from its diameter, combine it with the straight segment by length-weighted moments, and set the resulting centre-of-mass distance equal to 0.1 m." }, { "id": "cie-2019-s-52-q07ii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the required distance of the centre of mass by taking moments about A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=11", "qpPages": [ 12, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part was challenging for many and required taking moments about A after finding AC.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the needed geometric length first, take moments about A for the composite lamina, and solve for the centre-of-mass distance." }, { "id": "cie-2019-s-52-q07iii", "paperId": "cie-2019-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 6, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle determined by the composite semicircular lamina’s centre of mass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_52.pdf#page=11", "qpPages": [ 12, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found this part challenging; the report gives the moment equation and subsequent tangent relation needed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use moments about A to obtain the centre-of-mass coordinate, then relate it to the specified tangent ratio to find the angle." }, { "id": "cie-2019-s-53-q01i", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the string tension has the stated value for the particle moving in a horizontal circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the tension vertically to balance the particle's weight, using the right triangle formed by the string and its vertical offset." }, { "id": "cie-2019-s-53-q01ii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle's speed in the horizontal circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the horizontal component of tension as the centripetal force, with the circle radius obtained from the string-length triangle." }, { "id": "cie-2019-s-53-q02", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the launch speed and angle from the particle's speed and direction four seconds later.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the velocity at four seconds into horizontal and vertical components. The horizontal component is unchanged; reverse the vertical motion equation to find the initial vertical component, then recover the launch speed and angle." }, { "id": "cie-2019-s-53-q03", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "data-analysis", "mechanics" ], "primaryTopicId": "data-analysis", "summary": "Find the centre of mass of a cylinder after a hemisphere is removed from one end.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat the removed hemisphere as negative mass. Take moments about A using the cylinder and hemisphere volumes and their centre positions, then divide the net moment by the remaining volume." }, { "id": "cie-2019-s-53-q04i", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the trajectory equation for a projectile launched at 30° with speed 25 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the initial velocity into horizontal and vertical components, write x(t) and y(t) under constant gravity, eliminate t, and state y as a function of x." }, { "id": "cie-2019-s-53-q04ii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate where the parametric curve has the specified tangent gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was more challenging for many candidates; the report describes the required implicit differentiation and tangent-gradient substitution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t, form dy/dx, set it equal to the stated tangent value, and solve for x." }, { "id": "cie-2019-s-53-q05i", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the initial extension of an elastic string when the particle released from rest reaches O and stops there.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of energy between the release point and O, accounting for the change in gravitational potential energy and the elastic energy stored at the start." }, { "id": "cie-2019-s-53-q05ii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greatest speed of the oscillating particle using the energy equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The part was identified as more challenging; candidates needed to recognise the equilibrium position and include all three energy terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Recognise that maximum speed occurs at equilibrium and set up the three-term energy balance between the given position and equilibrium." }, { "id": "cie-2019-s-53-q06i", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "data-analysis", "coordinate-geometry" ], "primaryTopicId": "data-analysis", "summary": "State the centre-of-mass distances from the two perpendicular sides of the uniform triangular lamina.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the centroid location of a uniform right-angled triangle: each coordinate is one third of the corresponding perpendicular side measured from the right-angle vertex." }, { "id": "cie-2019-s-53-q06ii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle between side AB and the horizontal when the lamina hangs freely from B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass relative to B using the centroid distances, then use the right triangle formed by its horizontal and vertical offsets to obtain the orientation angle." }, { "id": "cie-2019-s-53-q06iii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Calculate the lamina's weight when the applied force holds AB vertical.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the suspension point B. Find the perpendicular moment arm of the 12 N force along AC, equate its moment to the weight's moment through the centroid, and solve for the weight." }, { "id": "cie-2019-s-53-q07i", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the differential relation between velocity and displacement before the elastic string becomes taut.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use acceleration = v dv/dx and apply Newton's second law while the string is slack, including gravity and the displacement-dependent resisting force with the stated sign convention." }, { "id": "cie-2019-s-53-q07ii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the upward velocity when the elastic string first becomes taut.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity-displacement equation from the launch point to the natural-length position, use the initial speed, and take the positive root for upward motion." }, { "id": "cie-2019-s-53-q07iii", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find an expression for the particle's upward acceleration after the elastic string becomes taut.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton's second law with upward positive, including gravity, the elastic tension based on extension beyond the natural length, and the opposing resistance." }, { "id": "cie-2019-s-53-q07iv", "paperId": "cie-2019-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 4, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Use the integrated force equation with correct limits to reach the final motion conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_53.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was challenging for most; very few used the correct limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the equation from part (iii), substitute the physical endpoints for the displacement, and interpret the resulting change in kinetic energy or speed." }, { "id": "cie-2019-s-61-q01i", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the two coded-data totals requested for the nine swimming times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Question 1 as challenging; candidates often ignored the stated instructions or failed to use the coded variance relation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate each observation’s deviation from 120, then add the deviations and their squares." }, { "id": "cie-2019-s-61-q01ii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the coded totals to calculate the variance of the swimming times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Question 1 as challenging; candidates often ignored the stated instructions or failed to use the coded variance relation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the variance formula for n = 9 to the deviation totals from part (i), accounting for the squared mean deviation." }, { "id": "cie-2019-s-61-q02", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the number of plums in Rosa's box from the probability of choosing plums from both boxes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the probability of a plum from each independent box, set the product equal to 1/4, and solve the resulting equation for x." }, { "id": "cie-2019-s-61-q03", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether the two events from the pair of die throws are independent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the equally likely ordered outcomes, count each event and their intersection, then compare P(X ∩ Y) with P(X)P(Y)." }, { "id": "cie-2019-s-61-q04", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare the central tendency and spread of the two examination-mark distributions using their cumulative-frequency graphs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=6", "qpPages": [ 6, 7 ], "msPages": [ 6, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read and compare the medians for central tendency and the interquartile ranges for spread, estimating each from the 25th, 50th and 75th percentiles." }, { "id": "cie-2019-s-61-q05i", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that more than 12 of 15 sampled children own a bicycle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the count as binomial with n = 15 and p = 0.65, then add the probabilities for 13, 14 and 15 owners." }, { "id": "cie-2019-s-61-q05ii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a suitable approximation for the probability that fewer than 179 of 250 children own a bicycle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Approximate the binomial count by a normal distribution with mean np and variance np(1−p), apply a continuity correction for X ≤ 178, and standardise." }, { "id": "cie-2019-s-61-q06i", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show the probability that Amy fails on both attempts and loses her original dollar.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=8", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Each attempt fails with probability 0.8; use independence to multiply the failure probabilities for the two attempts." }, { "id": "cie-2019-s-61-q06ii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Complete the probability distribution for Amy's net gain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the three outcomes: success on the first attempt, success on the second, or failure twice. Multiply the corresponding independent attempt probabilities and pair them with the stated net gains." }, { "id": "cie-2019-s-61-q06iii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate Amy’s expected gain from the probability distribution developed earlier.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A common difficulty was using the crucial bell probability correctly when evaluating expected gain.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Multiply each possible gain by its probability from the completed table, include the bell outcome correctly, and add the weighted values." }, { "id": "cie-2019-s-61-q07i", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the expected number of female giraffes below 700 kg.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=9", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise 700 using the stated normal mean and standard deviation, find the lower-tail probability, then multiply by the reserve population of 430." }, { "id": "cie-2019-s-61-q07ii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the upper boundary of a central 90% interval for the normal distribution from its standardised diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this part very challenging; candidates often did not recognise the central interval or interpret the z-value as the left-side area.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the tail area and corresponding z-value on a labelled standard normal curve, then solve the standardisation equation for the interval width." }, { "id": "cie-2019-s-61-q07iii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation of male giraffe weights from the proportion above 950 kg.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the upper-tail probability to a standard-normal z-value, then solve (950−1190)/σ = z for σ." }, { "id": "cie-2019-s-61-q08i", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(i)", "marks": 1, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count the ways to choose four toys from Freddie's nine different toys.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=10", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose four objects from nine without regard to order, using a binomial coefficient." }, { "id": "cie-2019-s-61-q08ii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count the four-toy selections that include both favourite toys.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=10", "qpPages": [ 14, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Fix the favourite car and bus in the selection, then choose the remaining two toys from the other seven." }, { "id": "cie-2019-s-61-q08iii", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count the arrangements in which the three buses are consecutive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat the three distinct buses as one block, arrange that block with the six distinct cars, and multiply by the internal arrangements of the buses." }, { "id": "cie-2019-s-61-q08iv", "paperId": "cie-2019-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iv)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements with a car at each end and no two buses adjacent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_61.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_61.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose and order the end cars, then arrange the remaining cars to form gaps. Select distinct gaps for the buses so no buses are adjacent, and account for all internal orders." }, { "id": "cie-2019-s-62-q01", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Use the dice outcomes to test whether the two defined events are independent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the context in Question 1 challenged many candidates; counting errors and circular reasoning were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Construct the complete equally likely outcome space, mark the outcomes in each event, and compare P(S∩T) with P(S)P(T)." }, { "id": "cie-2019-s-62-q02", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the expected monthly number of cartridges containing less than 28.9 ml of ink.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise 28.9 under the stated normal model, find the lower-tail probability for one cartridge, and multiply by eight cartridges per month." }, { "id": "cie-2019-s-62-q03i", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability Janice buys at most seven items in ten weeks.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the number of purchases as B(10, 0.35) and sum the probabilities for zero through seven purchases, or use the complementary upper tail." }, { "id": "cie-2019-s-62-q03ii", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the smallest number of weeks for which the chance of at least one purchase exceeds 0.99.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the complement: no purchase in n weeks has probability 0.65ⁿ. Solve 1−0.65ⁿ > 0.99 and check the least integer satisfying it." }, { "id": "cie-2019-s-62-q04", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of normally distributed male panda weights from two upper-tail probabilities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert each stated tail probability to a standard-normal quantile, form the two equations (121−μ)/σ and (102−μ)/σ, and solve simultaneously." }, { "id": "cie-2019-s-62-q05i", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Draw the tree diagram for two sweets drawn without replacement and update the box after the first draw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the context in Question 5 challenged many; candidates often treated the draws as if the original contents were unchanged.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Show each first-draw branch, update the contents before the second draw, and label the outcomes consistently with the context." }, { "id": "cie-2019-s-62-q05ii", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Form the distribution for the number of toffees drawn in the two-stage sweets experiment.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first-draw branches and the updated contents after three chocolates are added. Combine branch probabilities that give the same total number of toffees." }, { "id": "cie-2019-s-62-q05iii", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 1, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the expected number of toffees taken.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each possible toffee count in the distribution from part (ii) by its probability and add the products." }, { "id": "cie-2019-s-62-q05iv", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iv)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the first sweet was chocolate given that the second sweet was toffee.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conditional probability: calculate the joint probability of first chocolate and second toffee, then divide by the total probability that the second sweet is toffee." }, { "id": "cie-2019-s-62-q06i", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give one advantage and one disadvantage of a box-and-whisker plot.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State a useful feature such as showing median, quartiles and spread clearly, and a limitation such as not showing individual values or detailed distribution shape." }, { "id": "cie-2019-s-62-q06ii", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Choose the most suitable measure of central tendency for the marathon times and explain why the other two are less suitable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the median because the extreme 768-minute time would distort the mean; note that there is no repeated value to make the mode informative." }, { "id": "cie-2019-s-62-q06iiia", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a box-and-whisker plot for the 33 marathon times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Order the data, find the minimum, lower quartile, median, upper quartile and maximum using the course convention, then mark the five-number summary on the supplied scale." }, { "id": "cie-2019-s-62-q06iiib", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the interquartile range of the 33 marathon times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the quartiles from the ordered data and subtract the lower quartile from the upper quartile." }, { "id": "cie-2019-s-62-q07a", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count the ways to allocate six teenagers to the three distinct boats with capacities three, two and one.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose three teenagers for the 3-seat boat, choose two of the remaining three for the 2-seat boat, and place the last teenager in the 1-seat boat." }, { "id": "cie-2019-s-62-q07bi", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count seven-digit arrangements where the odd digits form one block and the even digits form another.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat the odd and even groups as two blocks, arrange the blocks, and count internal permutations while dividing for the repeated 2s and 7s." }, { "id": "cie-2019-s-62-q07bii", "paperId": "cie-2019-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 4, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count seven-digit arrangements in which the two 2s are not adjacent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_62.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract arrangements with the two 2s joined as a block from the total permutations of the multiset, accounting for the repeated 7s in both counts." }, { "id": "cie-2019-s-63-q01i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability a ferry crossing takes between 79 and 91 minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise both limits using the normal mean 85 and standard deviation 6.8, then subtract the two cumulative probabilities." }, { "id": "cie-2019-s-63-q01ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the time that 96% of ferry crossings exceed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the lower 4th percentile of the stated normal distribution: find its standard-normal quantile and convert back to minutes." }, { "id": "cie-2019-s-63-q02i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Draw a tree diagram for message type and whether an immediate reply arrives.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the three message-type probabilities on the first branches, complete the remaining type probability, and attach the matching conditional reply and no-reply probabilities to each branch." }, { "id": "cie-2019-s-63-q02ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Given that no immediate reply arrives, find the probability the message was an email.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Bayes' rule: multiply the email probability by its no-reply probability, then divide by the total no-reply probability across all message types." }, { "id": "cie-2019-s-63-q03i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the arrangements of the nine people while accounting for the two named people’s possible order.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A significant number gave 9! and forgot that the two named people could be arranged in two ways.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Arrange all nine people and include both orders of the two specified people." }, { "id": "cie-2019-s-63-q03ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count line-ups where the five Keene children stay together and the two Uzuma children stay together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat each sibling group as a block, arrange the two blocks with the four adults, then multiply by the internal permutations within both child blocks." }, { "id": "cie-2019-s-63-q04i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count six-person committees with at least twice as many men as women.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the feasible numbers of women (zero, one or two), multiply the corresponding choices of women and men, and add the case counts." }, { "id": "cie-2019-s-63-q04ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "Count committees where two specified men are not both selected.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract committees containing both specified men from all six-person committees, counting the remaining members from the other ten people." }, { "id": "cie-2019-s-63-q05i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that at most two of fourteen theatre-goers are men.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with n = 14 and p = 0.34, and sum the probabilities for zero, one and two men." }, { "id": "cie-2019-s-63-q05ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation with continuity correction to find the probability for the specified binomial event.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number forgot the continuity correction or used the wrong boundary and tail.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stated binomial mean and variance, apply the correct continuity boundary, standardise with the standard deviation, and choose the correct tail area." }, { "id": "cie-2019-s-63-q06i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Construct the probability distribution for the product of the two spinner scores.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the equally likely spinner-face pairs by their product, accounting for the three faces labelled 1 on the five-sided spinner, and list each distinct score with its probability." }, { "id": "cie-2019-s-63-q06ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the mean and variance of the spinner-product score.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probability table to calculate E(X) and E(X²), then evaluate Var(X) = E(X²)−[E(X)]²." }, { "id": "cie-2019-s-63-q06iii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the spinner-product score is greater than its mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the score values above the mean from part (ii) and add their probabilities from the distribution table." }, { "id": "cie-2019-s-63-q07i", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a back-to-back stem-and-leaf diagram for the two schools' race times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the tens digits as ordered stems, place each school's units digits on its required side, and provide a key." }, { "id": "cie-2019-s-63-q07ii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the interquartile range of Thaters School's race times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Order the 13 times, identify the lower and upper quartiles using the median-of-halves convention, and subtract Q1 from Q3." }, { "id": "cie-2019-s-63-q07iii", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Calculate the sum of squared deviations of Whitefay Park times from 60 minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each of the 13 times calculate (x−60)² and add these squared deviations." }, { "id": "cie-2019-s-63-q07iv", "paperId": "cie-2019-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iv)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the coded-data totals to calculate the variance of the times for Whitefay Park School.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_63.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes that candidates often ignored the given sum and previous result, and those errors prevented the final variance calculation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the given sum of deviations and your sum of squared deviations from part (iii) in the coded variance formula, dividing by the sample size and subtracting the square of the coded mean." }, { "id": "cie-2019-s-71-q01i", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Find the mean and standard deviation of the charge paid by one computer user.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the usage time by five cents per minute: the mean scales by five and the standard deviation scales by the absolute value of five." }, { "id": "cie-2019-s-71-q01ii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of the total charges from 15 independent users.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For a sum of 15 independent payments, multiply the single-user mean by 15 and the single-user standard deviation by √15." }, { "id": "cie-2019-s-71-q02i", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at the 1% level whether John's mean journey time has increased on the new route.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-tailed hypotheses, use the known standard deviation and n = 150 to standardise the sample mean, compare with the 1% critical value, and state the contextual conclusion. Include the required population-distribution assumption." }, { "id": "cie-2019-s-71-q02ii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Decide whether the Central Limit Theorem is needed for the journey-time test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that journey times are stated to have a normal distribution, so the sample mean is normal directly; the Central Limit Theorem is not needed." }, { "id": "cie-2019-s-71-q03i", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a binomial model to test whether the die shows a six less often than a fair die.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set H0: p = 1/6 and H1: p < 1/6. Under H0 model the count as B(30, 1/6), find the lower-tail probability for two or fewer sixes, compare it with 5%, and conclude in context." }, { "id": "cie-2019-s-71-q03ii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error in the repeated die test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the null model B(30, 1/6) and add the probabilities of the outcomes in the rejection region established by the 5% test." }, { "id": "cie-2019-s-71-q04ai", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the uniform random variable is below 0.5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many found this part demanding because the key information was shown in a diagram and they did not infer the density function.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "The density is 1/a over 0 to a. Integrate this constant density from 0 to 0.5." }, { "id": "cie-2019-s-71-q04aii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 1, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the mean of the uniform random variable on 0 to a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many found this part demanding because the key information was shown in a diagram and they did not infer the density function.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate x/a from 0 to a to obtain the expected value." }, { "id": "cie-2019-s-71-q04aiii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)(iii)", "marks": 2, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Show that the variance of the uniform random variable is a²/12.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many found this part demanding because the key information was shown in a diagram and they did not infer the density function.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate E(X²) by integration, use E(X)=a/2 from part (ii), and simplify Var(X)=E(X²)−[E(X)]²." }, { "id": "cie-2019-s-71-q04b", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the upper quartile of the stated continuous random variable T.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate its density from the lower support limit to b, set the cumulative probability to 3/4, and solve for b within the specified support." }, { "id": "cie-2019-s-71-q05ai", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a Poisson(2.3) variable is between 2 and 4 inclusive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the Poisson probabilities for 2, 3 and 4 using parameter 2.3." }, { "id": "cie-2019-s-71-q05aii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the sum of two independent Poisson(2.3) values exceeds 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the fact that independent Poisson variables add to a Poisson variable with parameter 4.6, then take one minus the probability of 0, 1 or 2." }, { "id": "cie-2019-s-71-q05aiii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(iii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability that the sum of 50 independent Poisson(2.3) values is at most 110.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The sum is Poisson with parameter 115. Approximate it by a normal distribution with mean and variance 115, apply a continuity correction at 110.5, and standardise." }, { "id": "cie-2019-s-71-q05b", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the Poisson parameter when the probabilities of values 3 and 5 are equal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate e⁻λ λ³/3! and e⁻λ λ⁵/5!, cancel the common exponential factor, and solve for the positive parameter." }, { "id": "cie-2019-s-71-q06i", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Explain why sampling only tennis-club members is not a random sample of adults in the town.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The club members are a restricted group and adults outside the club have no chance of selection, so the sample is not drawn from the whole town population." }, { "id": "cie-2019-s-71-q06ii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Describe a valid way to select a random sample of adults in the town.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a complete list or sampling frame of town adults, assign each person a number, and select 350 distinct numbers with a random-number method." }, { "id": "cie-2019-s-71-q06iii", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate an approximate 90% confidence interval for the proportion who rate the facilities as good.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p̂ = 47/350 and the normal confidence interval p̂ ± z₀.₉₅√(p̂(1−p̂)/350), then report both endpoints." }, { "id": "cie-2019-s-71-q06iv", "paperId": "cie-2019-s-71", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/71", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iv)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Complete the final inference using the confidence or testing result from the previous parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_71.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_71.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found part (iv) very challenging; some reached 0.980 but then used the wrong count.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the required standardised value, evaluate the probability, and connect it to the correct count or parameter in the question context." }, { "id": "cie-2019-s-72-q01i", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a Poisson(5) variable equals 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = 2 and λ = 5 into the Poisson probability formula e⁻λ λˣ/x!." }, { "id": "cie-2019-s-72-q01ii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 1, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Write an equation for equal adjacent probabilities in the Poisson(5) distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate e⁻⁵ 5ⁿ/n! to e⁻⁵ 5ⁿ⁺¹/(n+1)! and simplify by cancelling common factors." }, { "id": "cie-2019-s-72-q01iii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(iii)", "marks": 1, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find n when adjacent Poisson(5) probabilities are equal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the simplified equation from part (ii), solve for n, and verify it is an allowed integer value." }, { "id": "cie-2019-s-72-q02i", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Describe the distribution of the mean of a sample of 36 values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the sample mean has mean 372 and standard deviation 54/√36, and describe it as approximately normal by the Central Limit Theorem." }, { "id": "cie-2019-s-72-q02ii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Explain why the stated distribution is exact only under the required population assumption.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few gave a correct answer; vague claims that the sample mean “should be Normal” did not earn credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "State that the population variable itself must be normally distributed, and distinguish this condition from a central-limit approximation." }, { "id": "cie-2019-s-72-q03i", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates of the train journey population mean and variance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use x̄ = Σt/n for the mean and s² = [Σt²−(Σt)²/n]/(n−1) for the unbiased sample variance." }, { "id": "cie-2019-s-72-q03ii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test the claim that the mean train journey time is less than 1.9 hours at the 5% level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0: μ = 1.9 and H1: μ < 1.9. Use the sample mean and unbiased variance to form the one-sample t statistic with 49 degrees of freedom, compare with the 5% lower-tail critical value, and give a contextual conclusion." }, { "id": "cie-2019-s-72-q04", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that one of two independent plant heights is at least 1.5 times the other.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each direction form a difference such as X₁−1.5X₂, use independence to obtain its normal mean and variance, find the relevant tail probability, and add the two disjoint cases." }, { "id": "cie-2019-s-72-q05i", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State suitable hypotheses for testing whether the offer rate is below 10%.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Let p be the proportion of packets with an offer. State H0: p = 0.10 and the one-sided alternative H1: p < 0.10." }, { "id": "cie-2019-s-72-q05ii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the packet-offer test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Under H0 use X ~ B(40, 0.10). A Type I error occurs when X is below the stated rejection threshold, so calculate P(X ≤ 1)." }, { "id": "cie-2019-s-72-q05iii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate an approximate 90% confidence interval from six offers in 80 packets.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p̂ = 6/80 and the normal interval p̂ ± z₀.₉₅√(p̂(1−p̂)/80), and report the endpoints." }, { "id": "cie-2019-s-72-q05iv", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(iv)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Use the confidence interval to assess whether the packet-offer proportion could be 10%.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check whether 0.10 lies inside the interval from part (iii). If it does, the interval is consistent with the manufacturer's claim at the stated confidence level." }, { "id": "cie-2019-s-72-q06i", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Show the upper support endpoint b in terms of a for the given density proportional to a/x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate a/x² from 1 to b, set the total probability to one, and rearrange to obtain b = a/(a−1)." }, { "id": "cie-2019-s-72-q06ii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the density parameters using the stated median.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density to form its cumulative distribution, set F(3/2)=1/2, and solve together with b=a/(a−1) for a and b." }, { "id": "cie-2019-s-72-q06iii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the expected value of X using the parameters obtained from its median.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x f(x) over the support using the values of a and b from part (ii), then evaluate the exact logarithmic expression." }, { "id": "cie-2019-s-72-q07i", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 1, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Explain what 320 represents in the daily no-show model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "It is the number of passengers who have bought seats for the flight each day, each independently failing to arrive with probability 0.005." }, { "id": "cie-2019-s-72-q07ii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability that the total number of no-shows over two days is between 3 and 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the two independent daily binomial counts and use a Poisson approximation with parameter 2×320×0.005 = 3.2; add the probabilities for 3, 4 and 5." }, { "id": "cie-2019-s-72-q07iii", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Justify the Poisson approximation to the two-day no-show count.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The combined binomial model has a large number of trials and a small individual no-show probability, so a Poisson model with the same mean is suitable." }, { "id": "cie-2019-s-72-q07iv", "paperId": "cie-2019-s-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Complete the final test using the correct probability model and give a justified contextual conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_72.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 7 was found very challenging; part (iv) was poorly attempted and candidates often used an invalid normal model or omitted terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the stated binomial-to-Poisson setup consistently, include all required probability terms, compare with the critical value, and state a qualified conclusion in context." }, { "id": "cie-2019-s-73-q01", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate an approximate 98% confidence interval for the probability of heads from 60 heads in 100 tosses.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p̂ = 0.60 and z₀.₉₉ = 2.326 in the normal interval p̂ ± z√(p̂(1−p̂)/100), and state both bounds." }, { "id": "cie-2019-s-73-q02", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Find the sample mean length from the squared-value total and unbiased variance estimate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use s² = [Σx²−(Σx)²/50]/49 with s² = 9.62 and Σx² = 4361, solve for Σx, then divide by 50." }, { "id": "cie-2019-s-73-q03i", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Explain why the stated die method does not select A, B, C and D equally often.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "A, B and C each correspond to one die face, whereas D corresponds to three faces, so D has a larger selection probability." }, { "id": "cie-2019-s-73-q03ii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Describe how one die throw can make the four people equally likely to be chosen.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A significant number ignored the instruction to use a six-sided die and proposed a four-sided die instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Assign one face to each of A, B, C and D; scores 5 and 6 make no selection (reject those outcomes). This gives each person one equally likely selecting face." }, { "id": "cie-2019-s-73-q03iii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(iii)", "marks": 2, "topicIds": [ "probability", "probability" ], "primaryTopicId": "probability", "summary": "List the six possible pairs from four people and assign a die outcome to each pair.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List AB, AC, AD, BC, BD and CD, then map the six distinct faces of the fair die one-to-one to those six pairs." }, { "id": "cie-2019-s-73-q04", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the combined mass of a box, its large bricks and its small bricks is below 1200 g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the independent normal masses: use means 8 + 10×60 + 20×30 and variances 10×1.2² + 20×0.7². Standardise 1200 using the resulting normal model." }, { "id": "cie-2019-s-73-q05", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Test the stated mean using the sample, then find the maximum significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a minority scored full marks; a significant number used the sample variance or omitted the square-root sample-size factor.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the known variance and sample size to standardise the sample mean, compare against the relevant two-tail probability, and compute twice the one-tail area." }, { "id": "cie-2019-s-73-q06i", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Show that the given function is a probability density for every positive a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check that the function is non-negative on its support and integrate 3x²/a³ from 0 to a to show the total area is one." }, { "id": "cie-2019-s-73-q06ii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Show that the median condition gives a = 2.52 to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the cumulative probability at x = 2 equal to 1/2. This gives 8/a³ = 1/2; solve for the positive a and round to three significant figures." }, { "id": "cie-2019-s-73-q06iii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the expected value of X using the density and parameter from the earlier parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the integral of x·f(x) over 0≤x≤a, then substitute the value of a found from the median condition." }, { "id": "cie-2019-s-73-q07i", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability of at least three missed morning appointments.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a Poisson approximation to B(70, 0.04) with mean 2.8 and calculate 1−P(X≤2)." }, { "id": "cie-2019-s-73-q07ii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability of exactly six missed appointments over one morning and afternoon.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Approximate the morning and afternoon counts by independent Poisson variables with means 2.8 and 3.0. Their sum is Poisson(5.8); evaluate its probability at six." }, { "id": "cie-2019-s-73-q07iii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability of more than 50 missed appointments over ten days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The ten-day total is approximately normal with mean and variance 58. Apply a continuity correction at 50.5, standardise, and find the upper-tail probability." }, { "id": "cie-2019-s-73-q08i", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the spinner's probability of landing on A exceeds one quarter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0: p = 0.25 and H1: p > 0.25. Under H0 use B(10, 0.25), calculate the upper-tail probability for six or more A outcomes, compare with 1%, and conclude." }, { "id": "cie-2019-s-73-q08ii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the Type I error probability for the later spinner test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the null model B(10, 0.25) and add the probabilities over the critical region for the 1% upper-tail test." }, { "id": "cie-2019-s-73-q08iii", "paperId": "cie-2019-s-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "May/June", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Calculate the type II error probability under the alternative binomial model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_qp_73.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_ms_73.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the progressive difficulty of the full significance-test question caused problems for some candidates, including this type II error part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s19_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the complement of the critical region under the alternative parameter and evaluate the cumulative binomial probability through the stated cutoff." }, { "id": "ial-2019-10-WMA11-01-q01a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact radius of the sector from its area and angle.", "answerGuide": "Use the sector-area formula and solve for the positive radius.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q01b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact perimeter of the sector in simplest form.", "answerGuide": "Find the arc length using the radius and angle, then add the two radii.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q02a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Determine a and b in the linear model for the tree's height.", "answerGuide": "Substitute the two observed heights and times, then solve the simultaneous equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q02b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "State the modelled height of the tree when it was planted.", "answerGuide": "Evaluate H at t = 0, so the height is the intercept b from part (a).", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q03a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Find the coordinates of the minimum point of the quadratic curve.", "answerGuide": "Complete the square or use the vertex formula to find the minimum coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q03b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of the second intersection of the curve with the line through O and M.", "answerGuide": "Find the line through the origin and M, substitute it into the curve, and solve for the other intersection.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson's October 2019 PEF (p. 4) says part (b) proved a discriminator, with many candidates unable to get started; the report notes that failing to use the line's passage through the origin commonly stopped progress.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { 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Keep the boundary conventions consistent with the diagram.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson's October 2019 PEF (pp. 3-4) says only the most able candidates gave all the required inequalities; the upper x bound was commonly omitted and many mixed strict and inclusive boundaries.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2019-10-WMA11-01-q04a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use a parallelogram's side lengths and area to determine its obtuse interior angle.", "answerGuide": "Set area = ab sin(theta) using the two adjacent sides. Solve for the acute reference angle, then choose its obtuse supplement for the interior angle shown.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson's October 2019 PEF (p. 4) reports that about a third of candidates earned no marks on Question 4 and identifies choosing the wrong acute angle from the diagram as a common cause.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-10-WMA11-01-q04b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the length of diagonal BD to one decimal place.", "answerGuide": "Apply the cosine rule to the triangle with sides 10 and 6 and included angle DAB.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q05a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the curve and simplify dy/dx.", "answerGuide": "Apply the power rule to each term and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q05b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the normal to the curve at P in the required integer-coefficient form.", "answerGuide": "Evaluate the derivative at P, take the negative reciprocal, and use point-slope form.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q06a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch y = k + 4/x and state its horizontal asymptote and x-intercept.", "answerGuide": "Translate the 4/x graph vertically by k; identify y = k and solve k + 4/x = 0.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q06b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the positive values of k for which the line is tangent to the curve.", "answerGuide": "Equate the line and curve to form a quadratic in x, require a zero discriminant, and solve for k.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q07a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the tangent at P.", "answerGuide": "Differentiate y = 2x² + 5 and evaluate the derivative at x = 2.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q07b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of chord PQ in terms of h.", "answerGuide": "Find y at x = 2 + h, then calculate the difference quotient between Q and P.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q07c", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Explain how a chord gradient approaches the tangent gradient as the horizontal separation tends to zero.", "answerGuide": "Use the chord-gradient expression from part (b), let h tend to zero, and state that its limit equals the tangent gradient found in part (a).", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson's October 2019 PEF (p. 5) says the limit mark was rarely awarded; many candidates did not connect the chord gradient with the tangent gradient.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-10-WMA11-01-q08", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an equation involving a variable and its square root, giving exact roots.", "answerGuide": "Set u = sqrt(x), solve the quadratic u^2 - 6u + 4 = 0 to obtain u = 3 +/- sqrt(5), then square both roots to give x = 14 +/- 6 sqrt(5).", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=14", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson's October 2019 PEF (p. 6) groups Question 8 with the paper's least well-answered questions: nearly half of responses were blank or earned no marks, and only about a third earned full credit.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-10-WMA11-01-q09a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "State the period of a sine curve whose input is scaled by a factor of one twelfth.", "answerGuide": "For sin(x/12), divide the base period 2 pi by 1/12 to obtain 24 pi.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson's October 2019 PEF (p. 6) says many candidates did not understand the period, often giving an interval or a value based on an incorrect period.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-10-WMA11-01-q09b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "State the coordinates of the minimum point M.", "answerGuide": "Use the period from part (a) to locate the sine minimum and its y-value.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q09ci", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express the negative solution closest to zero using α.", "answerGuide": "Use sine symmetry and the 24π period to reflect the least positive solution across zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 20, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson's October 2019 PEF (p. 6) reports that the linked sine and cosine transformations in part (c) were unusually difficult across all grades; common answers did not account for the full period or the cosine phase.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-10-WMA11-01-q09cii", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-10-WMA11-01-q10a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Deduce the x-values for which the factored cubic function is non-positive.", "answerGuide": "Use the sign of the linear factor and the repeated quadratic factor to identify the solution set.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q10b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Expand f(x) into a cubic polynomial.", "answerGuide": "Multiply the squared linear factor by the remaining linear factor and collect like terms.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q10ci", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)(i)", "marks": 1, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of the y-axis intercept P.", "answerGuide": "Set x = 0 in the expanded function.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q10cii", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)(ii)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the curve at P.", "answerGuide": "Differentiate the polynomial and evaluate at x = 0.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q10di", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(d)(i)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the factorised expression for the curve translated two units to the right.", "answerGuide": "Replace x by x − 2 in f(x) and simplify in factorised form.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q10dii", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(d)(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the y-intercept of the translated curve.", "answerGuide": "Evaluate g(0) using the factorised expression from part (d)(i).", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q11a", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at P using the given gradient.", "answerGuide": "Use the point P and gradient 5 in point-slope form, then simplify to y = mx + c.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=17", "qpPages": [ 26, 28 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA11-01-q11b", "paperId": "ial-2019-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 8, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find f(x) from the given second derivative and conditions at P.", "answerGuide": "Integrate f″ to obtain f′ and use the gradient condition; integrate again and use the point condition to determine both constants.", "qpUrl": "https://www.alevelcopilot.com/whfile/35/f5/af/47/e8/06/47/90/0a/ef/99/52/d6/0e/66/46.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/99/55/ed/b7/ee/27/d0/10/b7/55/b8/f1/d2/58/78/e1.pdf#page=17", "qpPages": [ 26, 28 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q01a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinates of the stationary points of the curve.", "answerGuide": "Differentiate the cubic and solve dy/dx = 0.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q01b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the intervals where the curve is increasing.", "answerGuide": "Use the sign of the derivative on the intervals separated by the stationary points.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q02a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the predicted adult population at the start of 2032.", "answerGuide": "Use the geometric growth model with 2% annual growth and the correct number of yearly intervals.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=10", "qpPages": [ 4, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q02b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the total charity contributions from 2019 to 2032 inclusive.", "answerGuide": "Sum the geometric sequence of annual donations over all 14 years.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=10", "qpPages": [ 4, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q03a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first four terms of the binomial expansion.", "answerGuide": "Use the binomial theorem through x³ and simplify each coefficient.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q03b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the constant term in a product containing negative powers of x and a binomial expansion.", "answerGuide": "Rewrite the prefactor as x^-3 + 8x^-5, then combine the x^3 and x^5 coefficients from the binomial expansion; their sum is 77/8.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson's October 2019 PEF (p. 4) calls Question 3 one of the most challenging questions and says the constant term was rarely found because very few candidates included the additional term required from the binomial expansion.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-10-WMA12-01-q04a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State the remainder when f(x) is divided by x − 3.", "answerGuide": "Apply the remainder theorem by evaluating f(3).", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=14", "qpPages": [ 8, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q04b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Fully factorise f(x) using algebra.", "answerGuide": "Substitute a from part (b), factor out the given linear factor, then factorise the remaining quadratic.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=14", "qpPages": [ 8, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q05a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Sketch the curve for 0 < a < 1 and mark its y-intercept.", "answerGuide": "Draw the decreasing exponential in quadrants I and II with y-intercept (0,1).", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=16", "qpPages": [ 12, 15 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q05b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the trapezium rule and all table values to estimate the integral.", "answerGuide": "Use width 0.5 and the tabulated ordinates in the trapezium-rule sum.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=16", "qpPages": [ 12, 15 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q05c", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the estimate from part (b) to estimate the related integral.", "answerGuide": "Separate the cubic term, integrate it exactly, and combine it with the estimated integral from part (b).", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=16", "qpPages": [ 12, 15 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson's October 2019 PEF (p. 5) says the final part challenged most candidates; only stronger candidates recognized that they should integrate the extra term exactly and subtract it from the estimate in part (b).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2019-10-WMA12-01-q06a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the radius line PN.", "answerGuide": "Use the fact that a radius is perpendicular to the tangent at P, then pass the line through the circle's centre.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=18", "qpPages": [ 16, 19 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q06b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle.", "answerGuide": "Find the centre-to-tangency distance for the radius, then use the centre and radius in the circle equation.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=18", "qpPages": [ 16, 19 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q07i", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Simplify the first logarithmic expression in terms of k.", "answerGuide": "Apply the quotient log law and substitute log_a b = k.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=20", "qpPages": [ 20, 21 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q07ii", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Simplify a ratio of logarithmic expressions given log_a(b) = k.", "answerGuide": "Use log laws to rewrite the numerator as 2 + k and the denominator as 3k, giving (2 + k)/(3k).", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=20", "qpPages": [ 20, 21 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson's October 2019 PEF (p. 6) describes this logarithm-law part as more challenging; a significant number of candidates misapplied the log rules or made simplification errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-10-WMA12-01-q07iii", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Evaluate a finite sum combining a logarithm of an integer power with a constant.", "answerGuide": "Replace log_a(b^n) with nk, so the summand is (n+1)k. Sum the arithmetic sequence 2k through 51k to get 1325k.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=20", "qpPages": [ 20, 21 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson's October 2019 PEF (p. 6) says candidates often wrote initial terms but struggled to separate the logarithms from the arithmetic sum; omitting the factor k cost accuracy marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2019-10-WMA12-01-q08i", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact area of the shaded region.", "answerGuide": "Integrate the function term by term and evaluate it between the given limits.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=22", "qpPages": [ 22, 25 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q08ii", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find k from the given definite-integral condition.", "answerGuide": "Integrate the constant term with the other terms, apply the limits, and solve the resulting equation for k.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=22", "qpPages": [ 22, 25 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q09i", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the sine equation for θ in the specified degree interval.", "answerGuide": "Isolate the sine term, find both principal-angle families, and retain only solutions in the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=24", "qpPages": [ 26, 29 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q09iia", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)(a)", "marks": 3, "topicIds": [ "trigonometry", "sequences" ], "primaryTopicId": "trigonometry", "summary": "Use three trigonometric expressions in an arithmetic sequence to prove a relation between sine and cosine.", "answerGuide": "Set twice the middle term equal to the sum of the first and third terms. Replace 1/tan(alpha) with cos(alpha)/sin(alpha), then multiply through by sin(alpha) to reach the required identity.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=24", "qpPages": [ 26, 29 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson's October 2019 PEF (p. 7) says most candidates struggled to score marks on the part combining trigonometry with an arithmetic sequence.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2019-10-WMA12-01-q09iib", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation derived from the sequence condition and select the angle in a stated interval.", "answerGuide": "Use sin^2(alpha) + cos^2(alpha) = 1 to form a quadratic in cos(alpha), solve it, and select the root that places alpha between pi and 2 pi; alpha is approximately 5.517 radians.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=24", "qpPages": [ 26, 29 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson's October 2019 PEF (p. 7) reports that most candidates earned the early method marks but only a small number found the angle satisfying the specified range.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2019-10-WMA12-01-q10a", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "algebra", "differentiation" ], "primaryTopicId": "algebra", "summary": "Find the cubic's coefficients from the given point and gradient.", "answerGuide": "Substitute the point into the curve and differentiate to use its gradient condition, then solve the two equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=26", "qpPages": [ 30, 32 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-10-WMA12-01-q10b", "paperId": "ial-2019-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "proof", "differentiation" ], "primaryTopicId": "proof", "summary": "Prove the cubic curve has no turning points after using point and gradient information to determine its coefficients.", "answerGuide": "Differentiate and substitute the coefficient found in part (a), giving 4x^2 - 6x + 3. Its discriminant is -12, so the derivative has no real roots and the curve has no turning points.", "qpUrl": "https://www.alevelcopilot.com/whfile/3e/5a/97/4f/8c/9e/24/75/28/70/c9/0f/9f/1c/2f/db.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/08/5f/c8/fc/b8/46/d4/c8/4b/8b/1a/63/c9/b0/19/f9.pdf#page=26", "qpPages": [ 30, 32 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson's October 2019 PEF (p. 7) says many candidates stopped after setting the derivative to zero or asserted there were no solutions without justification; the discriminant argument was needed for full credit.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20200123.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "cie-2020-s-32-q01", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder when the quartic is divided by the quadratic.", "answerGuide": "Use polynomial long division, then state the quotient and the linear remainder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q02", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Use two points on the graph of ln y against x to find A and k in y² = Aeᵏˣ.", "answerGuide": "Take logarithms so that 2 ln y = ln A + kx, use the gradient to find k and then solve for A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q03", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral involving x^(3/2) ln x.", "answerGuide": "Use integration by parts twice as needed, simplify the antiderivative, and apply the limits 1 and 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q04", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of y = cos x sin 2x in the given interval.", "answerGuide": "Differentiate by the product rule, use a double-angle identity to reduce the stationary condition to one trigonometric equation, then select the root in the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q05a", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express the linear combination of cos x and sin x in the form R cos(x + α).", "answerGuide": "Match coefficients using R cos α and R sin α, then give exact R and α to the requested accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q05b", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the result from part (a) to solve the trigonometric equation over the stated degree interval.", "answerGuide": "Convert the equation to a single cosine equation using R and α, then list all solutions in the open interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q06a", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the maximum point of the rational curve.", "answerGuide": "Differentiate by the quotient rule, set the derivative to zero and select the non-negative stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q06b", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the stated substitution to find the exact area under the curve from x = 0 to x = 1.", "answerGuide": "Substitute u = √3 x², transform dx and the limits, then integrate the resulting arctangent form exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q07", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 9, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Solve the separable differential equation with the stated initial condition.", "answerGuide": "Separate the y and x terms, use partial fractions on the rational x expression, integrate, apply the initial condition, and rearrange for y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q08a", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the equation involving a complex number and its conjugate in Cartesian form.", "answerGuide": "Write the unknown as x + iy, substitute it and its conjugate, equate real and imaginary parts, and solve the simultaneous equations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q08bi", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 4, "topicIds": [ "geometry", "coordinate-geometry" ], "primaryTopicId": "geometry", "summary": "Sketch and shade the Argand-plane region defined by a circle and an argument half-plane.", "answerGuide": "Plot the circle from its modulus condition, draw the directed half-line from the argument condition and shade their intersection.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q08bii", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 2, "topicIds": [ "geometry", "coordinate-geometry" ], "primaryTopicId": "geometry", "summary": "Find the least imaginary part of a point in the shaded region.", "answerGuide": "Locate the lowest point of the circle that lies in the permitted half-plane and evaluate its imaginary coordinate exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q09a", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Show that the point where the two curves touch satisfies the stated tangent equation.", "answerGuide": "Use equality of the curve values to relate k and p, differentiate both curves, equate gradients at the touching point and eliminate k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q09b", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use the given iteration to find p to 3 decimal places.", "answerGuide": "Iterate from a suitable starting value, show each value to 5 decimal places and demonstrate convergence to the rounded root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q09c", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use p to find the constant k to 2 decimal places.", "answerGuide": "Substitute the rounded root into either curve equation and calculate k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=12", "qpPages": [ 15, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q10a", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find a vector equation for the line through the midpoint M and the point N that divides AB in the stated ratio.", "answerGuide": "Find position vectors for M and N, then use one point vector and the direction MN as the line equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q10b", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "vectors", "algebra" ], "primaryTopicId": "vectors", "summary": "Find the position vector of the intersection of MN with OB.", "answerGuide": "Parametrise both lines, equate components and solve for the line parameters and the intersection position vector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=13", "qpPages": [ 17, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-32-q10c", "paperId": "cie-2020-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "vectors", "trigonometry" ], "primaryTopicId": "vectors", "summary": "Calculate angle OPM in degrees.", "answerGuide": "Form vectors PO and PM, use their scalar product and magnitudes in the cosine formula, then evaluate the angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_32.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_32.pdf#page=13", "qpPages": [ 17, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q01", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the inequality involving two absolute-value expressions.", "answerGuide": "Find the critical points where the expressions change sign, solve on the resulting intervals and combine the values satisfying the strict inequality.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q02", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral of a linear factor times a decaying exponential.", "answerGuide": "Integrate by parts, simplify the antiderivative and apply the limits 0 and 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q03a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Rewrite the logarithmic equation as a quadratic equation in eˣ.", "answerGuide": "Exponentiate carefully, set u = eˣ and rearrange to a quadratic in u.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q03b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation to 3 decimal places.", "answerGuide": "Solve the quadratic in u = eˣ, retain the positive root, then take its natural logarithm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q04a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Differentiate y = x arctan(x/2).", "answerGuide": "Apply the product rule and the chain rule to the inverse-tangent factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q04b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the y-intercept of the tangent at x = 2.", "answerGuide": "Evaluate the curve and gradient at x = 2, form the tangent line and set x = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q05", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Convert the trigonometric equation to a quadratic in tan θ and solve over the stated interval.", "answerGuide": "Use the double-angle identity for cot 2θ and the tangent addition formula, substitute tan 45° = 1, solve the quadratic and select solutions in the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q06a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "functions", "graphs" ], "primaryTopicId": "functions", "summary": "Show graphically that the equation x⁵ = x + 2 has exactly one real root.", "answerGuide": "Sketch y = x⁵ and y = x + 2 with their end behaviour and show that they intersect once.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q06b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Show that any convergent sequence from the given iteration converges to the root from part (a).", "answerGuide": "Set the iteration’s fixed-point value equal to x and rearrange it to the equation x⁵ = x + 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q06c", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration and starting value to find the root to 3 decimal places.", "answerGuide": "Iterate from x₁ = 1.5, show values to 5 decimal places and provide enough terms to justify the rounded root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q07a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the rational function in partial fractions.", "answerGuide": "Resolve the denominator into linear factors, equate coefficients and find the constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q07b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the decomposition to show the stated identity for the square of the function.", "answerGuide": "Square the partial-fraction expression and simplify it to the required terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q07c", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the identity from part (b) to evaluate the exact definite integral.", "answerGuide": "Integrate each rational term, apply the limits 1 and 2, and simplify to the stated exact logarithmic form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q08a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find the fourth vertex of the parallelogram and verify it is not a rhombus.", "answerGuide": "Use vector addition to obtain the missing vertex, then compare the lengths of adjacent sides.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q08b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "vectors", "trigonometry" ], "primaryTopicId": "vectors", "summary": "Find angle BAD in the parallelogram.", "answerGuide": "Form vectors AB and AD and use their scalar product and magnitudes in the cosine formula.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q08c", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find the area of the parallelogram to 3 significant figures.", "answerGuide": "Use the product of adjacent side lengths and the sine of angle BAD, or an equivalent cross-product method.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=14", "qpPages": [ 13, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q09a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find both complex numbers from their difference and product.", "answerGuide": "Eliminate one unknown to obtain a quadratic for the other, solve it and use the difference relation to recover its partner.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q09b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "geometry", "coordinate-geometry" ], "primaryTopicId": "geometry", "summary": "Sketch and shade the Argand-plane region defined by a disk, argument bounds and a real-part boundary.", "answerGuide": "Plot the circle, the two argument rays and the vertical line Re z = 3; shade the intersection satisfying all inequalities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=14", "qpPages": [ 15, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q10a", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation", "mechanics" ], "primaryTopicId": "differentiation", "summary": "Use the outflow model and volume formula to derive the differential equation for h and t.", "answerGuide": "Differentiate V with respect to h, use dV/dt proportional to −√h and apply the chain rule to obtain the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-33-q10b", "paperId": "cie-2020-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 8, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the differential equation and express t in terms of h and r.", "answerGuide": "Separate variables and integrate, use t = 0 when h = r to find the constant, then use the emptying condition t = 14 at h = 0 to determine the positive parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_33.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_33.pdf#page=15", "qpPages": [ 17, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q01", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the resultant of three coplanar forces and state its direction.", "answerGuide": "Resolve the two equal angled forces into components using the given cosine, combine them with the 100 N force and state the resultant direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q02a", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tow-bar tension while the car and trailer accelerate.", "answerGuide": "Apply Newton’s second law to the trailer alone, including its resistance force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q02b", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the engine power at the stated speed.", "answerGuide": "Find the driving force needed for the combined car–trailer system, then use power = force × speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q03a", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the greatest height above ground reached by a vertically projected particle.", "answerGuide": "Use v² = u² + 2as at the top of the motion, then add the initial height above ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q03b", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find how long the particle is above the stated height.", "answerGuide": "Use vertical displacement to form a quadratic in time, find the two crossing times and subtract them.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q04a", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest applied force for which the ring remains at rest on the rough rod.", "answerGuide": "Resolve forces along and perpendicular to the rod, set friction at its limiting value μR and solve for T.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q04b", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the ring’s acceleration when the applied force is 3 N.", "answerGuide": "Find the normal reaction and friction force, then apply Newton’s second law along the rod.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q05a", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the child’s speed halfway along the arc from the release point to the bottom, with no resistance.", "answerGuide": "Calculate the loss of gravitational potential energy to the halfway position and equate it to the gain in kinetic energy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q05b", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the work done against resistance from the initial position to the lowest point.", "answerGuide": "Use the work–energy equation with the given bottom speed, initial potential energy and initial kinetic energy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q06a", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 7, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find k and the displacement function from the given velocity and two displacement values.", "answerGuide": "Integrate v = k(t² − 10t + 21), include the integration constant, substitute the two time–displacement pairs to solve for k and the constant, then state s(t).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q06b", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the displacement when the particle’s velocity is at its minimum.", "answerGuide": "Differentiate the velocity to locate its minimum, then substitute that time into the displacement function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q07a", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find Q’s speed after P slides down a smooth incline and collides with Q.", "answerGuide": "Find P’s acceleration and impact speed from the incline motion, then apply conservation of momentum to the collision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-41-q07b", "paperId": "cie-2020-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction when P and Q coalesce on the rough horizontal plane.", "answerGuide": "Use momentum to find their common speed just after impact, then use work–energy over the given distance to solve for the friction coefficient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_41.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_41.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q01a", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity–time graph and state the total time in motion.", "answerGuide": "Draw the accelerating, constant-speed and decelerating sections with the final slope twice as steep; use the equal-velocity changes to find the total duration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q01b", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the tram’s constant speed V from its total distance.", "answerGuide": "Equate the area under the velocity–time graph to 2775 m and solve for V.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q01c", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the magnitude of the tram’s acceleration.", "answerGuide": "Use the change from rest to V over 20 seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q02", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 6, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find P and the angle for the system of coplanar forces in equilibrium.", "answerGuide": "Resolve the forces into two perpendicular directions, solve the resulting equations for P and the angle, and select the physically valid values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q03", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 8, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the greatest and least magnitudes of the force that holds the particle in equilibrium on the rough incline.", "answerGuide": "Resolve parallel and perpendicular to the plane, set friction to its limiting value in both possible directions and solve for the two values of T.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q04a", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find B’s speed after its collision with A.", "answerGuide": "Use conservation of momentum and the stated relation between the final speeds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q04b", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show that A and B collide again after B collides with C.", "answerGuide": "Use conservation of momentum for the B–C collision and compare the resulting speeds of A and B to establish their later meeting.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q04c", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the total kinetic-energy loss over the three collisions.", "answerGuide": "Track the speeds through each collision, use momentum for the final coalescence and subtract final kinetic energy from the initial total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q05ai", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Calculate the engine power at the stated constant speed on a level road.", "answerGuide": "Find the driving force balancing the resistance and multiply by the speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q05aii", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the instantaneous deceleration after the engine power is reduced.", "answerGuide": "Convert the power reduction to a change in driving force at the current speed, then apply Newton’s second law.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q05b", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the constant speed uphill when the engine supplies the stated power and resistance depends on speed.", "answerGuide": "Balance engine power against the total resistance and the component of weight down the slope, then solve for the positive speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q06a", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity–time graph for the piecewise-defined motion.", "answerGuide": "Plot each formula over its stated interval, joining the segments at t = 5 and t = 7 with the correct endpoint values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q06b", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration at t = 6.", "answerGuide": "Differentiate the applicable velocity branch for 5 ≤ t ≤ 7 and substitute t = 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-42-q06c", "paperId": "cie-2020-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled over the full time interval.", "answerGuide": "Integrate the absolute velocity over each piece, splitting further at any zero of v, and add the distances.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_42.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_42.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q01", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use momentum conservation to find the mass of P.", "answerGuide": "Take the initial direction as positive and equate total momentum before and after the collision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q02a", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the minibus driving force at the stated acceleration.", "answerGuide": "Use Newton’s second law with the opposing resistance force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q02b", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the power needed to maintain the stated constant speed.", "answerGuide": "At constant speed, the driving force balances resistance; multiply by speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q03", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction angle of the balancing force.", "answerGuide": "Resolve the four forces horizontally and vertically, then combine the resultant components to obtain F and the angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q04a", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity–time graph for the car’s motion.", "answerGuide": "Draw a rising straight segment, a horizontal constant-velocity segment and a falling segment with twice the first slope magnitude.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q04b", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration a from the distance and constant-speed section.", "answerGuide": "Find the constant speed from 500 m in 25 s, then use v² = u² + 2as for the initial acceleration phase.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q04c", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total time for which the car is moving.", "answerGuide": "Find the acceleration and deceleration times from the speeds and accelerations, then add the constant-speed duration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q05a", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the decrease in kinetic energy as the block moves between P and Q.", "answerGuide": "Calculate the initial and final kinetic energies and subtract.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q05b", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the work done by the force pushing the block up the incline.", "answerGuide": "Add the increase in gravitational potential energy to the change in kinetic energy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q05c", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find how long the block takes to return to P after the pushing force is removed at Q.", "answerGuide": "Find its downhill acceleration, then use the displacement equation from rest at Q over the 10 m incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q06a", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the velocity when the particle’s acceleration is zero.", "answerGuide": "Differentiate the velocity to find acceleration, set it to zero and substitute the resulting time into v(t).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q06b", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the distance PQ until the particle first comes to rest at Q.", "answerGuide": "Solve v(t) = 0 for the positive stopping time, integrate the velocity from t = 0 to that time, and evaluate the displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q07a", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the plane angle that keeps the connected particles at rest when the plane is smooth.", "answerGuide": "Set the hanging particle’s tension equal to its weight and resolve the same tension along the plane for the heavier particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q07b", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Show the coefficient of friction for the stated rough-plane motion.", "answerGuide": "Resolve each particle’s forces along its direction of motion, use the shared acceleration and friction law, then solve for μ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-43-q07c", "paperId": "cie-2020-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find how long A continues moving up the plane after B reaches the floor.", "answerGuide": "Find A’s speed at that instant from its earlier motion, then use its post-release deceleration to calculate the time until it stops.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_43.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_43.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q01a", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Probability: Show that the probability of a total score of 4 is 1/12.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "There are 3 outcomes totaling 4 among the 36 equally likely ordered outcomes when two dice are thrown.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q01b", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the mean of X, the number of throws until a score of 4 first occurs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "X is geometric with success probability 1/12, so E(X)=1/(1/12)=12.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q01c", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the probability that a score of 4 is first obtained on the 6th throw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 6 to check the method and final answer for 1(c).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q01d", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find P(X < 8), where X is the number of throws until a score of 4 first occurs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric model: P(X<8)=1-(11/12)^7.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q02a", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the three Es are together and the two Ls are together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 7 to check the method and final answer for 2(a).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q02b", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the number of different arrangements that can be made from the 9 letters of the word JEWELLERY in which the two Ls are not next to each other.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Total number: → 3! in denominator → = 30 240 – 6720", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q03a", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Draw up the probability distribution table for the number of jellies that Jemeel chooses.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: P(0) = 3 → All probabilities correct → P(3) = 5", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q03b", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the probability that no more than 7 of these boxes contain more jellies than chocolates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 1 – P(8, 9, 10) = 1 – → 1 – (0.164156 + 0.064852 + 0.11529)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q04", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 8C2 × 4C2 × 6C3 = 28 × 6 × 20 = 3360 → Summing the number of ways for 3 or 4 correct scenarios → 8C4 × 4C1 × 6C2 = 70 × 4 × 15 = 4200", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q05a", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Draw a fully labelled tree diagram to represent this information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Fully correct labelled branches with correct probabilities for lateness with either 1 branch after W or 2 branches with the prob 0 (see linked mark scheme p.10 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q05b", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that Rani has some fruit.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 0.35 × 0.3 + 0.44 × 0.8 (+ 0) (see linked mark scheme p.10 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q05c", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that Rani does not have a burger given that she does not have any fruit.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: P(not B|not fruit) = ( (see linked mark scheme p.11 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q06a", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the probability that a randomly chosen female snake of this species has length between 50 cm and 60 cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Both values correct → (Correct area) → = 0.8375 + 0.7441 – 1", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q06b", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find estimates for the mean and standard deviation of the lengths of male snakes of this species.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: One appropriate standardisation equation with ߤ, ߪ, z-value (not probability) and 45 or 56. → σ = 4.65, μ = 49.6", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q07a", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Draw a histogram to represent this information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Class widths: 10, 5, 15, 20, 10 → Bar ends: 10.5, 15.5, 30.5, 50.5, 60.5 → Frequency density = frequency/their class width: 1.8, 4.8, 2, 1, 0.8", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q07b", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: What is the greatest possible value of the interquartile range for the data?", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Greatest IQR = 50 – 11 = 39", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-51-q07c", "paperId": "cie-2020-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Calculate estimates of the mean and standard deviation of the number of chocolate bars sold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_51.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Mean = 18 5.5 → Standard deviation = 15.0", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q01", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the value of n.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 50n + 144 = 944", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q02a", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that a randomly chosen student is at Canton college and prefers hockey.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 6 to check the method and final answer for 2(a).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q02b", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that a randomly chosen student is at Devar college given that he prefers soccer.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 6 to check the method and final answer for 2(b).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q02c", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Determine whether the events ‘the student prefers hockey’ and ‘the student is at Amos college or Benn college’ are independent, justifying your answer.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: P(hockey) = 220 → so not independent → P(hockey ∩ A or B) = 104", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q03a", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Data analysis: Find the median and the interquartile range for machine A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Median = 0.238 → So IQR = 0.245 – 0.231", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q03b", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Data analysis: Hence make two comparisons between the lengths of the rods produced by machine A and those", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Medians and quartiles correctly plotted for A or B → Completely correct, including scale (see linked mark scheme p.8 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q03c", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Data analysis: Draw box-and-whisker plots for machines A and B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the five-number summary for each machine to draw both box plots on a common scale.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q04a", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the probability that a randomly chosen tree is classified as short.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 9 to check the method and final answer for 4(a).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q04b", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Show that the probability that a randomly chosen tree is classified as tall is 0.298, correct to 3 decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 0.8944 divided by 3 (see linked mark scheme p.9 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q04c", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the height above which trees are classified as tall.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 0.2981 gives z = 0.53 → h = 46.4", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q05a", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Show that P(X=3)=7/15.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the 15 equally likely spinner-result pairs and count those that give X=3 under the rule in the question.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q05b", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Construct the probability distribution table for X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the 15 equally likely spinner-result pairs, apply the stated rule for X, and divide each frequency by 15.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q05c", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find E(X) and Var(X).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the distribution table: calculate E(X)=ΣxP(X=x) and Var(X)=Σx²P(X=x)-[E(X)]².", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q06a", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 11 to check the method and final answer for 6(a).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q06b", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Total number = → 3! Multiplied by 9C2 (OE) only (no additional terms) (see linked mark scheme p.12 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q06c", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the number of different selections if the four letters include at least one M and exactly one E.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 5C0 = 1 → Total = 16", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q07a", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that Moena messages Pasha on no more than 9 of 12 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the count as binomial with n=12 and the daily message probability given in the question.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q07b", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Given that Moena messages Pasha on 1 January, find the probability that the next message is on 5 January.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric waiting-time model: no message on the next three days, then a message on the fourth day.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-52-q07c", "paperId": "cie-2020-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Use an approximation to find the probability that she messages Pasha on fewer than 64 of 100 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Approximate the binomial count by a normal distribution with matching mean and variance, apply a continuity correction at 64, and evaluate the lower-tail probability.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-53-q01a", "paperId": "cie-2020-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Draw a fully labelled tree diagram to represent this information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Fully correct labelled branches with correct probabilities for lateness with either 1 branch after W or 2 branches with the (see linked mark scheme p.6 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-53-q01b", "paperId": "cie-2020-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that Juan goes to college by car given that he arrives early.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Summing three appropriate 2-factor probabilities (see linked mark scheme p.6 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-53-q02a", "paperId": "cie-2020-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that all 3 students own a car.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 0.223 = 0.0106", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-53-q02b", "paperId": "cie-2020-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that the number of these students who own a car is at least 2 and at most 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: P(2, 3, 4) = 16C2 → 0.179205 + 0.235877 + 0.216221", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-53-q03a", "paperId": "cie-2020-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the probability that a randomly chosen person from this town watches television for less than 21 hours in a week.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: P(X < 21) = P → = Φ(1.238)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-53-q03b", "paperId": "cie-2020-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find k such that P(X 15 (see linked mark scheme p.10 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-61-q06a", "paperId": "cie-2020-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Sketch the probability density function y=f(t) for the bus waiting time T.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the density shape and support specified in the question, including its symmetry about t=10.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-61-q06b", "paperId": "cie-2020-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Explain without calculation why E(T)=10.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The density is symmetric about 10, so its centre of symmetry and mean are both 10.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-61-q06c", "paperId": "cie-2020-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find Var(T).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Var(T)=E(T²)-[E(T)]², integrating with the density function given in the question.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-61-q06d", "paperId": "cie-2020-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find P(10 −a < T < 10 + a) in terms of p.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: (p – 0.5) × 2 or 1 – 2(1 – p) (see linked mark scheme p.11 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-61-q06e", "paperId": "cie-2020-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find P(8 2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 7 to check the method and final answer for 3(a)(iii).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q03b", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: find the largest possible value of n.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: largest n is 1918 (see linked mark scheme p.7 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q04a", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Distributions: Show that the variance of the spinner score X is 1.2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(X) and E(X²) from the five equally likely spinner scores, then subtract [E(X)]² from E(X²).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q04b", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Distributions: Given P(X̄>a)=0.1 for 200 spins, find a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal approximation for the sample mean, with the mean and variance from the spinner distribution, and find the 90th percentile.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q04c", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Distributions: Explain whether the Central Limit Theorem was needed in part (b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that the sample mean is treated as approximately normal because the sample size is large, although one spin is not normally distributed.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q04d", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Statistical testing: Carry out the spinner hypothesis test at the 5% significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the hypotheses, calculate the standardised test statistic using the sampling distribution, compare with the 5% critical value, and state the contextual conclusion.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q05ai", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: State the values that X can take.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 9 to check the method and final answer for 5(a)(i).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q05aii", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the parameter λ of the Poisson distribution for X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the information about the Poisson random variable in the question and solve for λ.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q05aiii", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(iii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find P(4 ≤X ≤6).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the prepared mark scheme on physical PDF page 9 to check the method and final answer for 5(a)(iii).", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q05b", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Use an approximation to find the parameter μ of the Poisson distribution for Y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Approximate the Poisson variable by a normal distribution with mean and variance μ, apply the stated probability and continuity correction, then solve for μ.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q06a", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Distributions: Show that the normalising constant k has the stated value in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the probability density over its support, set the result equal to 1, and solve for k.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q06b", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Distributions: Find E(X) in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x times the probability density over its support.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-62-q06c", "paperId": "cie-2020-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Distributions: Find the 60th percentile of X in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density up to the percentile, set the cumulative probability to 0.6, and solve.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q01", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Calculate unbiased estimates of the population mean and variance of X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Est μ = 15.56 → 48.4105 = 48.4 (3 sf)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q02", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that, on a particular day, Y is less than the total of W and X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Var(Y – W – X) = 1050 + 450 + 720 or 2220 → Φ(‘–1.910’) = 1 – Φ(‘1.910’)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q03", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Statistical testing: Use an approximating distribution to test at the 2.5% significance level whether the Poisson mean has increased from 5.2 customers per hour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For 20 days the null total has mean 104. Approximate by a normal distribution with mean and variance 104, apply a continuity correction, and compare the upper-tail statistic with the 2.5% critical value.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q04a", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Find P(A₁+A₂<2) for independent Poisson(1.5) variables.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "A₁+A₂ is Poisson(3), so P(A₁+A₂<2)=P(0)+P(1)=4e⁻³≈0.199.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q04b", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Given A₁+A₂<2, find P(A₁=1).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The possible totals are 0 and 1. Only A₁=1,A₂=0 contributes to the numerator, giving 1.5/4=0.375 after conditioning.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q04c", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Distributions: Explain why A₁−A₂ cannot have a Poisson distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "A₁−A₂ can be negative, while a Poisson random variable only takes non-negative integer values.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q05a", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Calculate an approximate 99% confidence interval for p.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: z = 2.576 → \"0.14\" ± z ×", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q05b", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Use the confidence interval from part (a) to comment on Sunita’s belief that the die is fair.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check whether the fair-die probability 1/6 lies in the 99% confidence interval; if it does, the sample does not provide evidence against her belief at that level.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q05c", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the value of !.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: z = 1.289 → α = ‘0.9013’ – (1 – ‘0.9013’)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q06a", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Distributions: State E(X) for the worm-length density function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The density is symmetric on 5≤x≤10, so its mean is the midpoint, 7.5 cm.", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q06b", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Distributions: Find Var(X) for the worm-length density function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(X²) by integrating x²f(x) over 5≤x≤10, then subtract [E(X)]².", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q06c", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Probability: Find the probability that exactly one of them has length less than 6 cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: 125 (–102 + 625 → 2 × ('0.104' × (1 – '0.104') (see linked mark scheme p.10 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q07a", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Hypothesis testing: Explain why this method is unsuitable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Later customers might spend times different from first ones (see linked mark scheme p.10 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q07b", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Hypothesis testing: State the probability of making a Type I error and explain what is meant by a Type I error in this context.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Concluding that μ ≠ 6.0, when actually μ = 6.0", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q07c", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Hypothesis testing: Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: H0: μ = 6.0 → Evidence that μ ≠ 6.0", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "cie-2020-s-63-q07d", "paperId": "cie-2020-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Hypothesis testing: Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s20_ms_63.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Prepared mark-scheme steps: Population distribution unknown (see linked mark scheme p.11 for the exact final value)", "administrationNote": "May/June 2020 exams were cancelled. This archived question was not administered and is provided for practice; no difficulty ranking is assigned." }, { "id": "ial-legacy-2019-january-WMA01-q1", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a line through two given points.", "answerGuide": "Calculate the gradient from the coordinate differences, then substitute one point to find the intercept.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q2a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Rewrite powers in terms of the given relation y = 2^x.", "answerGuide": "Express each power as a power of 2, then replace 2^x by y and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q2b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Rewrite powers in terms of the given relation y = 2^x.", "answerGuide": "Express each power as a power of 2, then replace 2^x by y and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q2c", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Rewrite a quotient of powers from the given x-to-y relation using y, then simplify.", "answerGuide": "Express each constant base as a power of 2, combine the exponents in the quotient, substitute y = 2ˣ and simplify the resulting fractional power.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report gives Question 2 a mean of 2.1/4 and says part (c) was notably weaker because candidates struggled to relate powers of 2 and simplify.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2019-january-WMA01-q3", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": null, "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of a curve at the specified point.", "answerGuide": "Differentiate the curve, substitute the point's x-coordinate, and simplify the surd.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q4a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use a recurrence relation and a given sum to find 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4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Use a recurrence relation and a given sum to find terms and determine its constant.", "answerGuide": "Apply the recurrence repeatedly, substitute the resulting terms into the stated sum, and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q5a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a fractional power with the binomial theorem, then use the expansion to approximate a value.", "answerGuide": "Use the generalized binomial coefficients through x^3, simplify each term, then substitute the required value of x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q5b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a fractional power with the binomial theorem, then use the expansion to approximate a value.", "answerGuide": "Use the generalized binomial coefficients through x^3, simplify each term, then substitute the required value of x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q6a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Sketch y = 1 + cos x on 0 to 2π and estimate an integral with equal-width trapezia.", "answerGuide": "Mark the intercepts and one full cosine cycle; apply the trapezium rule with width π/3 and include both endpoints.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q6b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the area under the cosine curve with six equal trapezium-rule strips.", "answerGuide": "Use the six equal x-intervals from 0 to 2π, include both endpoint ordinates and apply the trapezium rule with width π/3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says full marks were rare on the trapezium-rule question and highlights omitted endpoint values and incorrect strip counts or widths.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-legacy-2019-january-WMA01-q7", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the values of p for which a quadratic equation has no real roots.", "answerGuide": "Require its discriminant to be strictly negative and solve the resulting quadratic inequality in p.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q8", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": null, "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use the definite integral of 2x + 6/x² from 3 to k to establish the stated cubic equation, given k > 3.", "answerGuide": "Integrate to x² − 6/x, evaluate at k and 3, equate to 10k, and rearrange to k³ − 10k² − 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"paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a circle's centre and radius, then the tangent at a specified point.", "answerGuide": "Complete squares to read the centre and radius; use the radius gradient and the perpendicular-gradient rule for the tangent.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q9c", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a circle's centre and radius, then the tangent at a specified point.", "answerGuide": "Complete squares to read the centre and radius; use the radius gradient and the perpendicular-gradient rule for the tangent.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q10a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use a composite triangle-and-sector diagram to find angles, lengths, area and perimeter of a shop sign.", "answerGuide": "Use the sine rule and angle sums for the triangle, then combine the triangle with the circular sector and arc.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q10b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use a composite triangle-and-sector diagram to find angles, lengths, area and perimeter of a shop sign.", "answerGuide": "Use the sine rule and angle sums for the triangle, then combine the triangle with the circular sector and arc.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q10c", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the shop sign area by combining its circular sector with the attached triangle.", "answerGuide": "Use the major-sector angle 2π − 0.84 for the sector area, add the triangle area, and round to one decimal place.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report identifies sector-area calculation and confusion between the major and minor sector as common sources of error in parts (c) and (d).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2019-january-WMA01-q10d", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(d)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the shop sign perimeter from the major circular arc and the straight boundary lengths.", "answerGuide": "Calculate the major arc using radius 1.8 and angle 2π − 0.84, then add the two exposed straight sides and round to one decimal place.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report identifies incorrect major-sector arc lengths and adding or subtracting inappropriate sides or arcs as common errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-legacy-2019-january-WMA01-q11i", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(i)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation in which the unknown is the logarithm base, then give the result as a simplified surd.", "answerGuide": "Undo the logarithm to obtain x⁴ = 324, use the positive root because x is positive, and simplify it to 3√2.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=18", "qpPages": [ 24, 27 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this subpart caused many problems and very few candidates earned all three marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-legacy-2019-january-WMA01-q11ii", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(ii)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation and rearrange a second logarithmic equation for y.", "answerGuide": "Convert the logarithmic equations to exponential form, apply the log laws, and use the stated domain restrictions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=18", "qpPages": [ 24, 27 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q12a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find how many days are needed for increasing daily cycle distances to reach a stated total.", "answerGuide": "Model the daily distances as a geometric progression with first term 25 and ratio 1.1, set its sum equal to 1000, solve with logarithms, then interpret the integer day count.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=19", "qpPages": [ 28, 31 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report calls using the term formula instead of the geometric-series sum the most common and costly error in part (a).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-legacy-2019-january-WMA01-q12b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model daily cycling distances with a 10% increase and find a day count and cumulative total.", "answerGuide": "Treat the distances as a geometric progression with ratio 1.1; use its nth term and sum formula with the integer-day condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=19", "qpPages": [ 28, 31 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q12c", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(c)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model daily cycling distances with a 10% increase and find a day count and cumulative total.", "answerGuide": "Treat the distances as a geometric progression with ratio 1.1; use its nth term and sum formula with the integer-day condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=19", "qpPages": [ 28, 31 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q13a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a known linear factor to find a polynomial coefficient, factor the cubic, justify its real roots, and sketch transformed graphs.", "answerGuide": "Apply the factor theorem, divide by the linear factor, inspect the quadratic discriminant, then apply the stated horizontal or vertical transformations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=20", "qpPages": [ 32, 35 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q13b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a known linear factor to find a polynomial coefficient, factor the cubic, justify its real roots, and sketch transformed graphs.", "answerGuide": "Apply the factor theorem, divide by the linear factor, inspect the quadratic discriminant, then apply the stated horizontal or vertical transformations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=20", "qpPages": [ 32, 35 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q13c", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the quadratic factor from the preceding part to prove that the cubic equation has only one real solution.", "answerGuide": "Show the quadratic factor has negative discriminant and therefore no real roots; the only real root comes from the known linear factor.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=20", "qpPages": [ 32, 35 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies this as the least successful part and says many candidates did not explain why the quadratic factor contributes no additional real roots.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-legacy-2019-january-WMA01-q13di", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(d)(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a known linear factor to find a polynomial coefficient, factor the cubic, justify its real roots, and sketch transformed graphs.", "answerGuide": "Apply the factor theorem, divide by the linear factor, inspect the quadratic discriminant, then apply the stated horizontal or vertical transformations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=20", "qpPages": [ 32, 35 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q13dii", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(d)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a known linear factor to find a polynomial coefficient, factor the cubic, justify its real roots, and sketch transformed graphs.", "answerGuide": "Apply the factor theorem, divide by the linear factor, inspect the quadratic discriminant, then apply the stated horizontal or vertical transformations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=32", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=20", "qpPages": [ 32, 35 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q14i", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a sine equation over a symmetric degree interval, retaining every valid negative and positive solution.", "answerGuide": "Use the principal inverse-sine value and the second sine solution, subtract the phase shift, and retain all answers in −180° < x < 180°.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=36", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=22", "qpPages": [ 36, 39 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls this part difficult and says many candidates changed a negative inverse-sine angle to a positive one or omitted negative solutions in the requested interval.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-legacy-2019-january-WMA01-q14iia", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(ii)(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a sine equation on a degree interval and solve a related trigonometric equation using the required exact form.", "answerGuide": "Use the full sine solution families within the interval; convert the second equation to a common trigonometric form before solving.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=36", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=22", "qpPages": [ 36, 39 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q14iib", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 14, "questionPart": "14(ii)(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a sine equation on a degree interval and solve a related trigonometric equation using the required exact form.", "answerGuide": "Use the full sine solution families within the interval; convert the second equation to a common trigonometric form before solving.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=36", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=22", "qpPages": [ 36, 39 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q15a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 15, "questionPart": "15(a)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find where a straight line meets a curve, then calculate the exact area of the enclosed region.", "answerGuide": "Solve the intersection equation algebraically, identify which graph lies above, and integrate their difference with exact limits.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_que_20190109.pdf#page=40", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_msc_20190307.pdf#page=24", "qpPages": [ 40, 43 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA01-q15b", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": 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the incorrect area between the line and curve; the successful approach combined an integral under the curve with a triangle.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 9 ] } }, { "id": "ial-legacy-2019-january-WMA01-q16a", "paperId": "ial-legacy-2019-january-WMA01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA01/01", "variant": "Legacy paper", "questionNumber": 16, "questionPart": "16(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Optimize a container's surface area subject to its fixed volume, then find dimensions and verify the minimum.", "answerGuide": "Use the volume 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q1b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Convert trigonometric expressions to a single sine form, solve an equation, and find an exact maximum.", "answerGuide": "Use R sin(2θ−α), apply the interval to the sine solutions, then use the equivalent 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q2b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 6, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find a power-series expansion of a rational 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a small percentage gained full marks because many checked only one endpoint or only the vertex.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2019-january-WMA02-q3b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": 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acute-angle signs to simplify the exact ratios.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q5b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use secant identities to find possible m values, then evaluate exact trigonometric ratios for acute angles.", "answerGuide": "Replace sec² with 1+tan², solve the resulting quadratic in m, and use acute-angle signs to simplify the exact ratios.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q5c", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q6b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a 3D line equation, the acute angle to a line segment, and a point forming a right angle; calculate a triangle area.", "answerGuide": "Use the direction vector AB, a dot product for the angle, and a perpendicular dot-product condition for D before using the cross-product area formula.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q6c", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of a point on a 3D line that makes a right angle with a given point and line segment.", "answerGuide": "Parameterize D on AB, form vectors from C to A and C to D, set their dot product to zero, solve for the parameter, and substitute back for D.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 17 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls part (c) a good discriminator; most candidates did not form the correct vectors for the perpendicular dot-product condition.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5, 6 ] } }, { "id": "ial-legacy-2019-january-WMA02-q6d", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a 3D line equation, the acute angle to a line segment, and a point forming a right angle; calculate a triangle area.", "answerGuide": "Use the direction vector AB, a dot product for the angle, and a perpendicular dot-product condition for D before using the cross-product area formula.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q7a", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate a radical curve to find a shaded area and express the exact result in a simplified surd form.", "answerGuide": "Use the suggested substitution to remove the square root, change the limits consistently, and evaluate the definite integral.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q7b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the exact area under a radical curve using the suggested substitution.", "answerGuide": "Set u = 2x − 3, convert the limits and dx consistently, integrate each resulting power of u, and substitute the transformed bounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says fully correct responses were relatively rare and describes frequent substitution, differential, algebra, and integration errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6, 7 ] } }, { "id": "ial-legacy-2019-january-WMA02-q8a", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate parametric equations, find a tangent, and use algebra to find a second intersection.", "answerGuide": "Calculate dy/dx as (dy/dt)/(dx/dt), use the point's parameter for the tangent, then substitute the line into the curve and solve.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=20", "qpPages": [ 30, 33 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q8b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate parametric 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"January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the second intersection of a parametric curve with its tangent at a specified point.", "answerGuide": "Substitute the parametric equations into the tangent equation, form the cubic in t, factor out the known tangent-point root, then find the other parameter and coordinates.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=20", "qpPages": [ 30, 33 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was poorly answered, many candidates left it blank, and algebra errors in forming or solving the cubic commonly stopped progress.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-legacy-2019-january-WMA02-q9a", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate two integrals involving x and sin 2x, then find the area under the resulting curve.", "answerGuide": "Use integration by parts for the x sin 2x term, expand the square before integrating, then use exact intersection limits for the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=34", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=22", "qpPages": [ 34, 37 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q9b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate the square of a sum involving x and sin 2x.", "answerGuide": "Expand the square; use the result from part (a) for the mixed term and cos 4x = 1 − 2sin²2x for the remaining trigonometric integral.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=34", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=22", "qpPages": [ 34, 37 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "The report says only about half of candidates integrated the sin²(2x) term successfully.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-legacy-2019-january-WMA02-q9c", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate two integrals involving x and sin 2x, then find the area under the resulting curve.", "answerGuide": "Use integration by parts for the x sin 2x term, expand the square before integrating, then use exact intersection limits for the area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=34", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=22", "qpPages": [ 34, 37 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q10a", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use similar triangles and related rates to derive the water-height differential equation for a leaking cone.", "answerGuide": "Relate radius to water height, substitute that relation into the cone volume, differentiate with respect to time, and use the negative volume-loss rate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=23", "qpPages": [ 38, 41 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this a discriminating question and says very few candidates connected radius and height correctly before applying the chain rule.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-legacy-2019-january-WMA02-q10b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Model water draining from a cone using similar triangles and related rates, then find the emptying time.", "answerGuide": "Relate radius to water depth, express volume in terms of h, differentiate with respect to t, and integrate the resulting differential equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=38", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=23", "qpPages": [ 38, 41 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q10c", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Model water draining from a cone using similar triangles and related rates, then find the emptying time.", "answerGuide": "Relate radius to water depth, express volume in terms of h, differentiate 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"trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch the translated arccosine function on its stated restricted interval.", "answerGuide": "Start from the decreasing arccosine curve, shift it horizontally by one, and mark the endpoint values on 0 ≤ x ≤ 2.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=42", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=25", "qpPages": [ 42, 43 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": 10, "difficultyEvidence": "The report says part (a) was poorly done, many candidates scored zero, and very few earned both marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 9 ] } }, { "id": "ial-legacy-2019-january-WMA02-q11b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Sketch a restricted arccosine graph, bracket its equation's root, and use iteration to approximate it.", "answerGuide": "Use the domain and range to sketch the curve, substitute the interval endpoints to bracket the root, then iterate to the stated precision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=42", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=25", "qpPages": [ 42, 43 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q11c", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Sketch a restricted arccosine graph, bracket its equation's root, and use iteration to approximate it.", "answerGuide": "Use the domain and range to sketch the curve, substitute the interval 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"WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 12, "questionPart": "12(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch an absolute-value graph and solve an equation involving it to identify x-values for each positive k.", "answerGuide": "Mark the vertex and intercepts, split the equation at the absolute-value cases, and retain solutions consistent with each case.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=44", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=27", "qpPages": [ 44, 47 ], "msPages": [ 27, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q13a", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic model for an insect population to find its parameters, a time value, and an equivalent differential equation.", "answerGuide": "Substitute the initial population, solve the logarithmic time equation, differentiate the logistic expression, and eliminate t in favour of N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=48", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=29", "qpPages": [ 48, 52 ], "msPages": [ 29, 34 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WMA02-q13b", "paperId": "ial-legacy-2019-january-WMA02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA02", "unit": "Pure Mathematics 2 (P2)", "year": 2019, "session": "January", "paperCode": "WMA02/01", "variant": "Legacy paper", "questionNumber": 13, "questionPart": "13(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use a logistic model for an insect population to find its parameters, a time value, and an equivalent differential equation.", "answerGuide": "Substitute the initial population, solve the logarithmic 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"differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate a logistic population model and rewrite the rate of change entirely in terms of the population.", "answerGuide": "Differentiate the model, rearrange the population equation to eliminate the exponential in t, and simplify the derivative into the required quadratic expression in N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_que_20190116.pdf#page=48", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_msc_20190307.pdf#page=29", "qpPages": [ 48, 52 ], "msPages": [ 29, 34 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes this final part as much more challenging and says fully correct responses were rare, especially when converting the derivative from t to N.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WMA02_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-2019-1-WMA11-01-q1", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate a sum of powers and simplify the result.", "answerGuide": "Increase each power by one, divide by the new power, and include a constant of integration.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q2", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use laws of indices to express y as a simplified function of x.", "answerGuide": "Rewrite both sides with base 3, combine powers carefully, equate exponents, and isolate y in terms of x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates scored no marks, commonly because they could not apply the index laws to the right-hand side.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2019-1-WMA11-01-q3a", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the gradients and equations of two perpendicular lines through given points.", "answerGuide": "Rearrange the first line to find its gradient, use the negative reciprocal, then substitute the required point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q3b", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the gradients and equations of two perpendicular lines through given points.", "answerGuide": "Rearrange the first line to find its gradient, use the negative reciprocal, then substitute the required point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q4", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Identify the inequalities defining a shaded region bounded by a line, a curve, and a vertical boundary.", "answerGuide": "Find where the curve meets the positive x-axis, then state the vertical interval and which side of each line or 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q5ci", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Read a cosine minimum, sketch a sine curve, and count intersections over specified radian intervals.", "answerGuide": "Use the periodic graphs and endpoint positions to count the intersections, treating the two interval 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"difficultyEvidence": "The report calls part (c) one of the most challenging parts; very few candidates recognized that solutions correspond to intersections of the two sketched curves.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-pef-20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2019-1-WMA11-01-q6a", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Solve equations involving the first and second derivatives of a function for x > 0.", "answerGuide": "Differentiate term by term, solve the resulting power equations, and retain only positive solutions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q6b", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Solve equations involving the first and second derivatives of a function for x > 0.", "answerGuide": "Differentiate term by term, solve the resulting power equations, and retain only positive solutions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q7a", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=22", "qpPages": [ 18, 19 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q10a", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the area and perimeter of a sector to derive a quadratic, then find possible radii and angles.", "answerGuide": "Write the sector area and perimeter equations, eliminate r to form the quadratic in θ, then pair each 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"paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Sketch a quadratic and cubic, solve their intersections, and find the exact coordinates of a specified point.", "answerGuide": "Mark the roots and shapes, equate the curves and factor, then substitute the correct root into the cubic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20190109.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=26", "qpPages": [ 22, 25 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q11b", "paperId": "ial-2019-1-WMA11-01", "board": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-msc-20190307.pdf#page=26", "qpPages": [ 22, 25 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2019-1-WMA11-01-q12a", "paperId": "ial-2019-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2019, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": "12(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use a derivative and a point to find a tangent, then integrate the derivative to determine the function.", "answerGuide": "Evaluate the gradient at x=4, form the tangent through P, integrate f′(x), and use P to 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"paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply conservation of momentum to a direct collision, then find the impulse on one particle.", "answerGuide": "Choose a positive direction and use momentum conservation for the post-collision speed; use change in momentum for impulse.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q1ii", "paperId": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q2b", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 10, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Use both birds’ position vectors at t = 4 s 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-legacy-2019-january-WME01-q3a", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Write equations of motion for a person and an accelerating lift, then find both masses.", "answerGuide": "Apply Newton's second law separately to the person and lift, keeping the cable tension and normal reaction in the correct force balances.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q3b", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Write the equation of motion for the lift alone while it rises with constant acceleration.", "answerGuide": "Isolate the lift as the system: use cable tension upward and the lift weight plus the person’s downward reaction downward, equal to M times the upward acceleration.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates wrote the equation for the combined lift-and-person system instead of the required lift-only equation, earning no marks for this part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2019-january-WME01-q3c", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Write equations of motion for a person and an accelerating lift, then find both masses.", "answerGuide": "Apply Newton's second law separately to the person and lift, keeping the cable tension and normal reaction in the correct force balances.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q4a", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use equilibrium of a supported plank to find the boy's walking limit, explain a particle model, and calculate a counterweight mass.", "answerGuide": "Take moments about a support at the tipping point, state the modelling limitation, then balance moments for the rock at the minimum-mass condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q4b", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use equilibrium of a supported plank to find the boy's walking limit, explain a particle model, and calculate a counterweight mass.", "answerGuide": "Take moments about a support at the tipping point, state the modelling limitation, then balance moments for the rock at the minimum-mass condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q4c", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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says many candidates missed that the minimum rock mass requires zero reaction at P; the full question was among the most challenging and 31% scored zero.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2019-january-WME01-q5", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": null, "marks": 10, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use geometry and equilibrium to find the tensions in two ropes supporting a box.", "answerGuide": "Find the rope angle from the vertical geometry, resolve the two tensions horizontally and vertically, and solve the simultaneous equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q6a", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph and constant-acceleration stages to find the train's speed, travel times, and acceleration intervals.", "answerGuide": "Use the area under the velocity-time graph for total distance, then apply constant-acceleration equations separately to each stage.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q6b", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph and constant-acceleration stages to find the train's speed, travel times, and acceleration intervals.", "answerGuide": "Use the area under the velocity-time graph for total distance, then apply constant-acceleration equations separately to each stage.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q6c", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph and constant-acceleration stages to find the train's speed, travel times, and acceleration intervals.", "answerGuide": "Use the area under the velocity-time graph for total distance, then apply constant-acceleration equations separately to each stage.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q6d", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph and constant-acceleration stages to find the train's speed, travel times, and acceleration intervals.", "answerGuide": "Use the area under the velocity-time graph for total distance, then apply constant-acceleration equations separately to each stage.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q7a", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 11, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse two connected particles on a table and an inclined plane to find tension, motion, and pulley force.", "answerGuide": "Resolve forces along and perpendicular to the plane, apply Newton's law to both particles and the friction condition, then combine the string-force components at the pulley.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q7b", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A 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null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WME01-q7c", "paperId": "ial-legacy-2019-january-WME01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2019, "session": "January", "paperCode": "WME01/01", "variant": "Legacy paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse two connected particles on a table and an inclined plane to find tension, motion, and pulley force.", "answerGuide": "Resolve forces along and perpendicular to the plane, apply Newton's law to both particles and the friction condition, then combine the string-force components at the pulley.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_que_20190122.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WME01_01_msc_20190307.pdf#page=14", "qpPages": [ 20, 24 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q1a", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a three-set Venn diagram with independence and given probabilities to find regions and a conditional probability.", "answerGuide": "Use the mutually exclusive regions, independence for the L∩C overlap, and the total-probability constraints before forming the conditional ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q1b", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a three-set Venn diagram with independence and given probabilities to find regions and a conditional probability.", "answerGuide": "Use the mutually exclusive regions, independence for the L∩C overlap, and the total-probability constraints before forming the conditional ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q1c", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a three-set Venn diagram with independence and given probabilities to find regions and a conditional probability.", "answerGuide": "Use the mutually exclusive regions, independence for the L∩C overlap, and the total-probability constraints before forming the conditional ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q1d", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find an exact conditional probability for a student selected from the band-or-choir group.", "answerGuide": "Use P(L ∩ (B ∪ C)) / P(B ∪ C), account for the overlap correctly using the Venn regions, and present the exact fraction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls part (d) quite challenging; many candidates did not recognize the conditional probability and others omitted a numerator component or gave a decimal instead of an exact fraction.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2019-january-WST01-q2a", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a discrete probability distribution to find its mean, solve unknown probabilities from a second moment, and find a transformed variance.", "answerGuide": "Use sums of xP(X=x) and x²P(X=x), the total-probability equation, and Var(a+bX)=b²Var(X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q2b", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a discrete probability distribution to find its mean, solve unknown probabilities from a second moment, and find a transformed variance.", "answerGuide": "Use sums of xP(X=x) and x²P(X=x), the total-probability equation, and Var(a+bX)=b²Var(X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q2c", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a discrete probability distribution to find its mean, solve unknown probabilities from a second moment, and find a transformed variance.", "answerGuide": "Use sums of xP(X=x) and x²P(X=x), the total-probability equation, and Var(a+bX)=b²Var(X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q3a", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to find a lower-tail probability, a conditional probability, and an upper-tail cut-off.", "answerGuide": "Standardize with the given mean and standard deviation, divide the appropriate conditional probability by P(W<51), and invert the normal distribution for H.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q3b", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a conditional probability comparing Ren’s weight with a randomly selected opponent known to be lightweight.", "answerGuide": "Calculate P(49 < W < 51) divided by P(W < 51) using the normal distribution, then give an appropriate decimal probability.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report explicitly lists part 3(b) among the discriminating questions and notes that candidates often failed to state the conditional ratio.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-legacy-2019-january-WST01-q3c", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to find a lower-tail probability, a conditional probability, and an upper-tail cut-off.", "answerGuide": "Standardize with the given mean and standard deviation, divide the appropriate conditional probability by P(W<51), and invert the normal distribution for H.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q4a", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use grouped exercise-time data to draw a histogram, estimate summary statistics and skewness, and assess new information about zero values.", "answerGuide": "Use frequency density for bars, linear interpolation for the median, class midpoints for mean and standard deviation, then compare how low replacements affect each statistic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q4b", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q4c", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use grouped exercise-time data to draw a histogram, estimate summary statistics and skewness, and assess new information about zero values.", "answerGuide": "Use frequency density for bars, linear interpolation for the median, class midpoints for mean and standard deviation, then compare how low replacements affect each statistic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q4d", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use grouped exercise-time data to draw a histogram, estimate summary statistics and skewness, and assess new information about zero values.", "answerGuide": "Use frequency density for bars, linear interpolation for the median, class midpoints for mean and standard deviation, then compare how low replacements affect each statistic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q4e", "paperId": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q4f", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 4, "questionPart": "4(f)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain how replacing 18 low exercise times by zero affects the median, mean, and standard deviation.", "answerGuide": "State that the median stays the same, the mean decreases, and the standard deviation increases, with a reason tied to the unchanged rank position and the increased spread.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many candidates omitted reasons or directions; it gives the specific changes needed for each statistic.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2019-january-WST01-q5a", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Model a three-throw game with binomial and unequal independent probabilities, then compare two models with observed frequencies.", "answerGuide": "Calculate each outcome probability from independent throws, use complements to complete distributions, and compare predicted frequencies with the data for each model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=11", 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q5c", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Model a three-throw game with binomial and 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"Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(d)(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Model a three-throw game with binomial and unequal independent probabilities, then compare two models with observed frequencies.", "answerGuide": "Calculate each outcome probability from independent throws, use complements to complete distributions, and compare predicted frequencies with the data for each model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q5dii", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(d)(ii)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Show the probability of exactly one success when three independent throws have different success probabilities.", "answerGuide": "Add the three mutually exclusive outcomes with one success: success on throw 1 only, throw 2 only, or throw 3 only.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says part (ii) was more challenging; a common wrong response omitted the three separate cases and used a complement incorrectly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-legacy-2019-january-WST01-q5e", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Model a three-throw game with binomial and unequal independent probabilities, then compare two models with observed frequencies.", "answerGuide": "Calculate each outcome probability from independent throws, use complements to complete distributions, and compare predicted frequencies with the data for each model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q5f", "paperId": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q6c", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare exam results, assess a possible 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"Statistics 1 (S1)", "year": 2019, "session": "January", "paperCode": "WST01/01", "variant": "Legacy paper", "questionNumber": 6, "questionPart": "6(d)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare exam results, assess a possible outlier, recompute statistics without it, and use correlation and regression to justify exclusion.", "answerGuide": "Compare means and spreads, update sums after removing (38,0), calculate Sxy and r, find the regression line, then interpret the stronger linear association.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, 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"6(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare exam results, assess a possible outlier, recompute statistics without it, and use correlation and regression to justify exclusion.", "answerGuide": "Compare means and spreads, update sums after removing (38,0), calculate Sxy and r, find the regression line, then interpret the stronger linear association.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-legacy-2019-january-WST01-q6f", "paperId": 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trend.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_que_20190118.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_msc_20190307.pdf#page=12", "qpPages": [ 16, 20 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says few candidates gave the key interpretation that a correlation closer to 1 means points lie closer to a straight line.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2013/Exam%20materials/WST01_01_pef_20190307.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-legacy-2019-january-WST01-q6h", "paperId": "ial-legacy-2019-january-WST01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International 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null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2019-w-11-q02", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Use the derivative sign to find the least integer boundary for an increasing function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, 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"https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2019-w-11-q04i", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(i): Work backwards through the daily geometric progression to find the starting distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use r = 1.1 and the 21st-term relation a(1.1)^20 = 20 to find the first distance." }, { "id": "cie-2019-w-11-q04ii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(ii): Use the daily geometric progression to find 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"marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the trigonometric result from part (i) to find both solutions in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the direct route to both solutions was rarely seen and that other methods often produced only one solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the result from part (i), identify the third- and fourth-quadrant angle values, and list both solutions in the required interval." }, { "id": "cie-2019-w-11-q06", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2019-w-11-q07i", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the range of the function, including its lower limit.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The upper limit was often found, but the report says the existence of a lower limit was not apparent in most responses.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the function’s domain and its limiting behaviour to state both endpoints of the range, with correct strict or inclusive inequalities." }, { "id": "cie-2019-w-11-q07ii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "7(ii): Compose the two defined functions in the stated order.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the output of the inner function into the outer function, respect the given domains, and simplify." }, { "id": "cie-2019-w-11-q07iii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "7(iii): Compose the inverse of one defined function with the other function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the inverse of the indicated function, substitute it into the other function in the stated order, and simplify." }, { "id": "cie-2019-w-11-q08i", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(i)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(i): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2019-w-11-q08ii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(ii): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2019-w-11-q08iii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(iii)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(iii): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2019-w-11-q09i", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(i): Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2019-w-11-q09ii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(ii): Differentiate the given gradient function once more to obtain the second derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given gradient expression with respect to x once more and simplify to obtain d²y/dx²." }, { "id": "cie-2019-w-11-q09iii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(iii)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(iii): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2019-w-11-q10i", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(i): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2019-w-11-q10ii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(ii)", "marks": 3, "topicIds": [ "vectors", "coordinate-geometry", "proof" ], "primaryTopicId": "vectors", "summary": "Show that the indicated pair of sides are parallel.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This was the most frequently omitted part of the paper.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Find the relevant displacement vectors and show that one is a scalar multiple of the other, then state the parallel conclusion." }, { "id": "cie-2019-w-11-q10iii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(iii): Use the position vectors to calculate the area of the quadrilateral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the required side lengths and perpendicular height from the position vectors, then use the appropriate quadrilateral area formula." }, { "id": "cie-2019-w-11-q11i", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(i): Complete the square, then rearrange the curve to express x in terms of y.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=13", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square, use the branch consistent with x ≥ −2, and rearrange to make x the subject." }, { "id": "cie-2019-w-11-q11ii", "paperId": "cie-2019-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(ii)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume generated by rotating the stated curve region about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_11.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only a minority connected part (i) to the y-axis volume method, and the final correct answer was seen very occasionally.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange the curve to express x in terms of y, square it, integrate πx² with respect to y over the stated limits, and evaluate exactly or to the requested accuracy." }, { "id": "cie-2019-w-12-q01", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2019-w-12-q02", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "2: Use gradients or direction vectors to establish the stated line relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=5", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2019-w-12-q03", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Determine the constants in a derivative-defined curve from its gradient and two given points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates did not recognise that integration was needed and instead tried to equate the derivative to the line gradient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the derivative to obtain the curve equation with its constant, then use both points to form and solve the required equations." }, { "id": "cie-2019-w-12-q04i", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "4(i): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=6", "qpPages": [ 6, 6 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2019-w-12-q04ii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "4(ii): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2019-w-12-q05i", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "5(i): Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=7", "qpPages": [ 8, 8 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2019-w-12-q05ii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(ii): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2019-w-12-q06a", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the given tangent equation for all angles in the required radian interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes part (a) as challenging and lists incorrect double-angle expansions, degree mode, and wrong π shifts as common errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Use the tangent double-angle identity, solve in radians, and select all solutions in the stated interval without confusing π shifts." }, { "id": "cie-2019-w-12-q06bi", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b)(i): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=9", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q06bii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 2, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Find the range of the expression involving cos²x over the stated domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says some candidates substituted the domain endpoints, while stronger responses considered the maximum and minimum possible values of cos²x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Use the maximum and minimum values of cos²x on the allowed domain, then substitute them into the expression and state its range." }, { "id": "cie-2019-w-12-q07i", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "7(i): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2019-w-12-q07ii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "7(ii): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2019-w-12-q07iii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "7(iii): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2019-w-12-q08ai", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)(i)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a)(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=10", "qpPages": [ 14, 14 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q08aii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a)(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q08bi", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b)(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=11", "qpPages": [ 15, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q08bii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b)(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=11", "qpPages": [ 15, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q08biii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(iii)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "8(b)(iii): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=11", "qpPages": [ 15, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2019-w-12-q09i", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(i): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2019-w-12-q09ii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q09iii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(iii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(iii): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q09iv", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(iv)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(iv): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q10i", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(i): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=13", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-12-q10ii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(ii): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2019-w-12-q10iii", "paperId": "cie-2019-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the full shaded area, including the triangular region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_12.pdf#page=14", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was frequently omitted or used incorrect limits, and very few candidates found the whole area including the triangle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Find the relevant curve and line intersections, integrate each curved region over its own limits, and add the triangle area rather than omitting it." }, { "id": "cie-2019-w-13-q01i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2019-w-13-q01ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1(ii): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2019-w-13-q02", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse function and state its correct domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A large number gave x > −2 instead of recognising that the original function was defined for x > 4.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Complete the square, solve for the inverse branch consistent with the original domain, and state the range of the original function as the inverse domain." }, { "id": "cie-2019-w-13-q03", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2019-w-13-q04i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "4(i): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2019-w-13-q04ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "4(ii): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2019-w-13-q05i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "5(i): Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2019-w-13-q05ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "5(ii): Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2019-w-13-q06i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-13-q06ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(ii): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2019-w-13-q07i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-13-q07ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "List all four solutions of the trigonometric equation in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates omitted the ± when taking a square root or gave only two of the four solutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Take both signs when solving the squared cosine equation, then list every angle in the required interval." }, { "id": "cie-2019-w-13-q08i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-13-q08ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-13-q09i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "9(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-13-q09ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(ii): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2019-w-13-q09iii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(iii)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(iii): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2019-w-13-q10i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(i): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2019-w-13-q10ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(ii): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2019-w-13-q10iii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(iii)", "marks": 3, "topicIds": [ "vectors", "coordinate-geometry", "proof" ], "primaryTopicId": "vectors", "summary": "Find the area of the triangle formed by the stated vectors and point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates successfully answered this part; the report says only a minority used the perpendicularity from part (ii) to find the triangle area.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Use the perpendicular vectors established earlier, find the relevant base and height magnitudes, and apply one half times base times height." }, { "id": "cie-2019-w-13-q11i", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(i)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "11(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=14", "qpPages": [ 18, 18 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-13-q11ii", "paperId": "cie-2019-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2019, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(ii)", "marks": 8, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the total volume from the two curve sections rotated about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_13.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_13.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies common squaring, limit, and addition errors in this part and notes that the required substitution work had to be shown.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Square each curve expression separately, integrate over its own interval, show both limit substitutions, and add the two volumes." }, { "id": "cie-2019-w-21-q01i", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(i): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2019-w-21-q01ii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1(ii): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-21-q02", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-21-q03", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-21-q04i", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2019-w-21-q04ii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Complete the exact factorisation using the quadratic factor found in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A correct quadratic factor was often found, but complete correct factorisation was not commonly seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Divide by the known factor, factor the remaining quadratic exactly, and present the complete factorisation without decimal approximations." }, { "id": "cie-2019-w-21-q04iii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(iii): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2019-w-21-q05i", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(i): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2019-w-21-q05ii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the required trigonometric identity to solve the equation for α.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes frequent use of an incorrect sine identity, calculator mode errors, and several non-responses.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the correct triple-angle identity, keep the calculus angles in radians, track the sign change, and give all required solutions." }, { "id": "cie-2019-w-21-q05iii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(iii): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-21-q06a", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-21-q06b", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-21-q07i", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(i): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-21-q07ii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Show that the implicitly defined curve has no stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few complete and correct solutions were seen; many candidates stopped after reaching an intermediate expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Set the numerator of dy/dx to zero, use the resulting relation between x and y, and substitute into the original curve to show it gives an impossibility." }, { "id": "cie-2019-w-21-q07iii", "paperId": "cie-2019-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinates where the curve has a tangent parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_21.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not complete this part and often made little progress beyond an intermediate relation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Set the denominator of dy/dx to zero, use the resulting relation between x and y in the curve equation, and solve for both x-coordinates." }, { "id": "cie-2019-w-22-q01", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2019-w-22-q02i", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(i): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-22-q02ii", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "2(ii): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-22-q03", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Find the parameters in an exponential model from two points on its straight-line transformation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates were unable to progress much further than finding the gradient, and incorrect use of the given coordinates was common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain the linear model, use the gradient to determine the exponent, then substitute a point to find the scale parameter." }, { "id": "cie-2019-w-22-q04i", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(i): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-22-q04ii", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Use the iterative method to obtain the exact value requested from the iteration result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few correct solutions were seen; many reused the approximate answer from part (i) despite the exact-answer instruction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Use the correct iteration formula and its limiting equation, then distinguish the exact value from the numerical approximation in part (i)." }, { "id": "cie-2019-w-22-q05", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of the exponential function and give its exact coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates struggled to differentiate the exponential factor and some did not recognise that it could not equal zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Apply the product rule, set the derivative to zero, use that the exponential factor is nonzero, and substitute the exact x-value back into the curve." }, { "id": "cie-2019-w-22-q06a", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-22-q06b", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-22-q07i", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(i): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-22-q07ii", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(ii): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-22-q08i", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(i): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-22-q08ii", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(ii): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-22-q08iii", "paperId": "cie-2019-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(iii)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the extreme values of the trigonometric expression using its amplitude form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_22.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few correct responses were seen; candidates often expanded the expression or did not use the maximum and minimum values of the cosine term.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the expression using the amplitude result from part (i), then use the maximum and minimum possible cosine values to find the endpoints." }, { "id": "cie-2019-w-23-q01i", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(i): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2019-w-23-q01ii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1(ii): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-23-q02", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-23-q03", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-23-q04i", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2019-w-23-q04ii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Complete the exact factorisation using the quadratic factor found in part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A correct quadratic factor was often found, but complete correct factorisation was not commonly seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Divide by the known factor, factor the remaining quadratic exactly, and present the complete factorisation without decimal approximations." }, { "id": "cie-2019-w-23-q04iii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(iii): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2019-w-23-q05i", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(i): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2019-w-23-q05ii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the required trigonometric identity to solve the equation for α.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes frequent use of an incorrect sine identity, calculator mode errors, and several non-responses.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Use the correct triple-angle identity, keep the calculus angles in radians, track the sign change, and give all required solutions." }, { "id": "cie-2019-w-23-q05iii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(iii): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-23-q06a", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-23-q06b", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-23-q07i", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(i): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-23-q07ii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Show that the implicitly defined curve has no stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few complete and correct solutions were seen; many candidates stopped after reaching an intermediate expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Set the numerator of dy/dx to zero, use the resulting relation between x and y, and substitute into the original curve to show it gives an impossibility." }, { "id": "cie-2019-w-23-q07iii", "paperId": "cie-2019-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2019, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinates where the curve has a tangent parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_23.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not complete this part and often made little progress beyond an intermediate relation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Set the denominator of dy/dx to zero, use the resulting relation between x and y in the curve equation, and solve for both x-coordinates." }, { "id": "cie-2019-w-31-q01", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-31-q02", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2019-w-31-q03", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2019-w-31-q04i", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-31-q04ii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the exponential growth differential equation using the given boundary conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a minority reached the correct final answer after several errors in integration and processing.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 19, 20, 21 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate both sides, apply the boundary conditions to find the constant, and simplify the final expression carefully." }, { "id": "cie-2019-w-31-q04iii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": 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"tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-31-q05ii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(ii): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2019-w-31-q05iii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(iii): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-31-q06i", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(i): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-31-q06ii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 6, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(ii): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-31-q07i", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-31-q07ii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-31-q08i", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(i): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-31-q08ii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "8(ii): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-31-q09i", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "9(i): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2019-w-31-q09ii", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "9(ii): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant 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"sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-31-q10a", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "10(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=17", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-31-q10b", "paperId": "cie-2019-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the complex-number region satisfying the modulus, argument and imaginary-part conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_31.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A significant proportion did not attempt the question; the report describes recurring confusion over the circle centre, argument region and imaginary-part line.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 19, 20, 21 ], "tagConfidence": "reviewed", "answerGuide": "Draw the circle with its centre and radius, draw the argument boundary ray and the horizontal imaginary-part line, then shade their intersection in an Argand diagram." }, { "id": "cie-2019-w-32-q01", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-32-q02", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Differentiate the given gradient function once more to obtain the second derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given gradient expression with respect to x once more and simplify to obtain d²y/dx²." }, { "id": "cie-2019-w-32-q03", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine two polynomial coefficients from a factor condition and a remainder condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies the factors-and-remainders question as particularly challenging and describes widespread algebra and sign errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Use either exact polynomial division or the remainder theorem with the exact roots of the divisor, then equate coefficients to solve for both constants." }, { "id": "cie-2019-w-32-q04i", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(i): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-32-q04ii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation using the amplitude result from part (i).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates found this part challenging; inaccurate rounding and solving for 2θ rather than θ were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Use the exact amplitude and phase from part (i), solve for the angle in the stated interval, and retain enough accuracy throughout." }, { "id": "cie-2019-w-32-q05", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the y-coordinate where the curve has a horizontal tangent, rejecting the extraneous branch.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often misunderstood “parallel to the x-axis”; very few mentioned and rejected y = 0, and many stopped at y = x⁴.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly, set dy/dx to zero, examine both possible factors, reject y = 0 when it is not valid, then substitute into the original curve." }, { "id": "cie-2019-w-32-q06", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric 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"tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2019-w-32-q07bi", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 2, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the circle and horizontal line on one Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many found drawing the specified horizontal line challenging and drew a second circle instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Draw the circle with centre on the imaginary axis and radius 2, draw Im z = 3 as a horizontal line, and label the axes and intersection geometry." }, { "id": "cie-2019-w-32-q07bii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b)(ii): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=14", "qpPages": [ 13, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2019-w-32-q08i", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(i): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-32-q08ii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "8(ii): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2019-w-32-q09i", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "9(i): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2019-w-32-q09ii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "9(ii): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-32-q09iii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "9(iii): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-32-q10i", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(i): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=17", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2019-w-32-q10ii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(ii)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=18", "qpPages": [ 18, 18 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-32-q10iii", "paperId": "cie-2019-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(iii)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(iii): Use line and plane direction vectors to find the requested intersection, angle or plane equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_32.pdf#page=19", "qpPages": [ 18, 19 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the plane normal and line direction vector to solve for intersections and angles; form a plane equation from its normal and a known point." }, { "id": "cie-2019-w-33-q01", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2019-w-33-q02", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2019-w-33-q03", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-33-q04i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(i): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2019-w-33-q04ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(ii): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=7", "qpPages": [ 7, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2019-w-33-q05i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch both functions over their full domains, showing asymptotes and key intercepts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=7", "qpPages": [ 8, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Candidates often restricted the sketches to x > 0 and missed the full domains, asymptote, or correct exponential intercept.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Show the logarithm’s vertical asymptote and domain, then sketch the decreasing exponential across all real x with its correct intercept and horizontal asymptote." }, { "id": "cie-2019-w-33-q05ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=7", "qpPages": [ 8, 8 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-33-q05iii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(iii): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=8", "qpPages": [ 9, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-33-q06i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(i): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2019-w-33-q06ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(ii): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2019-w-33-q06iii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Show that the product expression has the required modulus and argument.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=8", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists this part among the questions candidates found most challenging and describes repeated arithmetic and algebra errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Use the modulus product rule and add the arguments of the factors, keeping the argument of the unit complex number exact." }, { "id": "cie-2019-w-33-q07i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=9", "qpPages": [ 12, 12 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-33-q07ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the perpendicular distance between two parallel planes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=9", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists this part among the most challenging and says incorrect signs and use of the point coordinates in the denominator were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Use the common normal vector and the difference between the planes’ constants in the parallel-plane distance formula." }, { "id": "cie-2019-w-33-q07iii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "vectors", "coordinate-geometry" ], "primaryTopicId": "vectors", "summary": "Find the vector equation of the line through the point perpendicular to the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates found the required line direction, and some used the wrong vector-equation notation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Use the plane’s normal vector as the line direction and write the line through the given point in vector form." }, { "id": "cie-2019-w-33-q08i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "8(i): Apply the stated numerical procedure to approximate the required root or value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2019-w-33-q08ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(ii)", "marks": 1, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Explain why the trapezium-rule estimate lies above or below the curve’s area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report lists this part among the most challenging; many gave only a gradient statement without showing how the trapezia compare to the curve.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Use the graph’s curvature and sketch the trapezia so the position of each straight chord relative to the curve is clear." }, { "id": "cie-2019-w-33-q08iii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(iii)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(iii): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2019-w-33-q09i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(i)", "marks": 9, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(i): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-33-q09ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(ii)", "marks": 1, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "exponentials", "summary": "State the limiting value of x as t becomes large.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "A fully correct response was not often seen because candidates did not use part (i) successfully in this linked deduction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Use the solution from part (i) to identify the value approached by x as t increases without bound." }, { "id": "cie-2019-w-33-q10i", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(i)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(i): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=13", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2019-w-33-q10ii", "paperId": "cie-2019-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2019, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(ii)", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral after making the stated substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_33.pdf#page=13", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report lists this as one of the most challenging parts and says many used the wrong transformed upper limit or did not use the requested substitution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 26, 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Apply the requested substitution, transform both limits correctly, then integrate by parts with brackets and signs kept consistent." }, { "id": "cie-2019-w-41-q01", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1: Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2019-w-41-q02", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-41-q03i", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(i): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2019-w-41-q03ii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(ii): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2019-w-41-q04i", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and string tension for two connected particles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 4 was one of the paper’s two most challenging questions; incomplete force diagrams and missing forces caused errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 29, 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Draw force diagrams for both particles, resolve along the slope and string, and use two consistent Newton’s second-law equations." }, { "id": "cie-2019-w-41-q04ii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(ii): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-41-q05i", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(i): Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2019-w-41-q05ii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(ii): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-41-q06i", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(i): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-41-q06ii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(ii): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-41-q06iii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(iii): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2019-w-41-q07i", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled by integrating the given velocity function until the particle is instantaneously at rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 7 was one of the most challenging, and the report notes that some candidates incorrectly used constant-acceleration formulae for the variable-velocity model.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 29, 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Solve v = 0 for the later rest time, integrate the velocity from the starting time to that time, and evaluate the displacement." }, { "id": "cie-2019-w-41-q07ii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-41-q07iii", "paperId": "cie-2019-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(iii): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_41.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2019-w-42-q01", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2019-w-42-q02i", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(i): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2019-w-42-q02ii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(ii): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2019-w-42-q03", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2019-w-42-q04i", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2019-w-42-q04ii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy equation to find the height reached by the lorry on the slope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says full credit required the energy method and many omitted the resistance work or used incorrect signs.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 32, 33, 34 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the kinetic-energy loss, include the work done against resistance over the distance travelled, and equate the balance to the gain in gravitational potential energy." }, { "id": "cie-2019-w-42-q05i", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find when two particles released one second apart collide and their collision height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This was identified as the most difficult part on the paper; many candidates ignored the launch delay or equated the two travelled distances.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 32, 33, 34 ], "tagConfidence": "reviewed", "answerGuide": "Write each particle’s displacement using its own elapsed time, account for the one-second launch delay, and solve when their separation closes." }, { "id": "cie-2019-w-42-q05ii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the particles’ velocities at the collision time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the most common error was using the same time value for both particles despite the one-second delay.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 32, 33, 34 ], "tagConfidence": "reviewed", "answerGuide": "Use the collision result from part (i) and apply v = u + at with the two different elapsed times." }, { "id": "cie-2019-w-42-q06i", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(i): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2019-w-42-q06ii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(ii): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2019-w-42-q06iii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(iii): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-42-q07i", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(i): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-42-q07ii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(ii): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2019-w-42-q07iii", "paperId": "cie-2019-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(iii): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_42.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2019-w-43-q01", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-43-q02", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to account for a non-constant resistance force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=5", "qpPages": [ 4, 4 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies the work-energy setup as more challenging and notes that candidates needed to distinguish the given work against resistance from a constant force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 35, 36, 37 ], "tagConfidence": "reviewed", "answerGuide": "Use the work-energy principle with the stated work against resistance, kinetic-energy change, and gain or loss in gravitational potential energy." }, { "id": "cie-2019-w-43-q03", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2019-w-43-q04i", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration from the two-stage journey using the given distance relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was more challenging and describes common confusion over the distance ratio and initial velocity.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 35, 36, 37 ], "tagConfidence": "reviewed", "answerGuide": "Apply a suitable constant-acceleration equation to each section, use the stated relationship between the distances, and solve for the shared acceleration." }, { "id": "cie-2019-w-43-q04ii", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(ii): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-43-q05i", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(i): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-43-q05ii", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2019-w-43-q05iii", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(iii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2019-w-43-q06i", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-43-q06ii", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the time when the particle’s acceleration is at a maximum or minimum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part was less well answered; some candidates differentiated acceleration but evaluated at the stated time instead of finding its stationary point.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 35, 36, 37 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the acceleration with respect to time, set its derivative to zero, and classify the stationary value using the resulting quadratic or completed square." }, { "id": "cie-2019-w-43-q07i", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(i): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2019-w-43-q07ii", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(ii): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=10", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-43-q07iii", "paperId": "cie-2019-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2019, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "kinematics", "functions" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph for the connected motion and the subsequent free motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_43.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was challenging and describes many incorrect curved or multi-segment graphs after the string broke.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 35, 36, 37 ], "tagConfidence": "reviewed", "answerGuide": "Show the two stages with straight-line segments, marking the break velocity and both relevant times on the axes." }, { "id": "cie-2019-w-51-q01", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "1: Use moments and the centre-of-mass geometry to determine the requested balance quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass using the shape’s symmetry or component moments, then take moments about the suspension or contact point." }, { "id": "cie-2019-w-51-q02i", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(i): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-51-q02ii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(ii): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-51-q03i", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-51-q03ii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(ii): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-51-q04i", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(i): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-51-q04ii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(ii): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-51-q05i", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(i): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-51-q05ii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the maximum speed of the particle using the spring energy equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists this among the most challenging parts; candidates needed to identify equilibrium and use a four-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Find the equilibrium position, then write the full energy balance between the stated position and equilibrium where speed is greatest." }, { "id": "cie-2019-w-51-q06i", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(i): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2019-w-51-q06ii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 7, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the unknown tension and natural length in the conical-pendulum setup.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this part quite challenging and describes the simultaneous force equations required before finding the natural length.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Resolve vertically for the tensions, apply Newton’s second law horizontally, then use Hooke’s law to determine the natural length." }, { "id": "cie-2019-w-51-q07i", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use similar triangles to find the required angle in the composite-lamina diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part appears in the report’s list of the more challenging parts and requires a two-step geometric construction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Establish the corresponding side lengths from the similar triangles, then use the right-triangle geometry to find the angle." }, { "id": "cie-2019-w-51-q07ii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass position by taking moments about point A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report lists part (ii) as challenging and specifies the geometry and moments calculation needed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the needed triangle length, then take moments of the component laminae about A and solve for the centre-of-mass distance." }, { "id": "cie-2019-w-51-q07iii", "paperId": "cie-2019-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction from horizontal and vertical equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_51.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report lists this part among the challenging ones and requires both force components before applying F = μR.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the forces to find the friction force and normal reaction, then divide friction by the normal reaction." }, { "id": "cie-2019-w-52-q01", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-52-q02i", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(i): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-52-q02ii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-52-q03", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3: Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-52-q04i", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(i): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-52-q04ii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(ii): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-52-q05i", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(i): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2019-w-52-q05ii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the two tensions for the particle moving in the vertical circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this part more challenging and needed both vertical resolution and horizontal Newton’s second law.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 40, 41 ], "tagConfidence": "reviewed", "answerGuide": "Resolve vertically for the forces and apply Newton’s second law horizontally, then solve the simultaneous equations for both tensions." }, { "id": "cie-2019-w-52-q06i", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-52-q06ii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(ii): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2019-w-52-q06iii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "integration", "mechanics" ], "primaryTopicId": "integration", "summary": "Find the range of projection speeds for which the particle’s minimum speed is positive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates stopped after finding v rather than v²/2 and then failed to use the value from part (ii) correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 40, 41 ], "tagConfidence": "reviewed", "answerGuide": "Integrate to obtain v²/2 as a function of x, use the zero-acceleration position from part (ii) to locate its minimum, then require that minimum value to be positive and express the result in terms of the projection speed." }, { "id": "cie-2019-w-52-q07i", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(i): Use moments and the centre-of-mass geometry to determine the requested balance quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass using the shape’s symmetry or component moments, then take moments about the suspension or contact point." }, { "id": "cie-2019-w-52-q07ii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(ii): Use moments and the centre-of-mass geometry to determine the requested balance quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass using the shape’s symmetry or component moments, then take moments about the suspension or contact point." }, { "id": "cie-2019-w-52-q07iii", "paperId": "cie-2019-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 5, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the limiting weight values from the two possible angular positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_52.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes this part as challenging and explains that candidates needed two separate moment equations for the two limiting positions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 40, 41 ], "tagConfidence": "reviewed", "answerGuide": "Take moments about A for each limiting position, use the corresponding angle with the vertical, and solve for both boundary values of W." }, { "id": "cie-2019-w-53-q01", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "1: Use moments and the centre-of-mass geometry to determine the requested balance quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass using the shape’s symmetry or component moments, then take moments about the suspension or contact point." }, { "id": "cie-2019-w-53-q02i", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(i): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-53-q02ii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(ii): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-53-q03i", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-53-q03ii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(ii): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-53-q04i", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(i): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2019-w-53-q04ii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(ii): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-53-q05i", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(i): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2019-w-53-q05ii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the maximum speed of the particle using the spring energy equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report lists this among the most challenging parts; candidates needed to identify equilibrium and use a four-term energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Find the equilibrium position, then write the full energy balance between the stated position and equilibrium where speed is greatest." }, { "id": "cie-2019-w-53-q06i", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(i): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2019-w-53-q06ii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 7, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the unknown tension and natural length in the conical-pendulum setup.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this part quite challenging and describes the simultaneous force equations required before finding the natural length.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Resolve vertically for the tensions, apply Newton’s second law horizontally, then use Hooke’s law to determine the natural length." }, { "id": "cie-2019-w-53-q07i", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use similar triangles to find the required angle in the composite-lamina diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part appears in the report’s list of the more challenging parts and requires a two-step geometric construction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Establish the corresponding side lengths from the similar triangles, then use the right-triangle geometry to find the angle." }, { "id": "cie-2019-w-53-q07ii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass position by taking moments about point A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report lists part (ii) as challenging and specifies the geometry and moments calculation needed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the needed triangle length, then take moments of the component laminae about A and solve for the centre-of-mass distance." }, { "id": "cie-2019-w-53-q07iii", "paperId": "cie-2019-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics 2", "year": 2019, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(iii)", "marks": 4, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction from horizontal and vertical equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report lists this part among the challenging ones and requires both force components before applying F = μR.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the forces to find the friction force and normal reaction, then divide friction by the normal reaction." }, { "id": "cie-2019-w-61-q01", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-61-q02i", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-61-q02ii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the value of the binomial parameter using the probability that at least one event occurs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was challenging for most candidates, especially recognising “at least one” as the complement of “none”.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 44, 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Write the required probability as one minus the probability of no occurrences, equate it to the stated value, and solve for the parameter." }, { "id": "cie-2019-w-61-q03i", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-61-q03ii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-61-q04i", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-61-q04ii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-61-q04iii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(iii)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a conditional probability from the discrete probability distribution table.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates found this part challenging; a common error was to quote the probability of X = 2 directly from part (i).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 44, 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Use the appropriate table probabilities for the numerator and denominator of the conditional probability, then simplify the ratio." }, { "id": "cie-2019-w-61-q05i", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(i): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2019-w-61-q05ii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(ii): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2019-w-61-q05iii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(iii): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2019-w-61-q05iv", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(iv)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "5(iv): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2019-w-61-q06i", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(i): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-61-q06ii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-61-q06iii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(iii)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count selections of letters satisfying the stated repeated-letter conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few fully correct solutions were seen; candidates often treated repeated letters as individually identifiable or omitted a valid case.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 44, 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "List the disjoint cases that meet the conditions, count each case while accounting for indistinguishable repeated letters, and add the counts." }, { "id": "cie-2019-w-61-q07i", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-61-q07ii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a probability by inverting a normal distribution condition and identifying a negative z-value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found this part challenging and often treated the given probability as a z-value instead of identifying the tail area.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 44, 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Sketch the normal curve, identify the correct tail area, use the corresponding negative standard score, and solve the standardisation equation." }, { "id": "cie-2019-w-61-q07iii", "paperId": "cie-2019-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(iii)", "marks": 3, "topicIds": [ "binomial", "distributions" ], "primaryTopicId": "binomial", "summary": "Use the result from the normal model in a binomial calculation for the required number of outcomes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_61.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates did not attempt this part; the report describes successful answers as using the complement of the earlier probability before applying the binomial model.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 44, 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Use the correct probability boundary from part (i), take its complement if required, and evaluate the corresponding binomial probability." }, { "id": "cie-2019-w-62-q01i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(i): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2019-w-62-q01ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-62-q02i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "2(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-62-q02ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(ii): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2019-w-62-q03i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a histogram using class boundaries and frequency densities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates found this part challenging; common errors included using class endpoints instead of boundaries and dividing class width by frequency.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 47, 48, 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Use continuous class boundaries, divide each frequency by its class width, choose a readable scale, and label both axes fully." }, { "id": "cie-2019-w-62-q03ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-62-q04i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-62-q04ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-62-q05i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-62-q05ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-62-q06i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-62-q06ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the symmetry of a normal distribution with the earlier probability to find a related probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not attempt this part, and probabilities greater than 1 were common among attempted solutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 47, 48, 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Draw and label the symmetric normal curve, reflect the relevant boundary across the mean, and use the probability from part (i) consistently." }, { "id": "cie-2019-w-62-q06iii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Solve for the unknown normal-distribution parameters from the stated probability conditions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was challenging and that inaccurate z-values and incorrect initial probability equations were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 47, 48, 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Form simultaneous standardisation equations from the two probability conditions, use the correct signed z-values, and solve the algebra carefully." }, { "id": "cie-2019-w-62-q07i", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(i)", "marks": 1, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements of the repeated-letter word that satisfy the stated placement condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates failed to apply the extra conditions and counted arrangements of all nine letters rather than the six distinct items.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 47, 48, 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Identify the distinct objects after grouping repeated letters, represent the placement condition with a simple arrangement, and count each valid case once." }, { "id": "cie-2019-w-62-q07ii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-62-q07iii", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(iii): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2019-w-62-q07iv", "paperId": "cie-2019-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(iv)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count selections meeting the stated repeated-letter composition requirement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_62.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few fully correct solutions were seen; candidates often treated repeated letters as individually identifiable or omitted a valid composition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 47, 48, 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate the possible valid compositions, account for repeated letters as indistinguishable, and combine the case counts." }, { "id": "cie-2019-w-63-q01i", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-63-q01ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-63-q02i", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(i): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-63-q02ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-63-q03i", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(i): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=6", "qpPages": [ 5, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2019-w-63-q03ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(ii): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2019-w-63-q04i", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-63-q04ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the number of people whose normally distributed heights lie within half a standard deviation of the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was more challenging; common errors included using z = ±1 or ±1.5 and mishandling the final truncation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Standardise the two boundaries to z = ±0.5, find the probability between them, multiply by the group size, and truncate to a whole number as requested." }, { "id": "cie-2019-w-63-q05i", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "5(i): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2019-w-63-q05ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(ii): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2019-w-63-q05iii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Read the requested upper-tail proportion from the cumulative-frequency graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This part was not well answered; most candidates confused “at least T minutes” with “fewer than T minutes”.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Interpret “at least T minutes” as the upper tail, use 80 rather than 120 observations for that tail, and read the corresponding time from the graph." }, { "id": "cie-2019-w-63-q05iv", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iv)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(iv): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2019-w-63-q06i", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-63-q06ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-63-q06iii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(iii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-63-q06iv", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iv)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(iv): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-63-q06v", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(v)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(v): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2019-w-63-q07ia", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(i)(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-63-q07ib", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)(b)", "marks": 2, "topicIds": [ "binomial", "probability" ], "primaryTopicId": "binomial", "summary": "Find the probability that none of the six people meet the stated supporter condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates were successful; many calculated the probability that all six were supporters instead of none.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with n = 6 and the probability of a supporter, then calculate the zero-success probability." }, { "id": "cie-2019-w-63-q07ii", "paperId": "cie-2019-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2019, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(ii): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_63.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-72-q01i", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(i): Identify the distribution and state its parameter values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the count model from the sampling setup and state its parameter using the given sample size and event probability." }, { "id": "cie-2019-w-72-q01ii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2019-w-72-q01iii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(iii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(iii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2019-w-72-q02ia", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)(a)", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "2(i)(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2019-w-72-q02ib", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(i)(b)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "2(i)(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2019-w-72-q02ii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(ii)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "State the population condition needed for the sample mean to have the stated exact distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates gave the correct answer; vague claims about the sample mean being normal were insufficient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 54, 55 ], "tagConfidence": "reviewed", "answerGuide": "State that the population itself must follow the specified normal distribution, distinguishing this condition from a large-sample approximation." }, { "id": "cie-2019-w-72-q03i", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(i)", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-72-q03ii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-72-q04i", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(i): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2019-w-72-q04ii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(ii): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2019-w-72-q04iii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(iii)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the probability between the mean and median of the given continuous distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This part was more challenging; candidates often used incorrect integration limits, while the concise symmetry-of-area method was rarely seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 54, 55 ], "tagConfidence": "reviewed", "answerGuide": "Find or use the median, integrate the density between the median and the mean, and use the correct limits and probability direction." }, { "id": "cie-2019-w-72-q05i", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-72-q05ii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(ii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-72-q06i", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(i)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Determine the critical region for the Poisson significance test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates calculated only one of the two required probabilities or did not clearly justify the boundary of the critical region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 54, 55 ], "tagConfidence": "reviewed", "answerGuide": "State the hypotheses, calculate both required tail probabilities, compare them with the significance level, and select the largest valid critical region." }, { "id": "cie-2019-w-72-q06ii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Explain a Type I error in the context of the Poisson test and find its probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates successfully explained the contextual Type I error or found its probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 54, 55 ], "tagConfidence": "reviewed", "answerGuide": "Describe rejecting the null hypothesis when it is true in the question context, then calculate the probability of the rejection region under the null model." }, { "id": "cie-2019-w-72-q06iii", "paperId": "cie-2019-w-72", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/72", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(iii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(iii): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_72.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_72.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2019-w-73-q01i", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(i): Apply expectation and variance rules to the transformed random variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use linearity of expectation and Var(aX+b) = a²Var(X); add means and variances for independent variables." }, { "id": "cie-2019-w-73-q01ii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(ii): Apply expectation and variance rules to the transformed random variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use linearity of expectation and Var(aX+b) = a²Var(X); add means and variances for independent variables." }, { "id": "cie-2019-w-73-q02i", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-73-q02ii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2019-w-73-q03i", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(i)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(i): Calculate or interpret the requested confidence interval for the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate z or t critical value with the sample standard error, then calculate and round both interval endpoints as requested." }, { "id": "cie-2019-w-73-q03ii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2019-w-73-q04", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "statistical-testing", "binomial" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the binomial significance test for the proportion of trains arriving on time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the tail probability and context were essential; candidates often used the wrong tail or a single point probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 56, 57, 58 ], "tagConfidence": "reviewed", "answerGuide": "State the independence assumption and hypotheses, calculate the correct lower-tail probability, compare it with the significance level, and conclude in context." }, { "id": "cie-2019-w-73-q05i", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(i): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2019-w-73-q05ii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2019-w-73-q05iiia", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(iii)(a): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2019-w-73-q05iiib", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(iii)(b)", "marks": 2, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "Solve the inequality involving factorial expressions for the integer n.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many different approaches were tried unsuccessfully; candidates commonly simplified the factorial ratio incorrectly or could not complete the logarithmic approach.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 56, 57, 58 ], "tagConfidence": "reviewed", "answerGuide": "Simplify the factorial ratio correctly, form the resulting inequality, and establish the integer boundary with sufficient working." }, { "id": "cie-2019-w-73-q06i", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(i)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2019-w-73-q06ii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(ii): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2019-w-73-q06iii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(iii)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the variance of the continuous distribution from its density and expected value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A common error was not squaring E(X), and some candidates stopped after evaluating only one part of the variance expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 56, 57, 58 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate E(X²) from the density, subtract [E(X)]², and include the full integral and limits." }, { "id": "cie-2019-w-73-q07i", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(i)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(i): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2019-w-73-q07ii", "paperId": "cie-2019-w-73", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2019, "session": "October/November", "paperCode": "9709/73", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error under the alternative distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_qp_73.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_ms_73.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not find the critical value for the acceptance region and instead used an unrelated boundary.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w19_er.pdf", "sourceEvidencePages": [ 56, 57, 58 ], "tagConfidence": "reviewed", "answerGuide": "Use the critical acceptance region from the earlier part and evaluate its probability under the stated alternative model." }, { "id": "cie-2020-m-12-q01", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Decide whether f(x) = 1/(3x + 2) + x² is increasing or decreasing for x < −1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and show f′(x) = 2x − 3/(3x + 2)² is negative throughout x < −1; f is decreasing." }, { "id": "cie-2020-m-12-q02", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the two transformations taking y = f(x) to y = 1 + f(x/2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply a stretch parallel to the x-axis by scale factor 2, then translate 1 unit upwards." }, { "id": "cie-2020-m-12-q03", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the region under y = x² + 1, between the y-axis and y = 5, about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use discs with radius x = √(y − 1): V = π∫₁⁵(y − 1)dy = 8π units³." }, { "id": "cie-2020-m-12-q04", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use the given rates of change on y = x² − 2x − 3 to find the point’s x-coordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with respect to time: dy/dt = (2x − 2)dx/dt. Set this to 4 with dx/dt = 6 to obtain x = 4/3." }, { "id": "cie-2020-m-12-q05", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the given trigonometric equation for all angles from 0° to 90° inclusive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Most candidates found one solution but did not consider the additional solution arising from sin θ = 0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Clear the denominators and rewrite tan θ as sin θ/cos θ. Solve the resulting equation while checking separately for sin θ = 0, which can be lost if the equation is divided by sin θ." }, { "id": "cie-2020-m-12-q06a", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the possible values of a when the coefficient of 1/x in (2x + a/x²)⁵ is 720.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Select the r = 2 term: 10(2x)³(a/x²)² = 80a²/x. Equating its coefficient to 720 gives a = ±3." }, { "id": "cie-2020-m-12-q06b", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Using the possible values of a from part (a), find the coefficient of x⁻⁷ in the expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says selecting the required term in part (b) proved more challenging, although many correct answers were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "For each valid value of a, identify the binomial term whose power of x is −7 and evaluate its coefficient. Apply the coefficient condition from part (a) consistently." }, { "id": "cie-2020-m-12-q07", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 6, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the area of the region between the two arcs in the 6 cm sector with angle 0.8 radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "First find OC = 6cos(0.8). Subtract the inner sector area from triangle OCA: ½·OC·AC − ½·OC²·0.8 = 2.01 cm² (3 s.f.)." }, { "id": "cie-2020-m-12-q08a", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Express the $600 first-year bonus as a percentage of the $30,000 basic salary.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate 600/30,000 × 100 = 2%." }, { "id": "cie-2020-m-12-q08b", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Find the bonus in year 24 as a percentage of the basic salary paid that year.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The bonus is 600 + 23×100 = $2,900; the salary is 30,000(1.03)²³ = $59,207.60. The percentage is 4.90% (3 s.f.)." }, { "id": "cie-2020-m-12-q09a", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Complete the square for 2x² + 12x + 11 in the form 2(x + a)² + b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor 2 from the quadratic terms and complete the square: 2(x + 3)² − 7, so a = 3 and b = −7." }, { "id": "cie-2020-m-12-q09b", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find f⁻¹(x) for f(x) = 2x² + 12x + 11 with x ≤ −4, and state its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write f(x) = 2(x + 3)² − 7 and choose the negative square-root branch: f⁻¹(x) = −3 − √((x + 7)/2), defined for x ≥ −5." }, { "id": "cie-2020-m-12-q09c", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "For k = −1, solve the composite-function equation f(g(x)) = 193.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only candidates who considered the original domain selected the correct root of the composite equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute g(x) into the completed-square form of f, solve the resulting quadratic, then use the original domain restriction x ≤ −4 to select the valid root." }, { "id": "cie-2020-m-12-q09d", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(d)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the largest k for which f∘g is defined when g(x) = 2x − 3 for x ≤ k and f has domain x ≤ −4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Require the largest possible g(x) to be at most −4: 2k − 3 ≤ −4. Thus k = −1/2." }, { "id": "cie-2020-m-12-q10a", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the positive a for which the stationary point has coordinates (a, 14), given dy/dx = 2√(x + 3) − x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set 2√(a + 3) − a = 0. Squaring gives a² − 4a − 12 = 0; the positive root is a = 6." }, { "id": "cie-2020-m-12-q10b", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Determine the nature of the stationary point at x = a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate again: d²y/dx² = 1/√(x + 3) − 1. At x = 6 this is −2/3, so the point is a local maximum." }, { "id": "cie-2020-m-12-q10c", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the curve whose gradient is 2√(x + 3) − x and which passes through (6, 14).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate to get y = (4/3)(x + 3)³ᐟ² − x²/2 + C; substituting (6,14) gives C = −4." }, { "id": "cie-2020-m-12-q11a", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the quadratic equation in tan x for every solution in the interval 0° to 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Some answers omitted the solution arising from the negative value of tan x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Factor or solve the quadratic for tan x, then use each positive and negative tangent value to list all angles in the stated interval." }, { "id": "cie-2020-m-12-q11b", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the values of k for which 3tan²x − 5tanx + k = 0 has no real solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat it as a quadratic in tan x. Its discriminant must be negative: 25 − 12k < 0, so k > 25/12." }, { "id": "cie-2020-m-12-q11c", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find k and all angles in 0° ≤ x ≤ 180° when the tangent equation has three solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set k = 0 so tan x(3tan x − 5) = 0. Include x = 0°, 180° from tan x = 0 and x = tan⁻¹(5/3) ≈ 59.0°." }, { "id": "cie-2020-m-12-q12a", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle whose diameter endpoints are (−3,−5) and (7,3).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The centre is (2,−1) and r² = 41, giving (x − 2)² + (y + 1)² = 41." }, { "id": "cie-2020-m-12-q12b", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle after C₁ is translated by vector (8,4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the centre (2,−1) to (10,3) and keep r² = 41: (x − 10)² + (y − 3)² = 41." }, { "id": "cie-2020-m-12-q12c", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the common chord RS of the two circles lies on y = −2x + 13.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the two circle equations to eliminate x² and y²; the resulting radical axis simplifies to y = −2x + 13." }, { "id": "cie-2020-m-12-q12d", "paperId": "cie-2020-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(d)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the line RS to show that the x-coordinates of R and S satisfy 5x² − 60x + 159 = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute y = −2x + 13 into either circle equation and simplify to the stated quadratic." }, { "id": "cie-2020-m-22-q01", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 2sin(θ + 30°) + 5cosθ = 2sinθ for 0° < θ < 90°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand the compound angle, collect sine and cosine terms, and solve for tanθ. The valid angle is approximately 87.4°." }, { "id": "cie-2020-m-22-q02a", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide 4x³ + 17x² + 9x by x² + 5x + 6 and show the remainder is 18.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Long-divide to get quotient 4x − 3 and remainder 18." }, { "id": "cie-2020-m-22-q02b", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Hence solve 4x³ + 17x² + 9x − 18 = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor as (x² + 5x + 6)(4x − 3) = 0. The roots are −3, −2 and 3/4." }, { "id": "cie-2020-m-22-q03", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find the positive a satisfying ∫ₐ³ᵃ 2/(2x − 5) dx = ln(7/2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate to ln|2x − 5|, apply the limits, combine logarithms and solve (6a − 5)/(2a − 5) = 7/2. This gives a = 25/2." }, { "id": "cie-2020-m-22-q04", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent to 3x² − y² − 4ln(2y + 3) = 26 at (3, −1).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly to find dy/dx = 3 at the point, then use y + 1 = 3(x − 3), giving y = 3x − 10." }, { "id": "cie-2020-m-22-q05a", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch y = |x + 2k| and y = |2x − 3k| and give their axis intercepts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the two V-shaped graphs. Their x-intercepts are −2k and 3k/2; their y-intercepts are 2k and 3k." }, { "id": "cie-2020-m-22-q05b", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the two intersection coordinates of y = |x + 2k| and y = |2x − 3k|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the two sign cases: the intersections are (5k, 7k) and (k/3, 7k/3)." }, { "id": "cie-2020-m-22-q05c", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the largest value of t satisfying the inequality between the two modulus expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a more challenging part and says few completely correct solutions were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the intersection information from part (b) to identify where the two absolute-value graphs have the required ordering, then solve the boundary equation and apply the positive-k condition." }, { "id": "cie-2020-m-22-q06a", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "Show that the x-coordinate of P on y = x³e^(0.2x), where the gradient is 15, satisfies the given iteration equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate to get dy/dx = e^(0.2x)x²(3 + 0.2x). Equating this to 15 and rearranging gives x = √(75e^(−0.2x)/(15 + x))." }, { "id": "cie-2020-m-22-q06b", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the x-coordinate of P lies between 1.7 and 1.8.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute 1.7 and 1.8 into the equation from part (a) and show the residuals have opposite signs (equivalently the iteration brackets the root)." }, { "id": "cie-2020-m-22-q06c", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "Use an iteration based on the equation from part (a) to find the point’s x-coordinate to 4 significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates omitted the final iteration needed to justify the result or did not give the final answer to the required 4 significant figures.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Choose an iterative formula from the displayed rearrangement, show successive values to 6 significant figures, and state the converged result to 4 significant figures." }, { "id": "cie-2020-m-22-q07a", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the exact x-coordinate where y = 4sin²x + 8sinx + 3 crosses the x-axis at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor as (2sinx + 1)(2sinx + 3) = 0. The valid root shown is sinx = −1/2, giving x = 7π/6." }, { "id": "cie-2020-m-22-q07b", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the exact gradient of y = 4sin²x + 8sinx + 3 at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate to get dy/dx = 8cosx(sinx + 1); at x = 7π/6 the gradient is −2√3." }, { "id": "cie-2020-m-22-q07c", "paperId": "cie-2020-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of the shaded region bounded by the curve, x = 0 and the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Errors in substitution and the upper limit meant that few correct final answers were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the root from part (a) as the upper limit, integrate the trigonometric terms exactly, and substitute both limits carefully." }, { "id": "cie-2020-m-32-q01a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch the graph of y = |x − 2|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Question 1 was among the more challenging questions; many graphs were incorrect, often a single straight line or one that missed the vertex or required domain.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Draw the symmetric V with vertex (2, 0), state its axis intercepts and show it across at least −1 ≤ x ≤ 5 rather than plotting isolated points." }, { "id": "cie-2020-m-32-q01b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the inequality |x − 2| < 3x − 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Few candidates who used the quadratic approach continued to state the final condition x > 3/2; errors also came from squaring only one side.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Split at x = 2 or use the graph from part (a), solve the two linear inequalities, and state the complete solution x > 3/2." }, { "id": "cie-2020-m-32-q02", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve ln 3 + ln(2x + 5) = 2ln(x + 2) in simplified exact form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms, respecting x > −2, then solve 6x + 15 = (x + 2)². The valid root is x = 1 + 2√3." }, { "id": "cie-2020-m-32-q03a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "functions", "trigonometry", "proof" ], "primaryTopicId": "functions", "summary": "Use a pair of sketches to show that sec x = 2 − ½x has exactly one root in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says sketching sec x was challenging and few candidates produced a representative graph; many stopped too early or extended beyond the asymptote.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Sketch y = sec x and y = 2 − ½x over 0 ≤ x < π/2, showing the correct intercept, asymptote and line behaviour so the single intersection is clear." }, { "id": "cie-2020-m-32-q03b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the root from part (a) lies between 0.8 and 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate sec x − (2 − x/2) at 0.8 and 1 and show the signs differ." }, { "id": "cie-2020-m-32-q03c", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use the given iteration to find the root to 2 decimal places, showing 4 d.p. iterates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Start with a value in (0.8,1), apply the stated inverse-cosine iteration repeatedly, and round only the converged root to 2 d.p." }, { "id": "cie-2020-m-32-q04", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate ∫ from π/6 to π/3 of x sec²x dx exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate by parts: ∫x sec²x dx = x tanx + ln|cosx|. Applying the limits gives 5π√3/18 − (ln 3)/2." }, { "id": "cie-2020-m-32-q05a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Show the stated identity relating the sum of two trigonometric fractions to 2 cot 2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "Candidates who expanded the triple-angle functions before finding a common denominator found it challenging to establish the identity.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Combine the two fractions over a common denominator first, then use angle-sum identities to simplify the numerator and denominator to the double-angle ratio." }, { "id": "cie-2020-m-32-q05b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the result from part (a) to solve the linked equation for 0 < x < π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 10, "difficultyEvidence": "The report identifies incorrect degree mode and incomplete or inaccurate answers as recurring errors in this challenging question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Replace the left side by 2 cot 2x, solve for cot 2x, and list every solution in the stated radian interval without switching to degrees." }, { "id": "cie-2020-m-32-q06a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 7, "topicIds": [ "exponentials", "integration", "trigonometry" ], "primaryTopicId": "exponentials", "summary": "Solve the separable differential equation using the condition y = 0 when x = 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates made progress beyond integrating the exponential term; successful candidates rarely reached the required parameter values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the y- and x-terms, integrate using the arctangent and exponential forms, apply the initial condition, and solve for y with the appropriate inverse-trigonometric branch." }, { "id": "cie-2020-m-32-q06b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "exponentials", "trigonometry" ], "primaryTopicId": "exponentials", "summary": "State the limit of y as x tends to infinity for the differential-equation solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says most candidates found this follow-through part challenging because errors in part (a) left few with the required exponential expression and finite limit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Let e⁻ˣ tend to zero in y = ½tan(2/e − 2e⁻ˣ): y tends to ½tan(2/e)." }, { "id": "cie-2020-m-32-q07a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Show that implicit differentiation of x³ + 3xy² − y³ = 5 gives the stated dy/dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides and collect y′ terms: dy/dx = (x² + y²)/(y² − 2xy)." }, { "id": "cie-2020-m-32-q07b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the points on the implicit curve where the tangent is parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 11, "difficultyEvidence": "Many candidates interpreted a tangent parallel to the y-axis as dy/dx equal to zero or one, and others omitted the solution with y = 0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative from part (a): a vertical tangent occurs when its denominator is zero, then solve the resulting condition together with the curve equation, including all valid points." }, { "id": "cie-2020-m-32-q08a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find vectors OM and MN in terms of i, j and k for the cuboid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use OA = 2i, OC = 3j and OD = 2k. Since AM is one-third of AB, OM = 2i + j; the midpoint N gives MN = −i + 2j + 2k." }, { "id": "cie-2020-m-32-q08b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Give a vector equation for the line through M and N.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 12, "difficultyEvidence": "The report includes this part among the more challenging questions and notes errors in attaching the parameter to the wrong vector and in the required vector notation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the position vector of M and the direction vector MN from part (a), and write the equation in vector form with a parameter on the direction vector." }, { "id": "cie-2020-m-32-q08c", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the position vector of the foot of the perpendicular from D to the line through M and N.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part particularly challenging; most candidates chose the wrong vector, and only very few used the minimum-distance method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Write a general point P on line MN using the vector equation from part (b). Set DP·MN = 0 to find the parameter, then give OP as the position vector of the foot." }, { "id": "cie-2020-m-32-q09a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra", "binomial" ], "primaryTopicId": "algebra", "summary": "Express (2 + 11x − 10x²)/((1 + 2x)(1 − 2x)(2 + x)) in partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve into −2/(1 + 2x) + 1/(1 − 2x) + 4/(2 + x)." }, { "id": "cie-2020-m-32-q09b", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "algebra", "binomial" ], "primaryTopicId": "algebra", "summary": "Expand the partial-fraction expression up to and including the x² term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use geometric expansions about x = 0 and combine coefficients to obtain 1 + 5x − (7/2)x²." }, { "id": "cie-2020-m-32-q10a", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the simultaneous equations for the complex numbers v and w, giving each in Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this question was challenging and many candidates used confused substitutions, including treating a complex unknown as real; few good attempts were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Eliminate one unknown to obtain a single complex equation, or write v and w in real and imaginary parts and equate real and imaginary components before solving." }, { "id": "cie-2020-m-32-q10bi", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(i)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Sketch the Argand locus |z − 2 − 3i| = 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw a circle of radius 1 centred at (2,3) on the Argand diagram, using the same scale on both axes." }, { "id": "cie-2020-m-32-q10bii", "paperId": "cie-2020-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the least value of the argument of a complex number on the stated circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes this part as challenging; few used the correct method and many confused which trigonometric ratio represented the tangent angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the circle centre and radius from the locus to form the right triangle from the origin to a tangent point. The tangent condition gives sin of the angle at the origin as radius divided by centre distance; subtract that angle from the centre argument." }, { "id": "cie-2020-m-42-q01a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the lorry engine power from 750,000 J of work done in 10 s.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Power = work/time = 750,000/10 = 75,000 W (75 kW)." }, { "id": "cie-2020-m-42-q01b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the lorry’s acceleration at speed 25 m s⁻¹ when resistance is 2400 N.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At constant power, driving force = P/v = 3000 N. Net force is 600 N, so a = 600/16,000 = 0.0375 m s⁻²." }, { "id": "cie-2020-m-42-q02a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the time for the particle to travel 1.44 m from rest with acceleration 2 m s⁻².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use s = ½at²: 1.44 = t², so t = 1.2 s." }, { "id": "cie-2020-m-42-q02b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction for the particle pulled by the angled 3 N force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the 3 N force using tanα = 3/4, giving horizontal 2.4 N and vertical 1.8 N. With g = 10, R = 2.2 N and friction = 2.4 − 0.8 = 1.6 N, so μ = 8/11 ≈ 0.727." }, { "id": "cie-2020-m-42-q03a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the rise h when the 0.2 kg particle slows from 5 to 3 m s⁻¹ with no resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Conserve mechanical energy: ½m(25 − 9) = mgh. With g = 10, h = 0.8 m." }, { "id": "cie-2020-m-42-q03b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle’s speed at C when it loses 3.1 J of work against resistance between A and C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part challenged many candidates; common errors reused the speed at B from part (a) or treated 3.1 J as a force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use an energy balance from A to C, setting the reference level at C. Include the initial kinetic energy, the net change in gravitational potential energy and the 3.1 J lost to resistance." }, { "id": "cie-2020-m-42-q04a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the cyclist’s constant acceleration from the travel times and the given AB-to-BC distance ratio.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this part challenging for many candidates and notes that a significant number incorrectly assumed the cyclist started from rest at A.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Let the speed at A be u and acceleration be a. Use the constant-acceleration displacement formula on AB and BC for two seconds each, set AB = 4/5 BC, and solve without assuming u = 0." }, { "id": "cie-2020-m-42-q04b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance AC for the cyclist.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find u = vB − 2a = 3.5 m s⁻¹, then use s = 4u + ½a(4²) = 18 m." }, { "id": "cie-2020-m-42-q05a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find F when α = 60° and the resultant of the four coplanar forces is horizontal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve vertically and set the resultant vertical component to zero: Fsin60° + 4sin150° − 6 = 0, giving F = 4.62 N (3 s.f.)." }, { "id": "cie-2020-m-42-q05b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find F and α when the four coplanar forces are in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set both horizontal and vertical force components to zero. Solving gives F ≈ 4.03 N and α ≈ 83.4°." }, { "id": "cie-2020-m-42-q06a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the force in the tow-bar while the car and trailer are braking.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies several common errors: using the whole car–trailer system where tow-bar force cancels, omitting trailer resistance, or giving the thrust as a negative magnitude.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Consider the trailer alone and apply Newton’s second law along the track, including its 200 N resistance and the −12 m s⁻² acceleration. Report the magnitude of the resulting thrust." }, { "id": "cie-2020-m-42-q06b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the braking force acting on the car.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates often omitted forces and commonly made sign errors when finding the braking force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law either to the whole car–trailer system with both resistance forces, or to the car using the tow-bar force from part (a). Keep the direction signs consistent." }, { "id": "cie-2020-m-42-q06c", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show that the car hits the stationary van at 8 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v² = u² + 2as with u = 22, a = −12 and s = 17.5; v² = 64, so v = 8 m s⁻¹." }, { "id": "cie-2020-m-42-q06d", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the post-collision speeds to find the mass of the van.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report notes a common error of omitting the trailer from momentum or using the initial 22 m s⁻¹ speed instead of the collision speed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply conservation of momentum to the car, trailer and van as one system, using the car and trailer’s combined mass and the speed just before collision from part (c)." }, { "id": "cie-2020-m-42-q07a", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the particle is instantaneously at rest in the first 6 seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate s = t² − 3t + 2: v = 2t − 3. Set v = 0 to get t = 1.5 s." }, { "id": "cie-2020-m-42-q07b", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the velocities immediately before and after the rebound at t = 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each displacement rule. Just before impact v = 9 m s⁻¹; just after, v = −5 m s⁻¹." }, { "id": "cie-2020-m-42-q07c", "paperId": "cie-2020-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance the particle travels in the first 10 seconds, including its rebound at t = 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part quite challenging; many candidates failed to account for the negative displacement at rest or confused distance with displacement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Find when s = 0 and when velocity is zero in the first interval. Split the motion at t = 1.5 and t = 6, track changes of direction, and add the absolute distances over each segment." }, { "id": "cie-2020-m-52-q01", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the coach selections when Ranuf must be in the coach and Saed in the car.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the other 34 coach members from the remaining 38: C(38,34) = 73,815." }, { "id": "cie-2020-m-52-q01a", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that the first 1 or 6 appears on throw 3, 4 or 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "With success probability 1/3, sum (2/3)²(1/3) + (2/3)³(1/3) + (2/3)⁴(1/3) = 76/243." }, { "id": "cie-2020-m-52-q01b", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Give the probability distribution for X, the number of 1s or 6s in three throws.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use X ~ B(3,1/3): probabilities for X = 0,1,2,3 are 8/27, 12/27, 6/27 and 1/27." }, { "id": "cie-2020-m-52-q01c", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find E(X) for the three throws.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the distribution or np: E(X) = 3(1/3) = 1." }, { "id": "cie-2020-m-52-q03a", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation when apple weights are N(82, σ²) and P(W > 87) = 0.22.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise 87 and use Φ(5/σ) = 0.78, so 5/σ ≈ 0.772 and σ ≈ 6.48 g." }, { "id": "cie-2020-m-52-q03b", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability an apple weight differs from 82 g by less than 4 g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use σ ≈ 6.48 and symmetry: P(|W − 82| < 4) = 2Φ(4/6.48) − 1 ≈ 0.463." }, { "id": "cie-2020-m-52-q04a", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count candle arrangements with a red candle at each end.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Fix the two red candles at the ends, then choose the positions of the three blue candles among the remaining nine: C(9,3) = 84." }, { "id": "cie-2020-m-52-q04b", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "Count the arrangements with all three blue candles together and the two red candles separated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates found the context challenging; a significant number failed to remove repeated arrangements for the identical green candles or did not keep the blue block together.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat the three blue candles as one block and arrange this with six identical green candles in 7!/6! ways. There are then eight gaps; choose two different gaps for the identical red candles using 8C2." }, { "id": "cie-2020-m-52-q05a", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that fewer than six adults in a sample of eight own a car.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that a significant number included 6 even though the condition was “less than 6”.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use X ~ B(8, 0.7) and sum P(X = 0) through P(X = 5), or subtract P(X ≥ 6) from 1; do not include six in the required event." }, { "id": "cie-2020-m-52-q05b", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use a normal approximation to find the probability that more than 75 of 120 adults own a car.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For B(120,0.7), use mean 84 and variance 25.2 with continuity correction P(X ≥ 76) ≈ P(N > 75.5), about 0.955." }, { "id": "cie-2020-m-52-q06a", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the tree diagram after transferring one ball from A to B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "First branch: red 7/8, blue 1/8. After red transfer, B has red 2/3 and blue 1/3; after blue transfer, B has red 3/5 and blue 2/5." }, { "id": "cie-2020-m-52-q06b", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the two selected balls have different colours.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add P(R then B) and P(B then R): (7/8)(1/3) + (1/8)(3/5) = 11/30." }, { "id": "cie-2020-m-52-q06c", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the ball transferred from A was blue, given that the ball drawn from B was blue.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Bayes’ rule: P(A blue and B blue)/P(B blue) = [(1/8)(2/5)]/[(7/8)(1/3) + (1/8)(2/5)] = 6/41." }, { "id": "cie-2020-m-52-q07a", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a cumulative-frequency graph for the grouped fish lengths.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the context was challenging and describes frequent errors in class boundaries, especially plotting the final class incorrectly or starting at an impossible negative fish length.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Plot cumulative frequencies 15, 63, 129 and 150 at upper class boundaries 9.5, 14.5, 19.5 and 30.5 cm; start at (0, 0), label both axes and draw a smooth increasing curve." }, { "id": "cie-2020-m-52-q07b", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the cumulative-frequency graph to estimate d when 40% of the fish have lengths of at least d cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates misinterpreted the percentage and read the length exceeded by 40% rather than the 60th percentile; some gave an unsupported graph reading.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert the statement to the 60th percentile, read the corresponding cumulative frequency of 90 from the graph, and draw a horizontal then vertical guide to estimate the length." }, { "id": "cie-2020-m-52-q07c", "paperId": "cie-2020-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the variance of the grouped fish lengths using the stated mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the real-life context was challenging; many candidates used an impossible negative fish length and chose the wrong first-class midpoint.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the mark-scheme midpoints 4.75, 12, 17 and 25 with their frequencies in the grouped-data variance formula, then subtract the square of the stated mean." }, { "id": "cie-2020-m-62-q01", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a suitable approximation to find the probability a 12,500-letter booklet has at least two incorrect letters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The Poisson mean is 12,500/30,000 = 5/12. Calculate 1 − P(0) − P(1) = 1 − e^(−5/12)(1 + 5/12) ≈ 0.0661." }, { "id": "cie-2020-m-62-q02", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the confidence percentage from the stated interval width, sample size and known population standard deviation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Many candidates found z ≈ 1.953 but then reported 97.46% as the confidence level instead of converting the upper-tail table value to the central two-sided percentage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use half-width 0.625 and standard error 3.2/√100 to find the two-sided critical z-value. Convert the central normal probability to a percentage, accounting for both tails." }, { "id": "cie-2020-m-62-q03a", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate unbiased estimates of the new journey-time population mean and variance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The mean is 1504/40 = 37.6 min. Use s² = [Σt² − (Σt)²/n]/(n−1) to get about 31.0 min²." }, { "id": "cie-2020-m-62-q03b", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% whether Freda’s mean journey time has decreased from 39.2 minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the one-sample test statistic (37.6 − 39.2)/√(31.0154/40) = −1.817 and compare with the 5% one-tailed critical value 1.645. Reject H₀ and conclude the mean time has decreased." }, { "id": "cie-2020-m-62-q04a", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of fewer than three accidents in 365 days when the rate is 0.4 per 50 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a Poisson model with λ = 0.4(365/50) = 2.92 and sum P(0) + P(1) + P(2)." }, { "id": "cie-2020-m-62-q04b", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the largest n for which the probability of no accidents in n days exceeds 0.95.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve e^(−0.4n/50) > 0.95, giving n < 6.41; the largest integer is 6 days." }, { "id": "cie-2020-m-62-q05a", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a bottle contains more than 305 ml of juice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates failed to use the correct limits when integrating the density in part (a).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the condition to X > 5 and integrate the given density from 5 to 10." }, { "id": "cie-2020-m-62-q05b", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Show that the cutoff p satisfies the stated cubic equation when 25% of bottles exceed 300 + p ml.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates used incorrect limits or omitted the limits entirely when forming the probability equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the density from p to 10, set the upper-tail probability equal to 0.25, and simplify the resulting equation to the given cubic." }, { "id": "cie-2020-m-62-q05c", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Given p = 3.47, find q so that half the bottles have volumes between 300 − q and 300 + q ml, and justify the value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says part (c) was poorly attempted; some omitted it and many assumed q = 2p or set up an incorrect integral instead of using symmetry.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the symmetry of the density about x = 0: equal central tail limits require q = p. State why the symmetric interval contains 50% rather than integrating over the two tails incorrectly." }, { "id": "cie-2020-m-62-q06a", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the combined volume of independent large and small tea cups is below 300 ml.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Using the stated normal parameter convention, the sum is N(310,50). Standardise 300 to obtain Φ(−10/√50) ≈ 0.0786." }, { "id": "cie-2020-m-62-q06b", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a large cup volume exceeds twice a small cup volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For D = L − 2S, use D ~ N(−20,110). Then P(D > 0) = 1 − Φ(20/√110) = 0.0283 (3 s.f.)." }, { "id": "cie-2020-m-62-q07a", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the rejection region for a one-sided test of whether fewer than 95% of the college students use social media.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls Question 7 demanding; candidates often used individual probabilities rather than a tail probability and many failed to show both cumulative probabilities needed to justify the cut-off.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Under H₀ use X ~ B(20, 0.95). Show P(X ≤ 16) ≤ 0.02 but P(X ≤ 17) > 0.02, so the rejection region is X ≤ 16." }, { "id": "cie-2020-m-62-q07b", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the test in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates incorrectly equated the Type I error probability with the nominal 2% significance level.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the null model X ~ B(20, 0.95) and calculate the probability that X falls in the rejection region; this is the test size, not the stated 2% level unless the discrete cut-off attains it exactly." }, { "id": "cie-2020-m-62-q07c", "paperId": "cie-2020-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Assuming the true college proportion is 0.70, find the probability of a Type II error.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes frequent confusion about which region produces a Type II error, as well as incorrect normal approximations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m20_er.pdf", "tagConfidence": "reviewed", "answerGuide": "Use X ~ B(20, 0.70) and sum the probabilities over the acceptance region determined in part (a)." }, { "id": "ial-2020-10-WMA01-01-q01a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Solve the cubic equation by algebra.", "answerGuide": "Factor out the common x term, then solve the remaining quadratic and list every root.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=2", "msUrl": "https://school.ninja/documents/166424/pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q01b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the algebraic roots to solve the related sine equation on the stated interval.", "answerGuide": "Set sin θ equal to each admissible algebraic root and list the angles in the required interval.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=2", "msUrl": "https://school.ninja/documents/166424/pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q02", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate a sum of powers of x and simplify each term.", "answerGuide": "Rewrite each term as a power, apply the power rule, and include the constant of integration.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=4", "msUrl": "https://school.ninja/documents/166424/pdf#page=10", "qpPages": [ 4, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q03a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the recurrence relation to find the next two terms in terms of k.", "answerGuide": "Substitute u₁ into the recurrence twice and simplify both expressions.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=6", "msUrl": "https://school.ninja/documents/166424/pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q03b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the given value of u₃ to evaluate the sum of the first three terms.", "answerGuide": "First solve for k from u₃=2, then substitute the terms into the finite sum.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=6", "msUrl": "https://school.ninja/documents/166424/pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q04i", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 4, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Simplify the given surd expression in terms of x.", "answerGuide": "Square or rationalize carefully as required, then combine like surd terms without using a calculator.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=8", "msUrl": "https://school.ninja/documents/166424/pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q04ii", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Solve the surd equation and express the result in the requested exact form.", "answerGuide": "Isolate the radical, square only after recording sign constraints, and simplify to the specified integer form.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=8", "msUrl": "https://school.ninja/documents/166424/pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q05a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the two definite integrals used in the shaded-region area argument.", "answerGuide": "Integrate the given powers over their stated limits and simplify both exact results.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=10", "msUrl": "https://school.ninja/documents/166424/pdf#page=15", "qpPages": [ 10, 13 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q05b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the stated area of the region to find the exact value of a.", "answerGuide": "Set the area integral equal to 9 and solve the resulting equation subject to a>4.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=10", "msUrl": "https://school.ninja/documents/166424/pdf#page=15", "qpPages": [ 10, 13 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q06a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the factored cubic curve and simplify.", "answerGuide": "Expand or use the product rule, then collect the derivative terms.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=14", "msUrl": "https://school.ninja/documents/166424/pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q06b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "differentiation", "quadratics" ], "primaryTopicId": "differentiation", "summary": "Find the x-values for which the derivative is at least 2.", "answerGuide": "Rearrange to a quadratic inequality, find its critical points, and select the correct intervals.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=14", "msUrl": "https://school.ninja/documents/166424/pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q07i", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation to three decimal places.", "answerGuide": "Isolate the exponential term, take logarithms, and round only at the end.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=16", "msUrl": "https://school.ninja/documents/166424/pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q07ii", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation exactly for x.", "answerGuide": "Use log laws to combine the terms, convert to an exponential equation, and retain only valid positive solutions.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=16", "msUrl": "https://school.ninja/documents/166424/pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q08a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "trigonometry", "geometry" ], "primaryTopicId": "trigonometry", "summary": "Find both possible sizes of the parallelogram angle.", "answerGuide": "Use the sine rule with the known side and angle, then retain both valid inverse-sine cases.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=18", "msUrl": "https://school.ninja/documents/166424/pdf#page=22", "qpPages": [ 18, 21 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q08b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "trigonometry", "geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the diagonal length when the included angle is obtuse.", "answerGuide": "Use the cosine rule with the obtuse angle and round the resulting length to two decimal places.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=18", "msUrl": "https://school.ninja/documents/166424/pdf#page=22", "qpPages": [ 18, 21 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q08c", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Find the area of the parallelogram.", "answerGuide": "Use adjacent side lengths and the sine of the included angle, then round to three significant figures.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=18", "msUrl": "https://school.ninja/documents/166424/pdf#page=22", "qpPages": [ 18, 21 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q09a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "functions", "graphs" ], "primaryTopicId": "functions", "summary": "Sketch the reciprocal curve and show its intercept and both asymptotes.", "answerGuide": "Find where the numerator is zero, identify the vertical and horizontal asymptotes, and draw the correct branches.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=22", "msUrl": "https://school.ninja/documents/166424/pdf#page=24", "qpPages": [ 22, 23 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q09b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "functions", "algebra", "quadratics" ], "primaryTopicId": "functions", "summary": "Find the range of k for which the line and curve do not intersect.", "answerGuide": "Set the two expressions equal, require the resulting quadratic to have no real roots, and apply k>0.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=22", "msUrl": "https://school.ninja/documents/166424/pdf#page=24", "qpPages": [ 22, 23 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q10a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand the expression to four terms in ascending powers of x.", "answerGuide": "Factor out the constant term and apply the generalized binomial series through x³, simplifying every coefficient.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=24", "msUrl": "https://school.ninja/documents/166424/pdf#page=27", "qpPages": [ 24, 27 ], "msPages": [ 27, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q10b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the zero coefficient of x to determine a as a simplified fraction.", "answerGuide": "Extract the linear coefficient from the expansion, set it equal to zero, and solve for a.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=24", "msUrl": "https://school.ninja/documents/166424/pdf#page=27", "qpPages": [ 24, 27 ], "msPages": [ 27, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q11a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Use the factor condition to verify one value of k and find the other.", "answerGuide": "Substitute the root implied by the factor into f(x)=0, solve for k, and identify both values.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=28", "msUrl": "https://school.ninja/documents/166424/pdf#page=30", "qpPages": [ 28, 31 ], "msPages": [ 30, 32 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q11b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "algebra", "quadratics", "proof" ], "primaryTopicId": "algebra", "summary": "For k=5, factor f(x) and show its equation has exactly one real solution.", "answerGuide": "Factor the polynomial, then inspect the remaining factor's discriminant or equivalent sign to establish the real-root count.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=28", "msUrl": "https://school.ninja/documents/166424/pdf#page=30", "qpPages": [ 28, 31 ], "msPages": [ 30, 32 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q12a", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": "12(a)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Derive the quadratic equation satisfied by the curve-intersection x-coordinates.", "answerGuide": "Equate the two curve expressions, multiply through by cos x, and use sin²x+cos²x=1.", "qpUrl": "https://school.ninja/documents/166499/pdf#page=32", "msUrl": "https://school.ninja/documents/166424/pdf#page=33", "qpPages": [ 32, 35 ], "msPages": [ 33, 34 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA01-01-q12b", "paperId": "ial-2020-10-WMA01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA01", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA01/01", "variant": "Standard paper", "questionNumber": 12, "questionPart": "12(b)", "marks": 4, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Find both intersection coordinates to two decimal places using algebraic working.", "answerGuide": "Solve the quadratic for sin x, select angles in 0 1/3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Most candidates missed the hint in part (b) and consequently could not answer this linked part successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 7, 8, 9, 10 ], "tagConfidence": "reviewed", "answerGuide": "Use the decomposition from part (b), f(x) = 2/3 + 2/[3(3x−1)]. Since x > 1/3, the second term is positive and can approach zero without reaching it, so the range is (2/3, ∞)." }, { "id": "cie-2020-w-13-q07a", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-13-q07b", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "7(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2020-w-13-q08a", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-13-q08b", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2020-w-13-q09a", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "9(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2020-w-13-q09b", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "9(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2020-w-13-q09c", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "9(c): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2020-w-13-q10a", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2020-w-13-q10b", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-13-q11a", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the specified point lies outside the given circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls Question 11 the most demanding on the paper and notes that candidates often stopped at a non-equality without proving the point was outside.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 7, 8, 9, 10 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the point into the circle’s centre-radius relation or compare its distance from the centre with the radius, then state the strict inequality and conclusion." }, { "id": "cie-2020-w-13-q11b", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact angle formed by the tangent geometry in the diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates used an unsuccessful gradient approach or assumed an unproved tangent length.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 7, 8, 9, 10 ], "tagConfidence": "reviewed", "answerGuide": "Use the centre, tangent and radius to form the appropriate right triangle, identify the sine ratio, and simplify the inverse-trigonometric value exactly." }, { "id": "cie-2020-w-13-q11c", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the line through the two tangent points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found a related gradient but could not continue; successful solutions recognised the square and used its midpoint property.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 7, 8, 9, 10 ], "tagConfidence": "reviewed", "answerGuide": "Use the tangent geometry to recognise the square, find the midpoint of the centre-to-external-point segment, and combine it with the perpendicular gradient to form the line." }, { "id": "cie-2020-w-13-q11d", "paperId": "cie-2020-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2020, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(d)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "11(d): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_13.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2020-w-21-q01", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2020-w-21-q02", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2020-w-21-q03", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2020-w-21-q04a", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-21-q04b", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-21-q05a", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(a): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2020-w-21-q05b", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-21-q06a", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2020-w-21-q06b", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the exact result from part (a) to obtain the exact value of sec θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates obtained a correct solution; the report explicitly points to using the exact value from part (a) with the reciprocal and Pythagorean identities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Write sec θ = 1/cos θ and use sin²θ + cos²θ = 1 with the exact value from part (a); retain the sign fixed by the given interval." }, { "id": "cie-2020-w-21-q06c", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2020-w-21-q07a", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2020-w-21-q07b", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Express the trigonometric linear combination in the required R-form exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates completed this standard process correctly; the report notes that an exact R was required and errors in tan α were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Equate the cosine and sine coefficients to R cos α and R sin α, calculate R exactly and select α in the required range." }, { "id": "cie-2020-w-21-q07c", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the R-form result to solve the stated trigonometric equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few correct solutions were seen because candidates often missed the instruction to use the preceding result.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Equate the R-form from part (b) to the given value, solve the resulting cosine equation, and list every solution in the required interval." }, { "id": "cie-2020-w-21-q08a", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2020-w-21-q08b", "paperId": "cie-2020-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the quotient-and-remainder expression obtained in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few complete correct solutions were seen; although many divided successfully, very few rewrote and integrated the remainder correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Separate the quotient and remainder, rewrite the remainder to expose the logarithmic form, integrate each term, and include the constant of integration." }, { "id": "cie-2020-w-22-q01", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-22-q02", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "2: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2020-w-22-q03a", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2020-w-22-q03b", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "3(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2020-w-22-q03c", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(c): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2020-w-22-q04a", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(a): Differentiate the given gradient function once more to obtain the second derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given gradient expression with respect to x once more and simplify to obtain d²y/dx²." }, { "id": "cie-2020-w-22-q04b", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Apply the trapezium rule to estimate the shaded area using the specified intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the trapezium rule is often challenging and points out that incorrect interval widths and missing ordinates were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Read the interval width from the diagram, use the four required ordinates for the three intervals, apply the trapezium rule and round to the requested decimal places." }, { "id": "cie-2020-w-22-q05a", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-22-q05b", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2020-w-22-q05c", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Show that the curve has no stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct solutions were seen and full marks were rare because candidates often checked the exponential factor but not y.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Set the numerator of the derivative to zero and explain why neither factor can vanish for the stated domain; address both y and the exponential factor." }, { "id": "cie-2020-w-22-q06a", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate the given rational-trigonometric expression and give the general antiderivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few completely correct solutions were seen; most candidates did not make the required secant rewrite and some omitted the constant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the denominator using the relevant secant identity before integrating, recognise the logarithmic derivative, and include the arbitrary constant." }, { "id": "cie-2020-w-22-q06b", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the specified trigonometric definite integral exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates often used a calculator despite the exact-answer instruction, and the report says this prevented full credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Use a double-angle identity to express the integrand in the required form, integrate term by term, apply the stated limits, and keep the exact radian result." }, { "id": "cie-2020-w-22-q07a", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-22-q07b", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-22-q07c", "paperId": "cie-2020-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_22.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2020-w-23-q01", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2020-w-23-q02", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2020-w-23-q03", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2020-w-23-q04a", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-23-q04b", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-23-q05a", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(a): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2020-w-23-q05b", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-23-q06a", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2020-w-23-q06b", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the exact result from part (a) to obtain the exact value of sec θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates obtained a correct solution; the report explicitly points to using the exact value from part (a) with the reciprocal and Pythagorean identities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Write sec θ = 1/cos θ and use sin²θ + cos²θ = 1 with the exact value from part (a); retain the sign fixed by the given interval." }, { "id": "cie-2020-w-23-q06c", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2020-w-23-q07a", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2020-w-23-q07b", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Express the trigonometric linear combination in the required R-form exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates completed this standard process correctly; the report notes that an exact R was required and errors in tan α were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Equate the cosine and sine coefficients to R cos α and R sin α, calculate R exactly and select α in the required range." }, { "id": "cie-2020-w-23-q07c", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the R-form result to solve the stated trigonometric equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few correct solutions were seen because candidates often missed the instruction to use the preceding result.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Equate the R-form from part (b) to the given value, solve the resulting cosine equation, and list every solution in the required interval." }, { "id": "cie-2020-w-23-q08a", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2020-w-23-q08b", "paperId": "cie-2020-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2020, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the quotient-and-remainder expression obtained in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few complete correct solutions were seen; although many divided successfully, very few rewrote and integrated the remainder correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Separate the quotient and remainder, rewrite the remainder to expose the logarithmic form, integrate each term, and include the constant of integration." }, { "id": "cie-2020-w-31-q01", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Solve the stated inequality and give the range that satisfies the original expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates found the quadratic roots, but the majority did not interpret them in the original inequality and lost the final mark.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 20, 21, 22, 23 ], "tagConfidence": "reviewed", "answerGuide": "Solve the boundary equation, then use a sketch or sign analysis in the original inequality to reject extraneous roots introduced by squaring." }, { "id": "cie-2020-w-31-q02", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2: Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2020-w-31-q03", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2020-w-31-q04", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-31-q05a", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "functions", "proof" ], "primaryTopicId": "functions", "summary": "Sketch the two graphs and use their intersections to show the equation has two roots in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates could not sketch the cosecant curve and commonly missed key asymptotic and intercept features needed to establish the two roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 20, 21, 22, 23 ], "tagConfidence": "reviewed", "answerGuide": "Sketch the cosecant and exponential curves with correct asymptotes, intercepts, symmetry and scale, then show that they intersect twice within the interval." }, { "id": "cie-2020-w-31-q05b", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2020-w-31-q06a", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2020-w-31-q06b", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Find all solutions of the cosine equation in the stated interval using the result from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The second solution was seen far less often, and the report notes that candidates often failed to replace θ by the required multiple of x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 20, 21, 22, 23 ], "tagConfidence": "reviewed", "answerGuide": "Use the amplitude result from part (a), solve the cosine equation for all admissible angles, divide by the coefficient of x, and check both interval solutions." }, { "id": "cie-2020-w-31-q07a", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-31-q07b", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-31-q08", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8: Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2020-w-31-q09a", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-31-q09b", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "9(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2020-w-31-q10a", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2020-w-31-q10b", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2020-w-31-q11a", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=19", "qpPages": [ 18, 18 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2020-w-31-q11b", "paperId": "cie-2020-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the unknown parameter from the acute angle between the two line directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_31.pdf#page=20", "qpPages": [ 18, 19 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The final mark was challenging; few candidates took the scalar-product modulus or allowed both signs, which is necessary to select the acute angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 20, 21, 22, 23 ], "tagConfidence": "reviewed", "answerGuide": "Use the two direction vectors in the scalar-product formula, take the absolute value or account for both signs to enforce the acute angle, then solve for the parameter." }, { "id": "cie-2020-w-32-q01", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2020-w-32-q02a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2020-w-32-q02b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "State the interval of x-values for which the binomial expansion in part (a) is valid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A significant number gave no response; some considered only one side of the inequality and others repeated part (a) instead of using it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 24, 25, 26, 27 ], "tagConfidence": "reviewed", "answerGuide": "Apply the binomial-series condition |6x| < 1 to obtain the open interval −1/6 < x < 1/6." }, { "id": "cie-2020-w-32-q03a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Show that the logarithmic relationship gives a straight-line graph and identify its gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Fully correct solutions were rarely seen; common errors included missing brackets and comparing an equation for ln y with the wrong straight-line form.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 24, 25, 26, 27 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms with brackets correctly retained, rearrange to make ln y linear in x (or match the required linear form), and read off the exact gradient." }, { "id": "cie-2020-w-32-q03b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2020-w-32-q04a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2020-w-32-q04b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange using the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2020-w-32-q05a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2020-w-32-q05b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinate of P, where the curve has its maximum gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only the strongest candidates answered correctly; many instead looked for where the original gradient was zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 24, 25, 26, 27 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the given gradient with respect to the parameter, set the derivative to zero, solve within the parameter interval, and use the diagram to select the point with negative x-coordinate." }, { "id": "cie-2020-w-32-q06a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2020-w-32-q06b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(b): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2020-w-32-q06c", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Use the given argument relation to determine the required exact value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The supplied result did not help many candidates; they often added arguments or replaced the exact relation with decimal approximations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 24, 25, 26, 27 ], "tagConfidence": "reviewed", "answerGuide": "Write the argument equation that follows from division of the complex numbers, use the correct subtraction of arguments, and work with exact angles rather than decimals." }, { "id": "cie-2020-w-32-q07a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Solve the separable differential equation by finding its general solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Integration of the trigonometric term proved much more difficult; candidates who did not recognise its secant-squared form made no further progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 24, 25, 26, 27 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate the y-side, rewrite the trigonometric factor in a standard secant-squared form, and carefully retain the coefficient from the chain rule." }, { "id": "cie-2020-w-32-q07b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-32-q08a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "8(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-32-q08b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "8(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2020-w-32-q08c", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "8(c): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-32-q09a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-32-q09b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-32-q10a", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=17", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2020-w-32-q10b", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "10(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=17", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2020-w-32-q10c", "paperId": "cie-2020-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the volume generated by rotating the stated curve over the given interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_32.pdf#page=18", "qpPages": [ 18, 19 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority used the correct volume integral and fewer integrated the squared cosine term correctly; many attempted the harder integral of x cos x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 24, 25, 26, 27 ], "tagConfidence": "reviewed", "answerGuide": "Set up the disk integral using πy², integrate the squared cosine term with a double-angle identity, and use the stated upper limit exactly." }, { "id": "cie-2020-w-33-q01", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Solve the stated inequality and give the range that satisfies the original expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates found the quadratic roots, but the majority did not interpret them in the original inequality and lost the final mark.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 28, 29, 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Solve the boundary equation, then use a sketch or sign analysis in the original inequality to reject extraneous roots introduced by squaring." }, { "id": "cie-2020-w-33-q02", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2: Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2020-w-33-q03", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2020-w-33-q04", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-33-q05a", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "functions", "proof" ], "primaryTopicId": "functions", "summary": "Sketch the two graphs and use their intersections to show the equation has two roots in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates could not sketch the cosecant curve and commonly missed key asymptotic and intercept features needed to establish the two roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 28, 29, 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Sketch the cosecant and exponential curves with correct asymptotes, intercepts, symmetry and scale, then show that they intersect twice within the interval." }, { "id": "cie-2020-w-33-q05b", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2020-w-33-q06a", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2020-w-33-q06b", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry", "numerical-methods" ], "primaryTopicId": "trigonometry", "summary": "Find all solutions of the cosine equation in the stated interval using the result from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The second solution was seen far less often, and the report notes that candidates often failed to replace θ by the required multiple of x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 28, 29, 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Use the amplitude result from part (a), solve the cosine equation for all admissible angles, divide by the coefficient of x, and check both interval solutions." }, { "id": "cie-2020-w-33-q07a", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-33-q07b", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-33-q08", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8: Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2020-w-33-q09a", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2020-w-33-q09b", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "9(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2020-w-33-q10a", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2020-w-33-q10b", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2020-w-33-q11a", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=19", "qpPages": [ 18, 18 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2020-w-33-q11b", "paperId": "cie-2020-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2020, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the unknown parameter from the acute angle between the two line directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_33.pdf#page=20", "qpPages": [ 18, 19 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The final mark was challenging; few candidates took the scalar-product modulus or allowed both signs, which is necessary to select the acute angle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 28, 29, 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Use the two direction vectors in the scalar-product formula, take the absolute value or account for both signs to enforce the acute angle, then solve for the parameter." }, { "id": "cie-2020-w-41-q01a", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-41-q01b", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2020-w-41-q02a", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-41-q02b", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-41-q03", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2020-w-41-q04", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4: Integrate the given acceleration to obtain velocity, then integrate velocity to find the displacement at the stated time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the given acceleration to obtain velocity, use the initial velocity to determine its constant, then integrate velocity from the start time to the stated time." }, { "id": "cie-2020-w-41-q05a", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2020-w-41-q05b", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-41-q06a", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy equation to find the work done by the driving force as the car and trailer climb the hill.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 6 was identified as one of the most challenging; many candidates omitted energy terms or combined them with the wrong signs.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 32, 33, 34 ], "tagConfidence": "reviewed", "answerGuide": "Balance driving-force work plus the loss of kinetic energy against the gain in potential energy and the work against resistance, keeping every term with the correct sign." }, { "id": "cie-2020-w-41-q06b", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(b): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2020-w-41-q07a", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 8, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the particle’s speed at C after moving across a smooth section and a rough section.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Question 7 was the most challenging on the paper; errors in potential energy and signs in the energy equation were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 32, 33, 34 ], "tagConfidence": "reviewed", "answerGuide": "Use energy conservation on the smooth segment, then account for the potential-energy change, kinetic-energy change and friction work on the rough segment." }, { "id": "cie-2020-w-41-q07b", "paperId": "cie-2020-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of friction for the particle to come to rest at C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_41.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates found this part challenging, despite a short energy solution being available.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 32, 33, 34 ], "tagConfidence": "reviewed", "answerGuide": "Since the particle starts and finishes at rest, equate potential-energy loss to work against friction and solve using the normal reaction on the rough section." }, { "id": "cie-2020-w-42-q01a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(a): Use the stated energy or momentum relationship to find the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2020-w-42-q01b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-42-q02a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2020-w-42-q02b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2020-w-42-q03", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2020-w-42-q04a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2020-w-42-q04b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2020-w-42-q05a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the height of the building from the projectile’s motion and the stated time above the roof.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates found this part challenging and commonly treated the four seconds above the building as the time to the roof.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 35, 36, 37, 38 ], "tagConfidence": "reviewed", "answerGuide": "Use the symmetry of the flight about maximum height to determine the time to the roof, then apply a constant-acceleration displacement equation." }, { "id": "cie-2020-w-42-q05b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-42-q06a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the range of friction coefficients for which the block moves up the plane under the stated force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Question 6 was one of the paper’s most challenging questions; candidates often failed to express the motion condition as an inequality throughout.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 35, 36, 37, 38 ], "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and perpendicular to the plane, use limiting friction, and express the positive net-force condition as an inequality for the coefficient." }, { "id": "cie-2020-w-42-q06b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the least coefficient of friction when the applied force acts horizontally and the block remains at rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies the changed normal reaction as a common difficulty; many candidates reused the reaction from part (a) or made sign and component errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 35, 36, 37, 38 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the horizontal force and weight both parallel and perpendicular to the plane, calculate the friction and changed normal reaction, then use limiting equilibrium to obtain the minimum coefficient." }, { "id": "cie-2020-w-42-q07a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Integrate the given acceleration, use the initial velocity, and solve for when the stated velocity is reached.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the given acceleration to obtain velocity, apply the stated initial velocity, then set velocity equal to the target value and solve for time." }, { "id": "cie-2020-w-42-q07b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Integrate the given acceleration to obtain velocity, then integrate velocity to find the displacement at the stated time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the given acceleration to obtain velocity, use the initial velocity to determine its constant, then integrate velocity from the start time to the stated time." }, { "id": "cie-2020-w-42-q08a", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "8(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2020-w-42-q08b", "paperId": "cie-2020-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use energy conservation for the connected particles to determine the required unknown.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_42.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_42.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates made a completely correct attempt because one or more energy terms were often omitted.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 35, 36, 37, 38 ], "tagConfidence": "reviewed", "answerGuide": "Include the work by the applied force, potential-energy loss, friction work and kinetic-energy gains for both particles in one energy equation." }, { "id": "cie-2020-w-43-q01a", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q01b", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q02a", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2020-w-43-q02b", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2020-w-43-q02c", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(c): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2020-w-43-q03a", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(a): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2020-w-43-q03b", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(b): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2020-w-43-q04", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q05a", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q05b", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2020-w-43-q05c", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(c): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q05d", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(d): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q06ai", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a)(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2020-w-43-q06aii", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of the car and caravan together and the tow-bar tension.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This was identified among the most challenging parts; candidates often omitted the resistance force or applied the driving force directly as the net force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 39, 40, 41 ], "tagConfidence": "reviewed", "answerGuide": "Find the driving force from power divided by speed, apply Newton’s second law to the full system including resistance, then use a separate vehicle equation to find tension." }, { "id": "cie-2020-w-43-q06b", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2020-w-43-q07a", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2020-w-43-q07b", "paperId": "cie-2020-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2020, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the time taken for the two particles on inclined planes to move through the stated vertical-height difference.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_43.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Finding the required time proved too challenging for the majority; many equated the path length or a single vertical change directly to one metre.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 39, 40, 41 ], "tagConfidence": "reviewed", "answerGuide": "Use the acceleration found from the force equations, relate the common path distance to both particles’ vertical changes using the two sine factors, then apply s = ½at²." }, { "id": "cie-2020-w-51-q01a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of the intersection of two events defined by the outcomes of two dice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says many candidates found this part challenging and often omitted outcomes when their outcome listings were inconsistent.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 42, 43, 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "List the ordered outcomes consistently, identify the outcomes satisfying each event and their intersection, then count over the complete equally likely sample space." }, { "id": "cie-2020-w-51-q01b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-51-q02a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-51-q02b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-51-q03a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-51-q03b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use a binomial approximation to find the probability of at least three successes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A significant number did not switch to the binomial model, and interpreting “at least three” was challenging for many.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 42, 43, 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Recognise that the condition changes from the geometric model in part (a) to a binomial model, calculate the probability of fewer than three successes, and subtract from one." }, { "id": "cie-2020-w-51-q04a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-51-q04b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2020-w-51-q05a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-51-q05b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2020-w-51-q05c", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Use the normal model and earlier probability to determine the number of days satisfying the stated condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report explicitly says most candidates found this part challenging and notes that many left the contextual count unfinished.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 42, 43, 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Use symmetry about the mean to obtain the required probability from part (a), then multiply by the number of days and interpret the result in context." }, { "id": "cie-2020-w-51-q06a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "6(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2020-w-51-q06b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2020-w-51-q06c", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the standard deviation from the grouped snowfall data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The majority found the unstructured part challenging; a frequent error was using class widths instead of midpoints.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 42, 43, 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Build a clear grouped-data table with class midpoints and frequencies, calculate the estimated mean and second moment, and take the square root of the variance." }, { "id": "cie-2020-w-51-q07a", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-51-q07b", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-51-q07c", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that an arrangement of SHOPKEEPER has an E at both ends.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found this part challenging and commonly treated identical letters as distinguishable.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 42, 43, 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Fix two of the three identical Es at the ends, count distinct arrangements of the remaining eight letters, and divide by the total number of distinct arrangements of SHOPKEEPER." }, { "id": "cie-2020-w-51-q07d", "paperId": "cie-2020-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the different four-letter selections containing exactly one P from SHOPKEEPER.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_51.pdf#page=16", "qpPages": [ 10, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates found this part challenging; the report notes that valid cases were often missed and repeated letters were overcounted.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 42, 43, 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Choose exactly one of the two Ps, then choose three letters from the five distinct non-P, non-E letters and the three identical Es; sum the cases for zero through three Es." }, { "id": "cie-2020-w-52-q01a", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-52-q01b", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use the binomial approximation to find the probability of at least three occurrences.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A significant number did not recognise the switch from a geometric to a binomial approximation, and the boundary wording was challenging.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 46, 47, 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Use the binomial model for the changed condition, calculate the probability of zero, one or two occurrences, and subtract this total from one." }, { "id": "cie-2020-w-52-q02a", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-52-q02b", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-52-q02c", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-52-q03a", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-52-q03b", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2020-w-52-q03c", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(c): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2020-w-52-q04a", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-52-q04b", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-52-q04c", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-52-q04d", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(d): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2020-w-52-q05a", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-52-q05b", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2020-w-52-q05c", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare the distributions using the spread and typical snowfall at the two locations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found this part challenging; simple numerical comparisons or definitions without contextual interpretation were insufficient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 46, 47, 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Use the summary statistics to make contextual comparisons of both variation and central tendency, stating which resort has greater spread and which generally has more snow." }, { "id": "cie-2020-w-52-q06a", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-52-q06b", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that the three Baker children are included in a randomly selected group of five.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates found this part challenging; a common error was multiplying by two as if the selection order mattered.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 46, 47, 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Count all five-person groups, count the groups containing the three fixed children by choosing the remaining two from the others, and form the ratio." }, { "id": "cie-2020-w-52-q06c", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-52-q06d", "paperId": "cie-2020-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements with exactly one person between Mr Ahmed and Mr Baker.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_52.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report describes subtraction approaches as very complex and commonly unsuccessful; the key omission was treating the men’s order as irrelevant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 46, 47, 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "Choose one of the seven position-pairs separated by one place, arrange Ahmed and Baker in either order, and permute the other seven people in the remaining positions." }, { "id": "cie-2020-w-53-q01a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-53-q01b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2020-w-53-q02a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-53-q02b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-53-q02c", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-53-q03a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-53-q03b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count committees that meet the condition involving a brother and sister.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A significant number used a less efficient case-by-case method and were rarely successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 50, 51, 52 ], "tagConfidence": "reviewed", "answerGuide": "Count all committees, subtract those containing both siblings, or add the disjoint cases with neither and with exactly one sibling." }, { "id": "cie-2020-w-53-q04a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-53-q04b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the normal approximation, including the continuity correction, to find the required probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies selecting the correct normal tail as the most challenging part; a significant number subtracted the desired area from one.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 50, 51, 52 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the binomial mean and variance, apply the stated continuity correction, standardise, and choose the appropriate tail area." }, { "id": "cie-2020-w-53-q05a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2020-w-53-q05b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2020-w-53-q06a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-53-q06b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-53-q06c", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2020-w-53-q07a", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "7(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2020-w-53-q07b", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the greatest possible interquartile range from the grouped discrete data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only the strongest candidates used the discrete-data limits rather than the boundaries used to draw the histogram.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 50, 51, 52 ], "tagConfidence": "reviewed", "answerGuide": "Identify the classes containing the lower and upper quartiles and use the class limits for the maximum possible range, not the histogram boundaries." }, { "id": "cie-2020-w-53-q07c", "paperId": "cie-2020-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2020, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_53.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-61-q01a", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-61-q01b", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-61-q01c", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(c): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-61-q02a", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-61-q02b", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-61-q02c", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-61-q03", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q04a", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2020-w-61-q04b", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2020-w-61-q05a", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q05b", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-61-q05c", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-61-q05d", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(d): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q05e", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(e): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q06a", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q06bi", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b)(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q06bii", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b)(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-61-q06biii", "paperId": "cie-2020-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)(iii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b)(iii): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_61.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2020-w-62-q01", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-62-q02a", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Construct the confidence interval for a population proportion from the sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this was not well attempted and that the confidence-interval method caused problems for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 54, 55, 56 ], "tagConfidence": "reviewed", "answerGuide": "Use p̂ = 56/300 and q̂ = 244/300 in the proportion confidence-interval formula with the correct critical value, then state the interval." }, { "id": "cie-2020-w-62-q02b", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "statistical-testing", "proof" ], "primaryTopicId": "statistical-testing", "summary": "Use the interval to assess whether the stated die probability is consistent with the sample.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates gave a fully correct answer; even those who checked interval inclusion often used an unjustifiably definite conclusion.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 54, 55, 56 ], "tagConfidence": "reviewed", "answerGuide": "Check whether the hypothesised proportion lies within the interval, then phrase the conclusion with suitable uncertainty rather than making a definite claim." }, { "id": "cie-2020-w-62-q03a", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2020-w-62-q03b", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find P(X > 2) for the given triangular probability density function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates failed to find the correct line equation; common errors included using the wrong gradient sign or assuming a constant height.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 54, 55, 56 ], "tagConfidence": "reviewed", "answerGuide": "Find the density-line equation from its endpoints using the negative gradient, then integrate the density from 2 to 3." }, { "id": "cie-2020-w-62-q03c", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(c): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2020-w-62-q04", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 8, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-62-q05a", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2020-w-62-q05b", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-62-q05c", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-62-q05d", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-62-q06a", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-62-q06b", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-62-q06c", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-62-q07a", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the probability that the number of minor errors is at least 200 greater than the number of major errors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says finding the mean and variance was challenging for many and notes frequent unit and interpretation errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 54, 55, 56 ], "tagConfidence": "reviewed", "answerGuide": "Let D be minor errors minus major errors. Combine the independent normal models to obtain D ∼ N(170, 220), then standardise 200 and take the upper-tail probability." }, { "id": "cie-2020-w-62-q07b", "paperId": "cie-2020-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the probability that the total correction cost is less than $190.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_62.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates struggled to find the mean and variance; a common error was summing frequency times value rather than frequency times squared value for the variance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_er.pdf", "sourceEvidencePages": [ 54, 55, 56 ], "tagConfidence": "reviewed", "answerGuide": "For total cost C = 0.20X + 0.50Y, where X and Y are the minor- and major-error counts, combine the independent normal models to obtain mean 181 dollars and variance 25.6 dollars squared, then standardise 190." }, { "id": "cie-2020-w-63-q01a", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-63-q01b", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-63-q01c", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(c): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2020-w-63-q02a", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2020-w-63-q02b", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-63-q02c", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-63-q03", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q04a", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2020-w-63-q04b", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2020-w-63-q05a", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q05b", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-63-q05c", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2020-w-63-q05d", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(d): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q05e", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(e): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q06a", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q06bi", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b)(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q06bii", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b)(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2020-w-63-q06biii", "paperId": "cie-2020-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2020, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)(iii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b)(iii): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w20_ms_63.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-s-11-q01", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "1: Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2021-s-11-q02", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2021-s-11-q03a", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "3(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-s-11-q03b", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(b): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-s-11-q04", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Read the constants in the transformed tangent graph y = a tan(x − b) + c.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates obtained a fully correct answer; the report says the horizontal translation and vertical stretch were the most challenging constants.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 1, 2, 3, 4 ], "tagConfidence": "reviewed", "answerGuide": "Use the graph's period and asymptote locations to determine the horizontal scale and shift, then read the vertical translation and check the branch orientation." }, { "id": "cie-2021-s-11-q05", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "5: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-s-11-q06", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2021-s-11-q07a", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-s-11-q07b", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-11-q08a", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-s-11-q08b", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(b): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2021-s-11-q08c", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Show the exact area of the plate formed from the circular disc and two semicircular cut-outs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part challenged most candidates and fully correct solutions were rare; candidates needed an overall area strategy.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 1, 2, 3, 4 ], "tagConfidence": "reviewed", "answerGuide": "Break the plate into sectors, triangles and semicircles using the radii and angles from the diagram, then combine the exact areas." }, { "id": "cie-2021-s-11-q09a", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "9(a): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2021-s-11-q09b", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-11-q09c", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-11-q09d", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(d)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(d): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-11-q10a", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2021-s-11-q10b", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2021-s-11-q11a", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-11-q11b", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-11-q11c", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(c): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-11-q11d", "paperId": "cie-2021-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(d)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "11(d): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_11.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-s-12-q01a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q01b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q02a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-s-12-q02b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-12-q03a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q03b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-s-12-q03c", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Use the sequence of secant gradients to infer the curve's gradient at the limiting point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was more challenging, was often omitted, and very few candidates referred to limits or gradients of chords.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 5, 6, 7 ], "tagConfidence": "reviewed", "answerGuide": "Recognise that the chords join points increasingly close to E, so their gradients approach the tangent gradient; state the limiting value and justify it as a limit of secants." }, { "id": "cie-2021-s-12-q04", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "4: Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-s-12-q05a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q05b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q06", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6: Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-s-12-q07a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the specified line is tangent to the circle at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Relatively few candidates gave a complete solution; many verified only that A was on the circle or only that the radius was perpendicular to the line.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 5, 6, 7 ], "tagConfidence": "reviewed", "answerGuide": "Verify that A lies on the circle and that the radius from the centre to A is perpendicular to the given line." }, { "id": "cie-2021-s-12-q07b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the other circle of the given radius tangent to the same line at A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not understand what the second-circle part required and gave the original circle or a circle with the wrong centre.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 5, 6, 7 ], "tagConfidence": "reviewed", "answerGuide": "Place the second centre on the perpendicular to the tangent through A, on the opposite side, at the same distance from A as the first centre; then write the circle equation." }, { "id": "cie-2021-s-12-q08a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "8(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-s-12-q08b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "8(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-s-12-q09", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9: Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2021-s-12-q10a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-12-q10b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=17", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-12-q11a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-12-q11b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q11c", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(c): Differentiate the given gradient function once more to obtain the second derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given gradient expression with respect to x once more and simplify to obtain d²y/dx²." }, { "id": "cie-2021-s-12-q11d", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(d)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(d): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-12-q12a", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(a)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Show the angle between the two rope sections where they meet the outer pipe.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the seven-pipe question was challenging and notes that candidates often assumed an equilateral triangle or divided areas without justifying the geometry.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 5, 6, 7 ], "tagConfidence": "reviewed", "answerGuide": "Use the symmetry of the six outer pipe centres and the radii perpendicular to the rope sections to determine the angle at the pipe centre." }, { "id": "cie-2021-s-12-q12b", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "12(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-12-q12c", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(c)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "12(c): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=21", "qpPages": [ 16, 17 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-s-12-q12d", "paperId": "cie-2021-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(d)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the area of the complete region enclosed by the tightly wrapped rope around seven pipes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_12.pdf#page=22", "qpPages": [ 16, 17 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this continuation was difficult for candidates who struggled with the preceding hexagon-area part and that weaker approaches often failed to account for the uncovered regions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 5, 6, 7 ], "tagConfidence": "reviewed", "answerGuide": "Combine the central hexagon area with the six straight rectangular sections and the six circular sectors at the corners, using the radius and angle found earlier." }, { "id": "cie-2021-s-13-q01", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-13-q02", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-13-q03", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2021-s-13-q04a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-13-q04b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-13-q04c", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(c): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-13-q05a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2021-s-13-q05b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(b): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2021-s-13-q06a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-13-q06b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(b): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-s-13-q07a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-s-13-q07b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "7(b): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-s-13-q08a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "8(a): Compose the two defined functions in the stated order.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the inverse of the indicated function, substitute it into the other function in the stated order, and simplify." }, { "id": "cie-2021-s-13-q08b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "8(b): Compose the inverse of one defined function with the other function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the inverse of the indicated function, substitute it into the other function in the stated order, and simplify." }, { "id": "cie-2021-s-13-q09a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(a): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-s-13-q09b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=14", "qpPages": [ 13, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2021-s-13-q10a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-s-13-q10b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-s-13-q10c", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(c): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-s-13-q10d", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(d)", "marks": 5, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the tangent at E, where BE is a diameter of the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the most frequently omitted question part and says candidates rarely used the circle-and-line intersection route to obtain the tangent directly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 8, 9, 10 ], "tagConfidence": "reviewed", "answerGuide": "Use the diameter condition to find E, determine the radius DE and its gradient, then use the perpendicular gradient in a line equation through E." }, { "id": "cie-2021-s-13-q11a", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-13-q11b", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "11(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-s-13-q11c", "paperId": "cie-2021-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "11(c): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_13.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-s-21-q01", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-s-21-q02", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on the stated full angle interval after expanding the compound angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates could not expand the compound-angle expression correctly, and errors in the subsequent algebra were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Expand sin(θ + 30°), replace cosec θ by 1/sin θ, rearrange to an equation in tan θ, then list all solutions in the interval." }, { "id": "cie-2021-s-21-q03a", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Expand the squared secant-and-cosine expression into the stated double-angle form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few correct responses were seen; candidates commonly missed the double-angle identity and treated the square on cos x as a coefficient.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Expand the square, use sec x cos x = 1, and replace cos²x with (1 + cos 2x)/2 before collecting constants and the cos 2x term." }, { "id": "cie-2021-s-21-q03b", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-21-q04a", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "4(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-21-q04b", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the required parametric derivative at the point whose x-coordinate is specified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few correct responses were seen; the report notes frequent chain-rule and product-rule errors when differentiating the parametric equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate both parametric equations with respect to t, apply the product and chain rules, find the relevant parameter value, and form dy/dx = (dy/dt)/(dx/dt)." }, { "id": "cie-2021-s-21-q05a", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "5(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-21-q05b", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Use a sign change to locate the minimum point's x-coordinate within the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct methods were seen; the report says many candidates tested the wrong function and failed to demonstrate a sign change for the correct condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Define the function whose zero gives the required stationary-point condition, evaluate it at both interval endpoints and show that its sign changes." }, { "id": "cie-2021-s-21-q05c", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-21-q06a", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(a): Apply the stated numerical procedure to approximate the required root or value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-21-q06b", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-21-q06c", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "6(c): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-s-21-q06d", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Give the requested conclusion about the trapezium-rule approximation and justify it.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates gave both a correct answer and a valid reason; explanations usually referred to numerical accuracy rather than the curve's shape and the trapezia.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 11, 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Use the curve's shape and its position relative to the trapezia to decide whether the rule overestimates or underestimates the integral, then state the reason clearly." }, { "id": "cie-2021-s-21-q07a", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-s-21-q07b", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-s-21-q07c", "paperId": "cie-2021-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_21.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-s-22-q01a", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-22-q01b", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-22-q02", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve for the two roots of the modulus equation and evaluate the requested expression involving them.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few correct solutions were seen; many candidates squared terms before finding the constants and did not handle the modulus or root ordering correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Solve the equation piecewise for the absolute value, preserve both roots in order, then substitute each root into its corresponding absolute-value expression." }, { "id": "cie-2021-s-22-q03", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-22-q04a", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-22-q04b", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the indefinite integral involving tan²x and sin²2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few completely correct responses were seen; the report notes that candidates often missed the double-angle identity needed before integrating.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite tan²x as sec²x − 1 and sin²2x using a double-angle identity, then integrate each term in exact form." }, { "id": "cie-2021-s-22-q05a", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-s-22-q05b", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-s-22-q05c", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-22-q06a", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradients at the two x-intercepts of the logarithmic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few correct solutions were seen; the report highlights errors differentiating (ln x)² and candidates losing one intercept by dividing by ln x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Set the curve equal to zero to find both intercepts, differentiate using the chain rule, and evaluate the derivative at each intercept." }, { "id": "cie-2021-s-22-q06b", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinate of the minimum of the logarithmic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few fully correct solutions were seen; most candidates carried forward an incorrect derivative from the preceding part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate y = (ln x)² − 2 ln x, set the derivative to zero, and solve for the positive x-coordinate." }, { "id": "cie-2021-s-22-q07a", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-22-q07b", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show by calculation that the parameter p lies between the two stated bounds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct methods were seen; the report says many candidates used the wrong function when testing for the required sign change.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange the defining relation as a root equation and evaluate the corresponding function at each endpoint to demonstrate a sign change." }, { "id": "cie-2021-s-22-q07c", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-22-q07d", "paperId": "cie-2021-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(d)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(d): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_22.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-s-23-q01a", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-23-q01b", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-23-q02", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve for the two roots of the modulus equation and evaluate the requested expression involving them.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few correct solutions were seen; many candidates squared terms before finding the constants and did not handle the modulus or root ordering correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Solve the equation piecewise for the absolute value, preserve both roots in order, then substitute each root into its corresponding absolute-value expression." }, { "id": "cie-2021-s-23-q03", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-23-q04a", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-23-q04b", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the indefinite integral involving tan²x and sin²2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few completely correct responses were seen; the report notes that candidates often missed the double-angle identity needed before integrating.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite tan²x as sec²x − 1 and sin²2x using a double-angle identity, then integrate each term in exact form." }, { "id": "cie-2021-s-23-q05a", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-s-23-q05b", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-s-23-q05c", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-23-q06a", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradients at the two x-intercepts of the logarithmic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few correct solutions were seen; the report highlights errors differentiating (ln x)² and candidates losing one intercept by dividing by ln x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Set the curve equal to zero to find both intercepts, differentiate using the chain rule, and evaluate the derivative at each intercept." }, { "id": "cie-2021-s-23-q06b", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinate of the minimum of the logarithmic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few fully correct solutions were seen; most candidates carried forward an incorrect derivative from the preceding part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate y = (ln x)² − 2 ln x, set the derivative to zero, and solve for the positive x-coordinate." }, { "id": "cie-2021-s-23-q07a", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-23-q07b", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show by calculation that the parameter p lies between the two stated bounds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct methods were seen; the report says many candidates used the wrong function when testing for the required sign change.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange the defining relation as a root equation and evaluate the corresponding function at each endpoint to demonstrate a sign change." }, { "id": "cie-2021-s-23-q07c", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-23-q07d", "paperId": "cie-2021-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(d): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_23.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-s-31-q01", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-s-31-q02", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the real root of the exponential equation to three decimal places and establish uniqueness.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few candidates recognised the resulting quadratic in eˣ; a common invalid method treated the logarithm of a sum as a sum of logarithms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 20, 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Clear the denominator, set u = eˣ to form a quadratic, retain only positive u values, then use the monotonicity or equivalent argument to justify the unique root before taking logarithms." }, { "id": "cie-2021-s-31-q03a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-31-q03b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-31-q04a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-31-q04b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral obtained from the trigonometric identity in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes this definite integration part as challenging for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 20, 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (a) to rewrite the integrand as tan²θ, replace it with sec²θ − 1, and apply the stated limits." }, { "id": "cie-2021-s-31-q05a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-s-31-q05b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the condition on p and q when both distinct roots lie on the imaginary axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates recognised that the imaginary-root condition required a negative discriminant.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 20, 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Use the quadratic formula or discriminant to require distinct purely imaginary roots, then translate that condition into a relation between p and q." }, { "id": "cie-2021-s-31-q05c", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Express q in terms of p when the origin and the two roots form an equilateral triangle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates correctly applied the equilateral-triangle conditions to their complex roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 20, 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Use the root coordinates in the Argand plane and equate the lengths from the origin and between the two roots to impose the equilateral condition." }, { "id": "cie-2021-s-31-q06a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-s-31-q06b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2021-s-31-q07a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-31-q07b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-31-q07c", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-31-q08a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "8(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-s-31-q08b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the point on the given line that is equidistant from A and B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates made progress; the report says most gave no solution to this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 20, 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Parameterise a general point on the line, express its distances from A and B, equate the squared lengths and solve for the parameter." }, { "id": "cie-2021-s-31-q09a", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-31-q09b", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "9(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-31-q10", "paperId": "cie-2021-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10", "marks": 11, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_31.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-32-q01", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-s-32-q02", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the complex-number region satisfying the circle and argument inequalities.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the half-line boundary was more challenging than the circle; some candidates incorrectly made it pass through the circle's centre.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 23, 24, 25, 26 ], "tagConfidence": "reviewed", "answerGuide": "Draw the closed disk with centre −1 + i and radius 1, interpret the argument inequality as an angular half-line boundary from 1, then shade their intersection." }, { "id": "cie-2021-s-32-q03a", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-s-32-q03b", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-32-q04", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate exactly the definite integral of arctan(x/2) using integration by parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as a challenging question for many candidates, with errors in recognising and differentiating the inverse-tangent expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 23, 24, 25, 26 ], "tagConfidence": "reviewed", "answerGuide": "Take arctan(x/2) as the undifferentiated factor, integrate the remaining factor, and simplify the resulting rational integral before applying the limits." }, { "id": "cie-2021-s-32-q05", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5: Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2021-s-32-q06a", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-32-q06b", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-32-q07", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-s-32-q08", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8", "marks": 8, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8: Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-32-q09a", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-32-q09b", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "9(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-s-32-q10a", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(a): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-s-32-q10b", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-32-q10c", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "10(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-32-q10d", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "10(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-32-q11a", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-s-32-q11b", "paperId": "cie-2021-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the possible position vectors of P on OM under the stated length ratio condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_32.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates made progress; many used an incorrect midpoint vector and few formed the required relation between AP and the position vector of P.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 23, 24, 25, 26 ], "tagConfidence": "reviewed", "answerGuide": "Find the midpoint vector OM, parameterise P on the line through O and M, form AP from the coordinates, and solve the stated ratio using lengths." }, { "id": "cie-2021-s-33-q01", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1: Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-s-33-q02", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-33-q03a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-33-q03b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-s-33-q04a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-33-q04b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "4(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-s-33-q05a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-33-q05b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-s-33-q06a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-s-33-q06b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-33-q06c", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-s-33-q07ai", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 1, "topicIds": [ "proof", "differentiation" ], "primaryTopicId": "proof", "summary": "Show that the geometric ratio MN/y equals the derivative dy/dx at P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates justified the statement; many simply repeated the expression in the question or made an integration/sign error.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 27, 28, 29, 30 ], "tagConfidence": "reviewed", "answerGuide": "Write the gradient of the normal PN from its vertical and horizontal displacements, then use the negative reciprocal relation between normal and tangent gradients." }, { "id": "cie-2021-s-33-q07aii", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a)(ii): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2021-s-33-q07b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-33-q08a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2021-s-33-q08b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "integration", "proof" ], "primaryTopicId": "integration", "summary": "Use integration by parts to prove the stated bound for the logarithmic integral for every a > 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the conclusion in this part challenged most candidates, despite many showing good integration-by-parts technique.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 27, 28, 29, 30 ], "tagConfidence": "reviewed", "answerGuide": "Integrate by parts, simplify the boundary and residual terms, and use a > 1 to establish the required strict inequality." }, { "id": "cie-2021-s-33-q09a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the side direction vectors and show one pair of opposite sides is parallel." }, { "id": "cie-2021-s-33-q09b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the side direction vectors and show one pair of opposite sides is parallel." }, { "id": "cie-2021-s-33-q09c", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(c): Use the position vectors to calculate the area of the quadrilateral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the side direction vectors and show one pair of opposite sides is parallel." }, { "id": "cie-2021-s-33-q10a", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-33-q10b", "paperId": "cie-2021-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 7, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_33.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-s-41-q01", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1: Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-41-q02a", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q02b", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q03a", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q03b", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "3(b): Use the stated energy or momentum relationship to find the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2021-s-41-q04a", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q04b", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration of Isabella that makes both cyclists travel the same distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Candidates generally found this part challenging; the report says a common error was using an incorrect approach for Isabella's distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 31, 32, 33 ], "tagConfidence": "reviewed", "answerGuide": "Calculate each cyclist's distance over the three motion stages, express the totals in terms of a, equate them and solve." }, { "id": "cie-2021-s-41-q05a", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q05b", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q06a", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a): Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2021-s-41-q06b", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(b): Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2021-s-41-q07ai", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a)(i): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q07aii", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a)(ii): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-41-q07b", "paperId": "cie-2021-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(b): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_41.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2021-s-42-q01", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-42-q02", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2: Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2021-s-42-q03", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2021-s-42-q04", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4: Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2021-s-42-q05ai", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a)(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-s-42-q05aii", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a)(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-s-42-q05bi", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b)(i): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-42-q05bii", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(b)(ii): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2021-s-42-q06", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the difference between the two possible ground-impact times after two particles collide and coalesce.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found the question particularly challenging and often made sign errors in the vertical-motion equation for the collision time.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 34, 35, 36 ], "tagConfidence": "reviewed", "answerGuide": "Find the collision time from the particles' vertical motions, use conservation of momentum for each possible mass assignment, then solve the post-collision motion to the ground." }, { "id": "cie-2021-s-42-q07a", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2021-s-42-q07b", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2021-s-42-q07c", "paperId": "cie-2021-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(c): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_42.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2021-s-43-q01", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q02a", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q02b", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q03", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2021-s-43-q04a", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-s-43-q04b", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the height h that the particle exceeds for 3.6 seconds during its vertical flight.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was more challenging and produced a variety of attempted methods.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 37, 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Use the symmetry of the height-time parabola to relate the two times at height h, substitute into vertical displacement with u = 22 and solve for h." }, { "id": "cie-2021-s-43-q05a", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(a): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the work-energy relation, accounting for the work of each force and any change in kinetic or potential energy." }, { "id": "cie-2021-s-43-q05b", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q06a", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q06b", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2021-s-43-q06c", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(c): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q07a", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2021-s-43-q07b", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-s-43-q07c", "paperId": "cie-2021-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the total distance travelled after the applied force is removed and the particle comes to rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_43.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this continuation was more challenging and that errors mainly arose when calculating acceleration after removal of the applied force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 37, 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Use the friction coefficient and the speed reached after the first three seconds to find the subsequent deceleration, then add the distance for that final stage." }, { "id": "cie-2021-s-51-q01", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count the selections of five differently sized marbles containing at least four of one colour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says some candidates found this question particularly challenging.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Split the selection into exactly four red and one blue, and five red; count each case with combinations and add them." }, { "id": "cie-2021-s-51-q02", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2: Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-s-51-q03a", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-51-q03b", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-51-q03c", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(c): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2021-s-51-q04a", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-51-q04b", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a student succeeds in gaining a science-college place through the staged tests.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says the context was challenging for some candidates and weaker solutions often treated the written-test routes separately or combined their probabilities incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Add the mutually exclusive routes: passing the first written test then the practical, or failing once, passing the second written test, then passing the practical." }, { "id": "cie-2021-s-51-q04c", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-s-51-q05a", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "5(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2021-s-51-q05b", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-51-q05c", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the greatest possible interquartile range for the grouped puzzle times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the most challenging part; many candidates used an incorrect upper class bound or merely subtracted quartile positions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Locate the classes containing the first and third quartiles, then use the extreme positions allowed within those classes to maximise their difference." }, { "id": "cie-2021-s-51-q06a", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-51-q06b", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-51-q06c", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Justify the normal approximation to the binomial model used in part (b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the vast majority found this justification challenging and often compared the wrong quantity with 5.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Check that both np and nq satisfy the normal-approximation condition for the sample size and probability, and state the conclusion." }, { "id": "cie-2021-s-51-q07a", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-s-51-q07b", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-51-q07c", "paperId": "cie-2021-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the variance of the number of tins opened before the first carrot tin is found.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_51.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not attempt this part; the report notes that a table of xP and x²P supported the successful variance calculations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Complete the probability distribution from part (b), calculate E(X) and E(X²), and use Var(X) = E(X²) − E(X)²." }, { "id": "cie-2021-s-52-q01a", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-52-q01b", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-52-q01c", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-52-q02", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2: Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-s-52-q03a", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-52-q03b", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-s-52-q04a", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-52-q04b", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-52-q05a", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-52-q05b", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-52-q05c", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-52-q06a", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-52-q06b", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count TOMORROW arrangements with R at both ends and the three Os not all together.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part very challenging and says few fully correct solutions were seen; repeated Os and indistinguishable Rs were often mishandled.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with the two Rs fixed at the ends, subtract cases with the three Os in one block, and divide by the repeated-letter factorials at each stage." }, { "id": "cie-2021-s-52-q06c", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that a four-letter selection from TOMORROW contains at least one O and at least one R.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates found this part challenging and often omitted possible O/R cases or failed to account for repetition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate feasible counts of Os and Rs in the selection, account for the repeated letters as indistinguishable, and divide the favourable count by all four-letter selections." }, { "id": "cie-2021-s-52-q07a", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "7(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2021-s-52-q07b", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-52-q07c", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2021-s-52-q07d", "paperId": "cie-2021-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(d): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_52.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-53-q01a", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2021-s-53-q01b", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-53-q01c", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(c): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2021-s-53-q02", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-53-q03a", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-53-q03b", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-53-q04a", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-53-q04b", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-53-q05a", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-s-53-q05b", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-s-53-q06a", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-53-q06b", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-53-q06c", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count REQUIREMENT arrangements with exactly three letters between the two Rs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part particularly challenging; only the most careful approaches produced the correct count.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 48, 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Choose valid positions for the two Rs at the required separation, then arrange the remaining letters and divide by 3! for the repeated Es." }, { "id": "cie-2021-s-53-q06d", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(d): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2021-s-53-q07ai", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a)(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-53-q07aii", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a)(ii): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-s-53-q07bi", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b)(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-53-q07bii", "paperId": "cie-2021-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b)(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_53.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q01", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q02a", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q02b", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Explain whether the Central Limit Theorem is needed when finding the sample-mean probability from a normal population.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls this part particularly challenging and says many responses failed to identify the parent population as the distribution known to be normal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "State that the theorem is unnecessary because the parent population is already normally distributed, so the sample mean has a normal distribution for any sample size." }, { "id": "cie-2021-s-61-q03", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use symmetry of the density about x = 4 to find the probability between 3 and 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not know how to approach this question and many did not attempt it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Pair the equal areas on either side of x = 4 and use P(X < 5) to determine the two symmetric tail pieces, then combine the required intervals." }, { "id": "cie-2021-s-61-q04", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q05a", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-s-61-q05b", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the largest sample size for which the probability of no cases exceeds 0.9.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not know how to approach this part and many omitted it; trial-and-improvement attempts were rarely successful.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Use the valid Poisson or binomial expression for zero cases, solve the inequality for n, and round down to the largest integer satisfying it." }, { "id": "cie-2021-s-61-q06", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6: Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2021-s-61-q07a", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q07b", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q08a", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "8(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-61-q08b", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "8(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-61-q08c", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a binomial model to find the probability of a Type I error at the stated significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as particularly challenging; it says the cumulative probability at the adjacent critical boundary was needed for a full justification.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Under the null model B(50, 0.15), identify the lower-tail critical region from cumulative probabilities at the boundary counts, then state its Type I error probability." }, { "id": "cie-2021-s-61-q08d", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(d)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State the test conclusion when four of the 50 students are found out of uniform.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says some candidates found this conclusion part particularly challenging and often failed to compare with the critical region or significance level.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 51, 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Compare the observed count with the critical region found in part (c), or compare the relevant probability with 0.05, and state the conclusion in context." }, { "id": "cie-2021-s-61-q08e", "paperId": "cie-2021-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "8(e): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_61.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-62-q01a", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-62-q01b", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-s-62-q02a", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-62-q02bi", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b)(i): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-s-62-q02bii", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b)(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-s-62-q03a", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-s-62-q03b", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Use the density 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"cie-2021-s-62-q05b", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-62-q06a", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, 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"difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-62-q06c", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability that all four 99% confidence intervals contain the population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found this part more challenging and many omitted it or multiplied or exponentiated the wrong probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 54, 55, 56 ], "tagConfidence": "reviewed", "answerGuide": "Use 0.99 as the probability for one interval to contain the true mean, then multiply the four independent coverage probabilities." }, { "id": "cie-2021-s-62-q07a", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_62.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-62-q07b", "paperId": "cie-2021-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, 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"cie-2021-s-63-q01", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2021-s-63-q02a", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-63-q02b", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-63-q02c", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-s-63-q03ai", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a)(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-63-q03aii", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a)(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-63-q03b", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Explain how an increasing annual accident rate affects the validity of the Poisson test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Most candidates omitted some or all of the required comments about the changing mean rate, the Poisson assumption and the validity of the test.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_er.pdf", "sourceEvidencePages": [ 57, 58, 59 ], "tagConfidence": "reviewed", "answerGuide": "State that the Poisson model for the test assumes a constant mean rate, then explain that the upward trend undermines that assumption and hence the validity of the test." }, { "id": "cie-2021-s-63-q04a", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-63-q04b", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-s-63-q04c", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-s-63-q04d", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-63-q05a", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-s-63-q05b", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-63-q05c", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-s-63-q05d", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(d): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-s-63-q06a", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-s-63-q06b", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2021-s-63-q06c", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=12", "qpPages": [ 10, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2021-s-63-q06d", "paperId": "cie-2021-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(d): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s21_ms_63.pdf#page=13", "qpPages": [ 10, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "ial-2020-10-WMA11-01-q01", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Find the constants after simplifying a product of powers of p and q.", "answerGuide": "Apply the power rules to each factor, combine the exponents, and match the coefficient and powers with apᵇqᶜ.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q02a", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Write the quadratic function in completed-square form.", 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], "primaryTopicId": "functions", "summary": "Apply the translated parabola to find its two x-intercepts, then determine the area of the resulting triangle.", "answerGuide": "Form the translated equation, solve exactly for both roots, and use their separation as the base with the stated vertical distance as the height.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson lists Q2(b) among the paper’s most challenging parts; the report describes errors in the translation, roots and triangle dimensions.", "difficultySourceUrl": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q03b", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 8, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Find the badge area and perimeter from the sector and two congruent triangles.", "answerGuide": "Use the angle at O and the cosine rule for each triangle; add the triangle and sector areas, then add the exposed sides for the perimeter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q04", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "algebra", "quadratics", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": 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"variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 6, "topicIds": [ "transformations", "functions", "graphs" ], "primaryTopicId": "transformations", "summary": "Sketch the two translated/reflected versions of the given curve and label their axis intersections.", "answerGuide": "Apply the horizontal transformations to each known point, preserve the curve's turning/contact behavior, and label every intercept.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=15", "qpPages": [ 14, 17 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q05ii", "paperId": 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[ 14, 17 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls part (ii) a discriminator and says it was rarely answered correctly; many assumed the maximum lay on the y-axis.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WMA11-01-q06a", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the gradient of AB as a simplified fraction.", "answerGuide": "Use the change in y divided by the change in x and reduce the fraction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=17", "qpPages": [ 18, 21 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q06b", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the integer-coefficient equation of the perpendicular bisector of AB.", "answerGuide": "Find the midpoint, take the negative reciprocal gradient, then use point-slope form and clear fractions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=17", "qpPages": [ 18, 21 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q06c", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two possible positions of a point constrained to a line when the triangle area is fixed.", "answerGuide": "Use the area to find the required distance from the line or midpoint, then intersect the line with the corresponding circle to find both positions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=17", "qpPages": [ 18, 21 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates earned no more than two of the five marks; many stopped after finding one length or set up equations without reducing to one unknown.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5, 6 ] } }, { "id": "ial-2020-10-WMA11-01-q07a", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch a translated reciprocal curve and show its intercept and asymptotes.", "answerGuide": "Recognise the reciprocal shape and its translation; locate the intercept and both asymptotes, then place the two branches in the correct quadrants.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=19", "qpPages": [ 22, 25 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates often failed to recognise the reciprocal form, and very few labelled both asymptotes.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2020-10-WMA11-01-q07b", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A 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"difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q07c", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Find the range of k for which the line meets the curve at two distinct points.", "answerGuide": "Solve the quadratic inequality from part (b), retaining the strict inequality and both exterior intervals.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=19", "qpPages": [ 22, 25 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q08a", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the factored rational curve and show the stated cubic derivative.", "answerGuide": "Rewrite or expand the quotient, differentiate, and simplify to the required polynomial.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=21", "qpPages": [ 26, 29 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q08b", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent to the curve at x=6.", "answerGuide": "Evaluate the curve and derivative at x=6, then form the tangent using point-slope form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=21", "qpPages": [ 26, 29 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA11-01-q08c", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the point or points where the curve has a tangent with the specified gradient.", "answerGuide": "Differentiate the curve, set the derivative equal to the required gradient, solve for the relevant x-coordinate(s), and use the curve equation for coordinates if required.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=21", "qpPages": [ 26, 29 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates made no meaningful attempt at part (c), and successful methods equated the gradient to the stated value.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2020-10-WMA11-01-q09", "paperId": "ial-2020-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2020, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9", "marks": 6, "topicIds": [ "integration", "indices", "algebra" ], "primaryTopicId": "integration", "summary": "Recover f(x) from its derivative and the point (9,10).", "answerGuide": "Split the fractional-index terms, integrate each power, then substitute (9,10) to find the constant and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20201020.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_msc_20210113.pdf#page=23", "qpPages": [ 30, 32 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q01a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the binomial expression through its first four 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"marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the constant term in the product of two series.", "answerGuide": "Use the required terms from each expansion and collect the contributions with total power zero.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q02a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "functions", "data-representation" ], "primaryTopicId": "functions", "summary": "Complete the table with the missing function values to three decimal places.", "answerGuide": "Substitute each listed x-value into the given function and round to three decimal places.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q02b", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Estimate the definite integral with the trapezium rule using the completed table.", "answerGuide": "Use the equal step width and all five tabulated ordinates in the trapezium rule.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q03a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Use the remainder theorem to derive the stated relation between a and b.", "answerGuide": "Evaluate f(−4), set it equal to the given remainder, and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q03b", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Use the factor condition to find a and b.", "answerGuide": "Set f(½)=0, solve it together with the relation from part (a), and state both constants.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q03c", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the cubic polynomial.", "answerGuide": "Apply the power rule term by term.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q03d", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of both stationary points of a cubic curve.", "answerGuide": "Set f′(x)=0 and solve the quadratic to obtain x=−1 and x=4/3. Substitute each value into f(x), giving (−1, 9) and (4/3, −100/27).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that a significant number of candidates stopped after finding the x-values and earned only one of four marks; some used the second derivative or gave y=0 for both coordinates. (PEF p. 4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-10-WMA12-01-q04a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the centre and radius from the endpoints of a circle's diameter.", "answerGuide": "Average the endpoint coordinates for the centre and use half the diameter length for the radius.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q04b", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Write an equation for the circle.", "answerGuide": "Use the centre and radius in (x−h)²+(y−k)²=r².", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q04c", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent to the circle at Q in the requested integer-coefficient form.", "answerGuide": "Find the radius gradient 3/4, so the tangent gradient is −4/3. Use Q=(−3,12) in point-slope form and rearrange to 4x+3y−24=0.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson notes that a significant number earned only the first mark by using the radius gradient as the tangent gradient; other common errors were using P or the centre instead of Q. (PEF p. 4)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-10-WMA12-01-q05a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the amount saved in the 40th month of an arithmetic sequence.", "answerGuide": "Use the nth-term formula with first term £100 and common difference £5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q05b", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the total saved over 60 months when the monthly amount increases arithmetically.", "answerGuide": "Use the arithmetic sequence with first term 100 and common difference 5; calculate the 60th term and apply the arithmetic-series sum formula.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Pearson specifically notes that candidates lost marks by using 40 terms for the requested 60-month total.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2020-10-WMA12-01-q05c", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "sequences", "quadratics" ], "primaryTopicId": "sequences", "summary": "Form the quadratic equation for Lina's total savings and show it has the stated form.", "answerGuide": "Use the arithmetic-series sum with first term 600 and difference −10, set it to 18,200, and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q05d", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Solve the quadratic equation for n.", "answerGuide": "Factor the quadratic or use the quadratic formula and give both roots.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q05e", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 1, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Identify which root is not a sensible number of months.", "answerGuide": "Reject the root that is outside the stated 60-month saving period or is negative, explaining why.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q06a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Find the second intersection point of two curves, given one intersection.", "answerGuide": "Equate the curve expressions, factor out the known root, solve for the other x-value, then substitute to obtain y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q06b", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the exact area enclosed by the two curves.", "answerGuide": "Determine which curve lies above the other between the intersections and integrate their difference over that interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q07i", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the stated trigonometric identity.", "answerGuide": "Rewrite tanθ as sinθ/cosθ, combine fractions, and simplify to the required expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=15", "qpPages": [ 24, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q07ii", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the squared-cosine equation over the stated degree interval and give both answers to one decimal place.", "answerGuide": "Rearrange to cos²(2x+10°)=1/3, then use cos(2x+10°)=±1/√3. Apply the interval restriction to obtain x=22.4° and x=57.6°.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=15", "qpPages": [ 24, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson reports that many candidates omitted the square root of 1/3 or the negative root; 22.4° was the most common single answer, with some substituting 1/3 directly into inverse cosine. (PEF p. 6)", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2020-10-WMA12-01-q08a", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Prove the stated formula for the sum of the first n terms of a geometric series.", "answerGuide": "Write the series sum and its ratio-shifted version, subtract to cancel the middle terms, and rearrange to the required finite-sum expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes the standard geometric-series proof as a disappointing source of lost marks; several candidates used an incorrect shifted sum or skipped the subtraction step.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2020-10-WMA12-01-q08b", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the common ratio from the second and fifth terms.", "answerGuide": "Divide the fifth term by the second to obtain r³, then select the real cube root.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q08c", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the sum of the first 13 terms of the geometric series.", "answerGuide": "Use the known second term to find a, then apply the finite geometric-series formula and round to two decimal places.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q09i", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(i)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the logarithmic equation exactly for x.", "answerGuide": "Use logarithm laws to combine terms, equate arguments, solve the resulting equation, and check the domain.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA12-01-q09iia", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)(a)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show that the given exponential equation reduces to a specified power equation.", "answerGuide": "Rewrite each term with a common base, collect the powers of y, and simplify the resulting equation to the stated base raised to y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a minority completed this proof successfully; the report notes that few candidates gained the later proof mark with the logarithm method.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2020-10-WMA12-01-q09iib", "paperId": "ial-2020-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2020, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)(b)", "marks": 2, "topicIds": [ "exponentials", "numerical-methods" ], "primaryTopicId": "exponentials", "summary": "Find y to three decimal places after obtaining the exponential equation.", "answerGuide": "Take logarithms of the positive exponential equation and round y to three decimal places.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20201022.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210113.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q02b", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "exponentials", "numerical-methods" ], "primaryTopicId": "exponentials", "summary": "Use the model to estimate the covered area after 30 days.", "answerGuide": "Substitute t=30 into the exponential model and round to two significant figures.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q03a", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate a quotient involving fractional 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q04b", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve f(x)=5x on the stated domain.", "answerGuide": "Split at the absolute-value vertex, solve the resulting linear equations, and keep only valid roots.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q04c", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "functions", "graphs" ], "primaryTopicId": "functions", "summary": "State the values of k for which f(x)=k has exactly two 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"topicIds": [ "transformations", "functions" ], "primaryTopicId": "transformations", "summary": "Find a and b from the transformed graph's vertex.", "answerGuide": "Track the original vertex through y=af(x−b), then use its transformed coordinates to solve for a and b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q05a", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", 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"answerGuide": "Match coefficients with Rcos(x−α), use R²=1²+4², and determine α from tanα=4.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q07b", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Determine the minimum height in a periodic-motion model.", "answerGuide": "Use the model’s amplitude and phase information to find the minimum value of the sinusoid, then add the vertical offset and state the contextual units.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says only about a third obtained the correct minimum; common errors included mishandling the amplitude or interpreting the height and units.", "difficultySourceUrl": 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g(x)=e³ˣsec(2x).", "answerGuide": "Apply the product rule and chain rule to the exponential and secant factors.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q08ib", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)(b)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the stationary point.", "answerGuide": "Set the derivative to zero, solve the resulting tangent equation, and select the root in the given interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q08ii", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate an inverse relation involving x = ln(sin y) and express the result in terms of x.", "answerGuide": "Differentiate implicitly, use sin y = e^x and the given y-range to express cos y in terms of e^x, then rearrange the derivative into the required form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls part (ii) much more challenging; only about one third earned full marks, often due to difficulty eliminating y.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 8, 9 ] } }, { "id": "ial-2020-10-WMA13-01-q09a", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Use the identity to find P and prove that Q=5.", "answerGuide": "Expand the right-hand side, collect powers of x, and equate coefficients with the left-hand side.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA13-01-q09b", "paperId": "ial-2020-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2020, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent to the 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"marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the exact area under the specified rational curve over the given interval.", "answerGuide": "Use the earlier algebraic decomposition to express the integrand in integrable terms, integrate, substitute the limits carefully, and simplify logarithms with valid positive arguments.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20201009.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-msc-20210113.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only about one third were fully correct; the report identifies sign, modulus and limit-substitution errors in the final definite integration.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 10 ] } }, { "id": "ial-2020-10-WMA14-01-q01", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove the parity statement about n by contradiction.", "answerGuide": "Assume n is odd and write n = 2k + 1; expand the relevant cube, show it is odd, and state the contradiction and conclusion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports very few completely correct solutions, commonly because candidates did not establish the contradiction algebraically.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2020-10-WMA14-01-q02a", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": 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"ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the series coefficients to find k.", "answerGuide": "Multiply the two series through the x term and equate the constant coefficient with the stated value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q02c", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the coefficient m in the expansion.", "answerGuide": "Collect the coefficient of x after substituting k from part (b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q03", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find the exact volume of revolution and express it as a ln2+b.", "answerGuide": "Find the x-intercept, integrate πy² from zero to that intercept, and simplify the exponential integral.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q04a", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "vectors" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of A and verify B=(20,0) on the parametric curve.", "answerGuide": "Set y=0, solve t³−4t=0 for the nonzero parameter values, then substitute into x(t) and y(t).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q04b", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Show the tangent equation at B.", "answerGuide": "Evaluate dx/dt and dy/dt at the parameter for B, form dy/dx, and use the point-slope equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q04c", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the x-coordinate of the second intersection P of the tangent and curve.", "answerGuide": "Substitute the tangent relation into the parametric equations, solve for the repeated intersection parameter, and evaluate x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q05a", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the indefinite integral involving ln x divided by a power of x.", "answerGuide": "Use integration by parts with ln x and the power of x, then include the constant.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q05b", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "integration", "geometry" ], "primaryTopicId": "integration", "summary": "Find the exact shaded area between x=2 and x=4.", "answerGuide": "Use the curve and antiderivative from part (a), account for the sign of the graph, and evaluate at both limits.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies implicit differentiation as a source of difficulty; many stopped after taking logs or mishandled the product rule.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WMA14-01-q06b", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show that the stationary-point x-coordinates satisfy tan x+x ln x=0.", "answerGuide": "Set the derivative from part (a) to zero, use y>0 and the curve equation, then rearrange to the stated equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q07i", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=16", "qpPages": [ 22, 25 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes Q7 as highly discriminating and identifies part (ii) as the harder rational-integration step.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WMA14-01-q08a", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "vectors", "algebra" ], "primaryTopicId": "vectors", "summary": "Find the position vector of the intersection X of two lines.", "answerGuide": "Equate the vector components, solve for the two parameters, then substitute into either line.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20201014.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_msc_20210113.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WMA14-01-q08b", "paperId": "ial-2020-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2020, 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calls this a demanding, frequently omitted final part and notes that candidates needed to reason about the limiting form.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2020-10-WME01-01-q01a", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the particles' common post-collision speed using conservation of momentum.", "answerGuide": "Write momentum before and after the collision and use the given speed ratio to solve.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q01b", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the impulse exerted on P.", "answerGuide": "Use the change in momentum of P, taking care to distinguish pre- and post-collision velocities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q01c", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "State the direction of the impulse on P.", "answerGuide": "Compare P's velocity before and after the collision and state the direction of its momentum change.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q02a", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the ball's total flight time to the ground.", "answerGuide": "Use constant-acceleration equations with upward positive and solve the height equation, taking the positive time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q02b", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A 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"Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled during the flight.", "answerGuide": "Find the rise to the highest point, then add the full descent from that point to the ground.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q02d", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph and label its start and end coordinates.", "answerGuide": "Use constant acceleration to draw a straight line from the initial velocity to the negative impact velocity at the flight time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q03a", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction on the particle on the inclined plane.", "answerGuide": "Resolve perpendicular to the plane and use R=mgcosα.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q03b", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest possible applied force that maintains equilibrium.", 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"Differentiate each velocity component with respect to time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q05d", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "kinematics", "vectors" ], "primaryTopicId": "kinematics", "summary": "Find when the velocity is parallel to the specified direction vector.", "answerGuide": "Equate the component ratio to that of the target vector and solve for t, checking direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q06a", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of the truck on the incline.", "answerGuide": "Resolve forces on the truck alone parallel to the incline, equate the resultant force to mass times acceleration, and solve using the stated resistance and thrust.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls Q6 by far the most challenging question; the report says candidates who did not isolate the truck’s forces could not progress in part (a).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 5 ] } }, { "id": "ial-2020-10-WME01-01-q06b", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the acceleration to find the engine’s driving force.", "answerGuide": "Apply Newton’s second law to the complete system or to the engine, using the acceleration from part (a), then solve for the driving force.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The Q6 commentary says many candidates who had an incorrect acceleration could still earn marks from a correct system or engine equation, while force assignment caused widespread errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WME01-01-q07a", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension and the resultant force exerted on the pulley in a connected-particle system.", "answerGuide": "Write one equation of motion for each particle with consistent signs, solve for acceleration and tension, then combine the two string tensions to obtain the pulley force.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson ranks Q7 third in difficulty; the report notes that relatively few recognized the pulley force as the resultant of the two tensions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-2020-10-WME01-01-q07b", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Explain how pulley smoothness is used in the solution.", "answerGuide": "State that a smooth pulley gives equal tensions in the two hanging parts of the string.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q08a", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled during the first four seconds.", "answerGuide": "Use the area under the acceleration-time graph to obtain velocity, then the velocity-time area for distance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WME01-01-q08b", "paperId": "ial-2020-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2020, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Find total distance travelled during the first 26 seconds.", "answerGuide": "Integrate the piecewise constant acceleration to obtain velocity, split at velocity reversals, and sum positive distance areas.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20201017.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_msc_20210113.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q01", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find E(X) from the discrete probability model with an unknown constant k.", "answerGuide": "Use the total probability to solve for k, then sum xP(X=x) across all five values.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q02a", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability", "data-representation" ], "primaryTopicId": "probability", "summary": "Complete the missing probabilities in the choice tree.", "answerGuide": "Use complementary probabilities at each branch and label the conditional choices consistently.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q02b", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a student chooses water.", "answerGuide": "Add the probabilities of the three mutually exclusive water paths through the tree.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q02c", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use independence to form an equation relating the probabilities p and q.", "answerGuide": "Set the intersection probability equal to the product of the marginal probabilities, substitute the expression from part (b), and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "The report says part (c) proved more challenging; candidates often used the wrong independence equation or made consequential arithmetic errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2020-10-WST01-01-q02d", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use the conditional-probability information to determine p and q.", "answerGuide": "Use independence to identify the relevant marginal probability, translate the conditional probability into an equation, then solve the resulting relations for q and p.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few fully correct solutions were seen and describes conditional-probability reasoning as shaky for many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2020-10-WST01-01-q02e", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 3, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Assess the manager's claim about water drinkers' main-course choices.", "answerGuide": "Calculate the conditional probabilities of each main course given water and compare them.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q03a", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the proportion of students who receive a medal.", "answerGuide": "Standardize 4.3 using the normal mean and standard deviation, then take the upper-tail probability.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=7", "qpPages": [ 8, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q03b", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the distance threshold exceeded by the top 80% of students.", "answerGuide": "Use the 20th percentile of the stated normal distribution and convert back to distance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=7", "qpPages": [ 8, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q03c", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the threshold distance for the upper third of medal-winning long jumps.", "answerGuide": "Use the normal tail probability conditional on exceeding the medal threshold, set that tail to one third of the medal-winner proportion, then standardize and solve for the distance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=7", "qpPages": [ 8, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says part (c) was more challenging; few candidates recognized the conditional probability needed to identify the gold-medal threshold.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-10-WST01-01-q03d", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the exact probability that a group of three medal winners contains at least one gold winner.", "answerGuide": "Use the complement: calculate the probability that none of the three is a gold winner, then subtract from one.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=7", "qpPages": [ 8, 11 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report identifies this as the more challenging final part and says only a few candidates found the correct arrangement probability.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-10-WST01-01-q04a", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "data-analysis", "data-representation" ], "primaryTopicId": "data-analysis", "summary": "Identify the outliers using the quartiles and 1.5×IQR rule.", "answerGuide": "Find the upper quartile from Q3−Q1, calculate both fences, and compare the extreme observations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=8", "qpPages": [ 12, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q04b", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Draw the box plot and show the outliers.", "answerGuide": "Use the quartiles and whisker endpoints, plotting outlying values separately.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=8", "qpPages": [ 12, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q04c", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "data-analysis", "data-representation" ], "primaryTopicId": "data-analysis", "summary": "Describe skewness from the relative quartile spacing.", "answerGuide": "Compare the median's distance to Q1 and Q3 and state the corresponding skew direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=8", "qpPages": [ 12, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q04d", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Redraw the box plot using the new quartiles and unchanged median and upper quartile.", "answerGuide": "Place the updated lower quartile at 26 and retain the given median, upper quartile and outliers.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=8", "qpPages": [ 12, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q04e", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give the possible ranges for two added marks from the stated quartile changes.", "answerGuide": "Track how the two new values must fall relative to the median and quartiles; use the changed lower quartile and unchanged upper quartile to constrain each value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=8", "qpPages": [ 12, 15 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the final part was rarely correct and required careful reasoning about where each new value could sit relative to the quartiles.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2020-10-WST01-01-q05a", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate annual rent for an 800 m² shop from the regression line.", "answerGuide": "Convert the area into the model's x-units, substitute in the regression equation, and convert rent to dollars.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=9", "qpPages": [ 16, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q05b", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate Syy and Sxx from the supplied summary statistics.", "answerGuide": "Use Syy=Σy²−(Σy)²/n and Sxx=Σx²−(Σx)²/n.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=9", "qpPages": [ 16, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q05c", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the product-moment correlation coefficient.", "answerGuide": "Use the regression slope to obtain Sxy, then calculate r=Sxy/√(SxxSyy).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=9", "qpPages": [ 16, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q05d", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show that adding a shop at the original mean floor area leaves Sxy unchanged.", "answerGuide": "Use the updated mean and the added point x = 90; its centered x-term is zero, so the new observation contributes zero to the sum for Sxy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=9", "qpPages": [ 16, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls part (d) challenging; few candidates explained why the added centered x-term is zero and Sxy stays the same.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WST01-01-q05e", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the new regression line after adding the 11th shop.", "answerGuide": "Update the means and Sxx while using the unchanged Sxy, then compute the new slope and intercept.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=9", "qpPages": [ 16, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q05f", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(f)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Comment on predicting rent for a 3000 m² shop.", "answerGuide": "Compare the requested area with the observed sample range and identify extrapolation as a limitation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=9", "qpPages": [ 16, 19 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q06a", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Show that the spinner score has expectation 3.5.", "answerGuide": "Sum each score multiplied by its stated probability.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q06b", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the variance of the spinner score.", "answerGuide": "Calculate E(A²), then subtract [E(A)]².", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q06c", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Name the probability distribution for the four-sided die score.", "answerGuide": "Recognize that each of four values is equally likely and state the discrete uniform distribution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q06d", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find k from the equality of the die and spinner expectations.", "answerGuide": "Average the four equally likely die scores and set the result to 3.5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2020-10-WST01-01-q06e", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that Tim wins for the stated spinner and die results.", "answerGuide": "Compare the two upper-tail normal probabilities; standardize Sam’s result, derive the threshold for Tim’s random mean at standard deviation 4, then add the probabilities for qualifying spinner scores.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the final three parts were more challenging; many did not translate the z-score comparison into the condition on Tim’s spinner score.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WST01-01-q06f", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(f)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the largest possible probability of scoring above 3.5 in the game.", "answerGuide": "Choose the available mean and standard deviation that maximize the upper-tail probability, standardize, and evaluate the corresponding normal tail with sufficient working.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says candidates often looked up the positive z-value without converting it to the required upper tail, limiting marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2020-10-WST01-01-q06g", "paperId": "ial-2020-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2020, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(g)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability of obtaining the maximum probability identified in part (f).", "answerGuide": "Identify the unique spinner and die outcomes that produce the maximizing mean and standard deviation, then multiply their independent probabilities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20201023.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-msc-20210113.pdf#page=10", "qpPages": [ 20, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report notes that some candidates failed to connect this final part to the earlier distributions and instead attempted unnecessary normal calculations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210113.pdf", "tagConfidence": "reviewed", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "cie-2021-m-12-q01a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the first three ascending-power terms of (1 + x)^5.", "answerGuide": "Use the binomial coefficients to obtain 1 + 5x + 10x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q01b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the first three ascending-power terms of (1 − 2x)^6.", "answerGuide": "Expand to obtain 1 − 12x + 60x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q01c", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use both expansions to find the coefficient of x² in their product.", "answerGuide": "Combine the x² contributions: 10 + 5(−12) + 60 = 10.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q02", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Solve the equation involving (2x − 3)² and its reciprocal using a substitution.", "answerGuide": "Let u = (2x − 3)². Solve u − 4/u − 3 = 0, retain u = 4, then solve for x to obtain 1/2 and 5/2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q03", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the given trigonometric quotient equation for angles between 0° and 180°.", "answerGuide": "Use tanθ = sinθ/cosθ and simplify to cosθ = 1/4; θ ≈ 75.5°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q04", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Find the parameter values for which the line and quadratic curve intersect twice.", "answerGuide": "Equate the equations to form a quadratic in x. Require its discriminant (k − 5)(k + 3) to be positive, so k < −3 or k > 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q05a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the two transformations that produce the displayed graph from y = f(x).", "answerGuide": "Read the graph changes relative to the x- and y-axes, distinguish a stretch from a translation, and state the axis, scale factor, and direction for each transformation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this new topic drew fewer attempts and that, although transformations were often identified, their descriptions were often incorrect or did not name the relevant axes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q05b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "transformations", "functions" ], "primaryTopicId": "transformations", "summary": "Give an equation for the broken-line graph in terms of f, x and y.", "answerGuide": "The graph is shifted 4 units right and stretched vertically by factor 3: y = 3f(x − 4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q06a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "differentiation", "kinematics" ], "primaryTopicId": "differentiation", "summary": "Find the rate of change of the point’s x-coordinate at A from the given gradient and dy/dt.", "answerGuide": "At x = 1, dy/dx = 6. Use dy/dt = (dy/dx)(dx/dt) with dy/dt = 3 to get dx/dt = 1/2 units per second.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q06b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Find the equation of the curve from its gradient and the point A(1, −3).", "answerGuide": "Integrate 6(3x − 2)⁻³ and use A to find the constant: y = −2 − 1/(3x − 2)².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q07a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "functions", "quadratics" ], "primaryTopicId": "functions", "summary": "Complete the square for f and state its range on the given domain.", "answerGuide": "Write f(x) = (x + 1)² + 2. Since x ≤ −1, its range is f(x) ≥ 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q07b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "functions", "quadratics" ], "primaryTopicId": "functions", "summary": "Find the inverse function for the restricted quadratic f.", "answerGuide": "Choose the negative square-root branch for the domain x ≤ −1: f⁻¹(x) = −1 − √(x − 2), with domain x ≥ 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q07c", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "functions", "quadratics" ], "primaryTopicId": "functions", "summary": "Solve the composite-function equation g(f(x)) = 13.", "answerGuide": "Set 2f(x) + 1 = 13, giving f(x) = 6. Solve the quadratic and retain x = −3 under the given domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q08a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle through the three stated points.", "answerGuide": "Use the right angle at B: the centre is the midpoint of AC, (4, 5), and the radius is 5. Thus (x − 4)² + (y − 5)² = 25.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q08b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the tangent to the circle at B.", "answerGuide": "Use the centre and radius found in part (a), find the gradient of the radius to B, take the perpendicular gradient, and form the line through B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as the most frequently omitted part of a question; many candidates who found the centre did not continue to the tangent gradient and equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q09ai", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)(i)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Show that the second term of the stated geometric progression is cosθ sin²θ.", "answerGuide": "Use the given sum-to-infinity relation to find the common ratio, then multiply it by the first term cosθ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q09aii", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)(ii)", "marks": 2, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the sum of the first 12 terms of the geometric progression at θ = π/3.", "answerGuide": "Substitute first term 1/2 and ratio 3/4 into S₁₂ = a(1 − r¹²)/(1 − r); the sum is about 1.937.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q09b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the 85th term of the arithmetic progression at θ = π/3.", "answerGuide": "The first term is 1/2 and the second is 3/8, so d = −1/8 and u₈₅ = 1/2 + 84(−1/8) = −10.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q10a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Find k when the sector angle is π/6 and the chord divides the sector into equal areas.", "answerGuide": "Equate the triangle area ½k²a²sinθ to half the sector area ¼a²θ. At θ = π/6, k = √(π/6) ≈ 0.7236.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q10b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "trigonometry", "proof", "algebra" ], "primaryTopicId": "trigonometry", "summary": "For the variable-angle case, find all values of k that satisfy the stated area condition.", "answerGuide": "Use the sector and triangle area relation established in part (a), express it with the variable angle, and solve the resulting inequality while respecting the angle domain and excluding inadmissible negative values of k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was the most frequently omitted part; the general-case relationship was rarely identified and few complete solutions were seen, with negative k values often included.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q11a", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Find where the given curve crosses the x-axis.", "answerGuide": "Set 9(x⁻¹ᐟ² − 4x⁻³ᐟ²) = 0 and solve x − 4 = 0, giving x = 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q11b", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent to the curve at its x-axis crossing.", "answerGuide": "Differentiate the curve, evaluate its gradient at x = 4 to get 9/8, then use the point (4, 0): y = (9/8)(x − 4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q11c", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "differentiation", "indices" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the curve’s maximum point.", "answerGuide": "Set the derivative to zero; the stationary point occurs at x = 12 and is the maximum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-12-q11d", "paperId": "cie-2021-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(d)", "marks": 4, "topicIds": [ "integration", "indices" ], "primaryTopicId": "integration", "summary": "Find the area bounded by the curve, x-axis and x = 9.", "answerGuide": "Integrate 9x⁻¹ᐟ² − 36x⁻³ᐟ² from x = 4 to 9; the area is 6 square units.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q01a", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "graphs", "functions" ], "primaryTopicId": "graphs", "summary": "Sketch y = |3x − 5| and y = x + 2 on the same axes.", "answerGuide": "Draw the V-shaped absolute-value graph with vertex (5/3, 0), and the straight line with y-intercept 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q01b", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the intersections by solving |3x − 5| = x + 2.", "answerGuide": "Solve separately on either side of x = 5/3; the roots are x = 3/4 and x = 7/2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q02", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve sec²θ cotθ = 8 for 0 < θ < π.", "answerGuide": "Rewrite as 2/sin(2θ) = 8. The solutions in the interval are θ ≈ 0.126 and 1.445 radians.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q03", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the parametric curve at t = 0.", "answerGuide": "Evaluate dy/dt = 6cos(2t) and dx/dt = e²ᵗ(2cos4t − 4sin4t) at 0; dy/dx = 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q04a", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "differentiation", "graphs" ], "primaryTopicId": "differentiation", "summary": "Differentiate the curve and find the x-coordinate of its maximum.", "answerGuide": "For y = 5x/(4x³ + 1), differentiate by the quotient rule and set 1 − 8x³ = 0; x = 1/2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q04b", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Approximate the shaded area from x = 1 to 3 with two trapezia.", "answerGuide": "Use width 1 and ordinates y(1) = 1, y(2) = 10/33 and y(3) = 15/109; the area is about 0.87.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q04c", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Decide whether the trapezium-rule result is an over- or under-estimate.", "answerGuide": "The curve is convex over the interval, so the straight chords lie above it and the trapezium rule overestimates the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q05a", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Rearrange the logarithmic equation into the given iterative form.", "answerGuide": "Combine logarithms, use x > 0, and simplify to x²(x + 2) = 9; take the positive square root to obtain x = √(9/(x + 2)).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q05b", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Show by calculation that the root of the iteration lies between 1.5 and 2.0.", "answerGuide": "Evaluate x − √(9/(x + 2)) at 1.5 and 2.0; the results have opposite signs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q05c", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Iterate to find the root to 3 significant figures.", "answerGuide": "Starting with a value in the stated interval, repeatedly apply xₙ₊₁ = √(9/(xₙ + 2)); the root is 1.58 to 3 s.f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q06a", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the factor and remainder conditions to find a and b in p(x) = x³ + ax + b.", "answerGuide": "Set p(−2) = 0 and p(3) = 5. Solving gives a = −6 and b = −4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q06b", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the factorisation to solve p(e²ʸ) = 0 exactly.", "answerGuide": "Factor p(x) = (x + 2)(x² − 2x − 2). Since e²ʸ > 0, use x = 1 + √3 to obtain y = ½ln(1 + √3).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q07a", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write 5√3 cosx + 5 sinx as R cos(x − α).", "answerGuide": "Compare coefficients to get R = 10 and α = π/6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q07b", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the least value of 4 + 5√3 cosx + 5 sinx and its x-value in the interval.", "answerGuide": "Use 10cos(x − π/6); the minimum is −6 when x = −5π/6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-22-q07c", "paperId": "cie-2021-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of the squared trigonometric expression over its stated limits.", "answerGuide": "Expand the square using double-angle identities, integrate each sine and cosine term, then substitute the given limits exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_22.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q01", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation and give the answer to 3 significant figures.", "answerGuide": "Combine logarithms and respect x³ − 3 > 0. The equation gives x³ = 9/2, so x = (9/2)¹ᐟ³ ≈ 1.65.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q02", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find a and b from the factor and remainder conditions on the cubic polynomial.", "answerGuide": "Use p(−2) = 0 and p(−1) = 2; solving the simultaneous equations gives a = 3 and b = −4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q03", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "trigonometry", "quadratics" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation for 0° < x < 180° using a quadratic in tanx.", "answerGuide": "Set t = tanx, expand tan(x + 45°), and solve t² + t − 1 = 0. Convert both tangent values to angles in the interval, about 38.2° and 121.7°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q04a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Solve the differential equation using y(π) = 4.", "answerGuide": "Separate variables and integrate to obtain y = C(1 − cosx); the condition gives y = 2(1 − cosx).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q04b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "Sketch the solution graph for the stated interval of x.", "answerGuide": "Use the solution and boundary values established in part (a), then draw the curve through the required intercepts with the symmetry stated by the differential equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says most candidates found this graph part challenging; few sketches passed through both required endpoints and were symmetric about x = π.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q05a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express √7 sinx + 2 cosx as R sin(x + α), giving R exactly and α to 2 d.p.", "answerGuide": "Compare coefficients: R = √11 and tanα = 2/√7, so α ≈ 37.05°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q05b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the result from part (a) to solve the linked equation for the stated interval.", "answerGuide": "Write the left side as √11 sin(2θ + α), with α = tan⁻¹(2/√7), and solve both sine branches; θ ≈ 62.7° and 170°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q06a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Resolve the rational function into partial fractions.", "answerGuide": "Set 5a/((2x − a)(3a − x)) = 2/(2x − a) + 1/(3a − x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q06b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the partial fractions to evaluate the definite integral.", "answerGuide": "Integrate to ln|2x − a| − ln|3a − x| and substitute x = a and x = 2a to get ln 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q07a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the two given lines are skew.", "answerGuide": "Show the lines do not intersect, then compare their directional vectors and establish that neither is a scalar multiple of the other.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few candidates successfully showed skewness: many used position or full line vectors instead of directional vectors, or tested equality rather than whether one was a multiple of the other.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q07b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "vectors", "coordinate-geometry" ], "primaryTopicId": "vectors", "summary": "Find the acute angle between the directions of the two lines.", "answerGuide": "Use direction vectors (2, −1, 3) and (1, −1, 4); cosθ = 15/(√14√18), so θ ≈ 19.2°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q08a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express u/v in Cartesian form for u = −4 + 2i and v = 3 + i.", "answerGuide": "Multiply by the conjugate of the denominator to obtain u/v = −1 + i.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q08b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Express the given complex quotient in exact exponential form reiθ.", "answerGuide": "Find the modulus and principal argument of the quotient from its Argand-plane location, give the argument in radians, and retain an exact expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says correct argument answers were rarely seen; candidates often used degrees or a decimal although an exact radian answer was required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q08c", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "State the geometric relationship between OA and BC.", "answerGuide": "Since C represents 2u + v and B represents v, vector BC = 2u = 2 vector OA; they are parallel in the same direction and BC is twice as long.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q08d", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(d)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove that angle AOB is 3π/4.", "answerGuide": "Use the arguments of u and v, or their direction vectors, and the tangent subtraction identity to show the angle is 3π/4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q09a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Verify numerically that the unique root of x = f(x) lies between 1 and 1.5.", "answerGuide": "Evaluate x − f(x) at 1 and 1.5 and show the signs differ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q09b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Iterate to find the root to 2 decimal places.", "answerGuide": "Apply xₙ₊₁ = (e²ˣⁿ + 1)/(e²ˣⁿ − 1) to convergence; a ≈ 1.20.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q09c", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate the exponential quotient and find the exact x for which its derivative equals −8.", "answerGuide": "Apply the quotient rule, set the derivative equal to −8, substitute u = eˣ to obtain a quadratic, retain the valid positive value of u, and give x in exact form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls forming the correct quadratic the main challenge; candidates often kept the negative solution or gave a decimal when an exact value was required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q10a", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area under y = sin2x cos²x and above the x-axis.", "answerGuide": "Set u = sinx, so the integral becomes ∫₀¹2u(1 − u²)du = 1/2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-32-q10b", "paperId": "cie-2021-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinate of M on the trigonometric curve.", "answerGuide": "Differentiate the curve expression correctly with the chain rule, form the trigonometric condition for M, and solve it over the interval given in the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_32.pdf", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report highlights this part as requiring the chain rule and describes frequent errors differentiating the squared sine and cosine terms, including omitted factors of 2.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q01", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find P’s speed after the collision when Q comes to rest.", "answerGuide": "Conserve momentum along the line: 0.2(0.5) − 0.3(1) = 0.2v, giving speed 0.75 m s⁻¹ in P’s original direction reversed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q02a", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the car’s speed on the first incline.", "answerGuide": "At constant speed, engine force balances resistance and the component of weight: F = 600 + 1400, so v = 22500/2000 = 11.25 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q02b", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the initial acceleration on the 2° incline with unchanged power and resistance.", "answerGuide": "Use driving force P/v = 2000 N, subtract 600 N resistance and 14000sin2° N, then divide by 1400 kg; a ≈ 0.651 m s⁻².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q03", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the tensions in the two strings supporting the particle in equilibrium.", "answerGuide": "Resolve horizontally and vertically with weight 2 N; the QR tension is 1 N and the PQ tension is √3 N.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q04a", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the elevator’s acceleration during its first 1.5 seconds.", "answerGuide": "Use the gradient of the velocity-time graph: a = 2/1.5 = 4/3 m s⁻² upwards.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q04b", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the downward speed V from the equal start and finish levels.", "answerGuide": "Equate upward and downward areas under the velocity-time graph: 11.5 = 7.75V, so V ≈ 1.48 m s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q04c", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find cable tension while the elevator decelerates on the upward journey.", "answerGuide": "The acceleration is 2 m s⁻² downward; with g = 10, T = 1500(10 − 2) = 12 000 N.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q05a", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the block’s acceleration from its displacement in the first 5 seconds.", "answerGuide": "Use s = ½at² from rest: 2 = ½a(25), giving a = 0.16 m s⁻².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q05b", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Given the coefficient of friction, find the magnitude X of the applied force.", "answerGuide": "Resolve the force perpendicular to the floor to find the reaction, use F = 0.4R, then apply Newton's second law along the floor using the acceleration from part (a) and the correct component of X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report names this among the paper's most challenging parts and notes that candidates often omitted the normal component of X or used the wrong component or left out acceleration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q05c", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction coefficient when the block with X = 25 N is in limiting equilibrium.", "answerGuide": "Resolve vertically to get R = 50 − 25sin30° = 37.5 N, then μR = 25cos30°; μ ≈ 0.577.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q06a", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find displacement from O at t = 1 from the given velocity function.", "answerGuide": "Integrate v(t) from 0 to 1; the displacement is 32/15 m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q06b", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Show that the stated value is the particle's minimum velocity.", "answerGuide": "Differentiate v(t) to find the stationary times, evaluate the velocity at the relevant time, and verify it is a minimum, for example by checking the change in acceleration or the second derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists this among the paper's most challenging parts; many candidates found stationary times but finished without showing that the velocity was a minimum.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q07a", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension for m = 0.3 kg when both planes are smooth.", "answerGuide": "Apply Newton’s second law along each plane with P moving down; solve the two equations to obtain T ≈ 2.56 N.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-42-q07b", "paperId": "cie-2021-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy method to find the mass of Q after the two particles move 1 m on the rough planes.", "answerGuide": "Account for the kinetic energy gained, the vertical potential-energy changes, the work by the driving force, and work against friction. If treating particles separately, first find the new string tension for this situation rather than reusing part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_42.pdf", "qpPages": [ 12, 13 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this particularly challenging and says most candidates did not complete it successfully; common errors omitted energy terms, used the 1 m travel as a vertical height, or reused the tension from part (a).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q01a", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the first score of 3 occurs on spin 8.", "answerGuide": "There are seven non-3 outcomes followed by a 3: (4/5)⁷(1/5).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q01b", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that fewer than six spins are needed for the first 3.", "answerGuide": "Use the geometric model: P(T < 6) = 1 − (4/5)⁵ = 0.67232.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q02a", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability Georgie wears a hat on a randomly chosen day.", "answerGuide": "Use total probability across the scarf colours: 0.2(1) + 0.45(0.4) + 0.35(0.3) = 0.485.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q02b", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability she wore yellow given that she did not wear a hat.", "answerGuide": "Use conditional probability: (0.35×0.7)/(1−0.485) ≈ 0.476.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q03a", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability a shopper spends between 85 and 100 minutes.", "answerGuide": "Standardise both endpoints using N(96,18²) and subtract the lower cumulative probability from the upper; the result is about 0.318.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q03b", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find t when 88% of shoppers spend more than t minutes.", "answerGuide": "Use the 12th percentile of N(96,18²): t = 96 + 18Φ⁻¹(0.12) ≈ 74.9 minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q04a", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Write the probability distribution table for X in terms of k.", "answerGuide": "For x = 1,2,3,4, the probabilities are 4k, 6k, 6k and 4k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q04b", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the distribution to show Var(X) = 1.05.", "answerGuide": "Normalise to find k = 1/20; calculate E(X) = 2.5 and E(X²) = 7.3, so Var(X) = 7.3 − 2.5² = 1.05.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q05a", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-analysis", "data-representation" ], "primaryTopicId": "data-analysis", "summary": "Draw a cumulative-frequency graph for the grouped journey distances.", "answerGuide": "Use cumulative frequencies 12, 28, 60, 126, 146, 150 at continuous upper boundaries 4.5, 10.5, 20.5, 30.5, 40.5 and 60.5 km; begin at (0,0).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q05b", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Use the cumulative-frequency graph to estimate the distance d exceeded by 30% of journeys.", "answerGuide": "Locate 70% of the total journey count on the cumulative-frequency axis, read across to the curve, then down to estimate d and show the graph construction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates found this part challenging; the most common error was using 30% of the journeys rather than the complementary 70% on the graph.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q05c", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean journey distance from the grouped data.", "answerGuide": "Use class midpoints 2, 7.5, 15.5, 25.5, 35.5, 50.5 with the frequencies; the estimate is 21.6 km.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q06a", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "sequences", "probability" ], "primaryTopicId": "sequences", "summary": "Count the distinct arrangements of CATERPILLAR.", "answerGuide": "Divide 11! by 2! for each repeated letter A, L and R: 11!/(2!2!2!) = 4 989 600.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q06b", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "sequences", "probability" ], "primaryTopicId": "sequences", "summary": "Count arrangements with R at both ends and the two As separated.", "answerGuide": "Fix the Rs at the ends, then subtract arrangements with adjacent As from all interior arrangements: 9!/(2!2!) − 8!/2! = 70 560.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q06c", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the six-letter selections from CATERPILLAR that contain both Rs and at least one A and one L.", "answerGuide": "Fix both Rs, enumerate the valid remaining four-letter cases with at least one A and one L, and count each multiset once while accounting for repeated letters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this part challenging; extra invalid scenarios were regularly included, and the RRAALL case was not uncommon to omit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q07ai", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 1, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Find the probability a randomly selected student prefers swimming.", "answerGuide": "There are 104 + 31 = 135 such students out of 400; the probability is 135/400 = 0.3375.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q07aii", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Determine whether being male and preferring swimming are independent events.", "answerGuide": "State a valid independence condition, calculate the relevant joint and marginal probabilities from the table, compare them numerically, and state a conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "In the question the report says many candidates lacked sufficient rigour to justify independence; successful solutions stated the condition, identified the probabilities and made a numerical comparison.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q07bi", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability at least 3 of 10 students do not like any sports.", "answerGuide": "For X ~ B(10,0.3), calculate 1 − P(X=0) − P(X=1) − P(X=2), about 0.617.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-52-q07bii", "paperId": "cie-2021-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use an approximation to find the probability that fewer than 32 of 90 students do not like any sports.", "answerGuide": "Model the count with the stated binomial parameters, use the appropriate normal approximation and continuity correction at 31.5, then standardise and evaluate the probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says Question 7's context challenged many candidates and notes that weaker solutions to the approximation omitted the continuity correction or misread the boundary as 32.5.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q01a", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate a 98% confidence interval for the population mean build time.", "answerGuide": "Use the sample mean 4820/60 and the unbiased sample variance from Σt². With z = 2.326, the interval is 77.6 to 83.1 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q01b", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Explain why the Central Limit Theorem was needed for the confidence interval in part (a).", "answerGuide": "Refer to the population distribution explicitly: it is not specified as normal, but the large sample size makes the sample mean approximately normal by the Central Limit Theorem.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says part (b) proved challenging and that answers were often not fully accurate; it stresses referring unambiguously to the population distribution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q02a", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find k from the area under the triangular probability density.", "answerGuide": "The triangle has base 2k and height 1/(2k) as labelled; set its area equal to 1 to obtain k = 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q02b", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the density for 0 ≤ x ≤ k and then its expectation.", "answerGuide": "The straight-line density is f(x) = 1 − x/2 for 0 ≤ x ≤ 2. Integrate xf(x) over this interval to get E(X) = 2/3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q02c", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find p such that P(p < X < 1) = 0.25.", "answerGuide": "Integrate 1 − x/2 from p to 1 and solve the quadratic, retaining the valid root p = 2 − √2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q03a", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Choose whether the architect’s question needs a one-tail or two-tail test.", "answerGuide": "Use a one-tailed test because the stated alternative asks whether the city’s buildings are higher on average.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q03b", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the one-tailed test about whether the city's buildings are higher on average at the 1% level.", "answerGuide": "State hypotheses in context, compare the given z statistic with the 1% one-tailed critical value, and write a contextual conclusion with appropriate uncertainty.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was less well attempted: few candidates stated correct hypotheses and many made no attempt, while conclusions often lacked context or uncertainty.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q04a", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the distribution of the number of faulty microchips in the sample, including its parameters.", "answerGuide": "Model the 1000 independent chips as Bernoulli trials with fault probability 1/400, so X has a binomial distribution with n = 1000 and p = 1/400.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says candidates found choosing the binomial model and a valid approximation challenging, and identifies parts (a) and (b) as particularly challenging.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q04b", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "State a valid approximating distribution for the number of faulty microchips and its parameter values.", "answerGuide": "Recognise X as binomial with large n and small p, then use the Poisson approximation with mean np and state the parameter clearly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found choosing a binomial model and a valid approximation challenging, and identifies parts (a) and (b) as particularly challenging.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q04ci", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the Poisson approximation to find P(X = 4) for the faulty-chip count.", "answerGuide": "With λ = 2.5, calculate e⁻²·⁵(2.5⁴/4!) ≈ 0.134.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q04cii", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the approximation to find P(2 ≤ X ≤ 4).", "answerGuide": "For X ~ Po(2.5), add P(2), P(3) and P(4), giving about 0.604.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q04d", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance at least one of 700 chips is faulty.", "answerGuide": "The Poisson mean is 700/400 = 1.75; use 1 − P(0) = 1 − e⁻¹·⁷⁵ ≈ 0.826.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q05a", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability the combined volume of 3 large and 4 small bottles is below 25.5 L.", "answerGuide": "The sum is normal with mean 25.34 and variance 0.045; standardise 25.5 to obtain about 0.775.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q05b", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability a large bottle contains at least twice the volume of a small bottle.", "answerGuide": "For L − 2S, the mean is 0.08 and variance 0.0246. Standardise at zero; the probability is about 0.695.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q06a", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the proportion of adults owning a Chantor car increased after advertising.", "answerGuide": "Under H₀ use X ~ B(25,0.08). P(X ≥ 4) ≈ 0.135 > 0.05, so there is insufficient evidence of an increase.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q06b", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Identify the possible testing error given the result in part (a).", "answerGuide": "Because H₀ was not rejected, the possible error is Type II: failing to reject a false null hypothesis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2021-m-62-q06c", "paperId": "cie-2021-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the stated binomial test.", "answerGuide": "Use the null distribution B(25, 0.08), identify the rejection region from the 5% test, and calculate its tail probability, showing that the relevant binomial probability is below 0.05.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates gained full marks; the report says they often found the relevant P(X ≥ 5) but did not show its comparison with 0.05 or state the full binomial expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m21_er.pdf", "tagConfidence": "reviewed" }, { "id": "cie-2022-m-12-q01", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-12-q02", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the range of c for which the given curve and line meet at two distinct points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Solving the discriminant inequality in c was challenging for some candidates; sketches helped those candidates choose the correct regions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Equate the curve and line to obtain a quadratic in x. Require its discriminant to be strictly positive, solve the resulting inequality in c, and state the interval endpoints correctly." }, { "id": "cie-2022-m-12-q03a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(a): Identify the constant term in the stated binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-m-12-q03b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(b): Identify the constant term in the stated binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-m-12-q04", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use equalities between selected terms of a geometric and an arithmetic progression to find the sum of the first 20 arithmetic terms in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes considerable sign and algebra errors in eliminating the GP/AP variables; these often prevented candidates from using the arithmetic-progression sum formula successfully.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Write the relevant GP and AP terms in terms of a, r and d. Use the third-GP/second-AP and fifth-GP/sixth-AP equalities, apply the positive and non-identical-term conditions to select the valid relationship, then substitute the resulting first term and difference into the AP sum formula." }, { "id": "cie-2022-m-12-q05a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-12-q05b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(b): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-m-12-q06a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-m-12-q06b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-m-12-q07a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-m-12-q07b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-m-12-q08a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-m-12-q08b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(b): Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2022-m-12-q09a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-12-q09b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-12-q10a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(a): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2022-m-12-q10b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=13", "qpPages": [ 16, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2022-m-12-q11a", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find, in terms of k, the x-values where the given curve has stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=14", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Algebra errors left few fully correct solutions; common slips included expanding the squared expression too early and omitting the negative square-root branch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set dy/dx to zero, and solve for x while retaining both valid square-root branches and the restrictions on the expression inside the negative power." }, { "id": "cie-2022-m-12-q11b", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Find the stationary-point location for the specified function and determine whether its stationary value is a maximum or minimum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=14", "qpPages": [ 18, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Although the second derivative was usually used, only the strongest responses gave a correct process and clear explanation of the stationary value’s nature.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate and solve f′(x)=0 for the allowed domain, then use the second derivative or a sign change in f′ to justify the nature of the stationary value." }, { "id": "cie-2022-m-12-q11c", "paperId": "cie-2022-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(c): Use the derivative sign to find the least integer boundary for an increasing function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_12.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor the derivative, determine where it is positive, and choose the smallest integer n that makes the whole domain x > n lie in that interval." }, { "id": "cie-2022-m-22-q01", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-22-q02", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-m-22-q03a", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-m-22-q03b", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-m-22-q04a", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-m-22-q04b", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-m-22-q04c", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-m-22-q05a", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-m-22-q05b", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-m-22-q06a", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-m-22-q06b", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-22-q06c", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(c): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-m-22-q07a", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-m-22-q07b", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-m-22-q07c", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "7(c): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-m-22-q07d", "paperId": "cie-2022-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_22.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-m-32-q01", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2022-m-32-q02", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2: Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-m-32-q03", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "algebra", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Use two points on the graph of ln y against ln x to determine n and C in the relation xⁿy² = C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates generally found this question challenging. Common errors included substituting the plotted logarithmic coordinates into x and y directly, or taking logarithms of the already-logarithmic coordinates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain 2 ln y = −n ln x + ln C. Use the plotted pairs as (ln x, ln y), find the line gradient m and intercept b, then use n = −2m and C = e^(2b)." }, { "id": "cie-2022-m-32-q04", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Verify the stated derivative for a curve given parametrically.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates generally found this question challenging; a common error was incorrect chain-rule work, including confusing θ with x or changing dy/dx without showing the parameter derivatives.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y separately with respect to θ, then form dy/dx=(dy/dθ)/(dx/dθ). Apply the chain rule to each trigonometric term and simplify to the required expression." }, { "id": "cie-2022-m-32-q05", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-m-32-q06", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Solve the complex equation subject to Re(w) ≤ 0 and give all roots in Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Candidates often treated the imaginary part of 1 as nonzero or cancelled x and lost a root; few included the second valid root after checking Re(w) ≤ 0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Set w=x+iy and w*=x−iy, expand the equation, and equate real and imaginary parts. Solve the resulting simultaneous equations and check both roots against the real-part restriction." }, { "id": "cie-2022-m-32-q07a", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "functions", "proof" ], "primaryTopicId": "functions", "summary": "Sketch the two functions to show that the equation has exactly one root in the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "Many candidates plotted only a few isolated points and joined them, rather than sketching the required smooth curves and identifying their intersection.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Draw smooth curves with the correct intercepts, end behaviour and gradient trends over almost the full interval, then show or explain the single intersection." }, { "id": "cie-2022-m-32-q07b", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "numerical-methods", "proof" ], "primaryTopicId": "numerical-methods", "summary": "Verify numerically that the unique root lies between 1 and 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Candidates often treated 1 and 2 as degree measures instead of radians; using the radian values would have enabled this verification.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the two sides of the equation at x=1 and x=2 in radians, or evaluate their difference, and show that the signs/values place the root within the interval." }, { "id": "cie-2022-m-32-q07c", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-m-32-q08a", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-m-32-q08b", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the quotient and remainder from part (a) to evaluate the stated definite integral exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not connect parts (a) and (b), often omitted the denominator from the remainder term, and then made errors in the logarithm/arctangent integral or gave 45° instead of the exact radian value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the rational integrand as the quotient plus the remainder over 4x²+1. Integrate the polynomial and arctangent terms separately, apply the limits, and express arctan(1) as π/4." }, { "id": "cie-2022-m-32-q09", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9", "marks": 9, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-m-32-q10a", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-m-32-q10b", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-m-32-q10c", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "10(c): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-m-32-q11a", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the maximum point on y = sin x cos 2x over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Most candidates knew the approach but had difficulty applying the chain rule to cos 2x; missing the factor 2 often prevented a solvable trigonometric equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the product using the product and chain rules, set the derivative to zero, solve within the given interval, and identify the maximum point rather than an endpoint." }, { "id": "cie-2022-m-32-q11b", "paperId": "cie-2022-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use u = cos x to find the exact area enclosed by the curve and the x-axis in the first quadrant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_32.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates struggled to form the correct integral in u; common errors included replacing cos 2x inaccurately or mixing the x-limits with the u-limits.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Set u=cos x and du=−sin x dx, rewrite cos 2x using 2u²−1, transform the intersection limits into u-values, and integrate with the correct order of limits." }, { "id": "cie-2022-m-42-q01a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(a): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-m-42-q01b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-m-42-q02a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-m-42-q02b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-42-q03a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(a): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the work-energy relation, accounting for the work of each force and any change in kinetic or potential energy." }, { "id": "cie-2022-m-42-q03b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tow-bar tension while the car and trailer travel down the hill at constant speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found this part challenging, often because they omitted the trailer’s weight component along the slope.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Consider the trailer alone and resolve forces parallel to the hill. With zero acceleration, balance the tow-bar tension and the downhill weight component against the 200 N resistance." }, { "id": "cie-2022-m-42-q04a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-m-42-q04b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-m-42-q05a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(a): Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2022-m-42-q05b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "With F fixed at 3 N, find G so the resultant is perpendicular to the 10 N force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates misunderstood the perpendicular-resultant condition and set the horizontal components to zero instead of the components parallel to the 10 N force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Resolve every force parallel to the 10 N force, which is the vertical direction in the diagram. Set that resultant component to zero and solve for G." }, { "id": "cie-2022-m-42-q06a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the time k at which the cyclist returns to rest from the given time-dependent acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the most common errors were in solving the resulting equation, with some candidates struggling with fractional powers.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Integrate a(t) to obtain velocity, use the initial rest condition to determine the integration constant, set v(k)=0, and solve carefully for the positive time." }, { "id": "cie-2022-m-42-q06b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the cyclist’s maximum speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Most candidates performed well, but the majority of the errors came from solving the acceleration equation with its fractional powers.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Set the acceleration to zero to find the time of maximum velocity, then substitute that time into the velocity expression from part (a)." }, { "id": "cie-2022-m-42-q06c", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(c): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-m-42-q07a", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(a): Use the stated energy or momentum relationship to find the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-m-42-q07b", "paperId": "cie-2022-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find when the two beads meet again after their collision on the inclined wire.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_42.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that many candidates incorrectly reused the earlier 0.45 m displacement in this part and commonly made errors finding B’s acceleration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Use conservation of momentum to find the post-collision speed of B, determine B’s acceleration from the force and friction data, then write both displacements from the collision point and equate them." }, { "id": "cie-2022-m-52-q01a", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-m-52-q01b", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-m-52-q02a", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-52-q02b", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly two of three independent seven-day periods each contain at least one day with more than 10 cm of rain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this two-stage question extremely challenging; incomplete attempts often counted no-rain days or applied the wrong success condition across the three periods.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "First use the daily rain probability to find the probability a seven-day period has at least one qualifying day. Then apply a binomial model with n=3 and that period-level probability, taking exactly two successes." }, { "id": "cie-2022-m-52-q03a", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "3(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2022-m-52-q03b", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2022-m-52-q03c", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-m-52-q04a", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-m-52-q04b", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-m-52-q04c", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-52-q05a", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-m-52-q05b", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2022-m-52-q05c", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the arrangements where the three boys stand together and an adult is at each end of the line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes this part as very challenging; weak solutions chose the two end adults as a combination or assumed only one end-adult selection, while others counted only the boys’ internal arrangements.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Choose and order two of the five adults for the ends. Inside, arrange the boys as one block with the four girls and three remaining adults (8! orders), then arrange the three boys within their block (3! orders)." }, { "id": "cie-2022-m-52-q06a", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-52-q06b", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the first lemon chocolate is selected only after at least six chocolates have been checked.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A significant number did not attempt this part; successful methods recognised it as the probability of six consecutive non-lemon selections.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Interpret the condition as no lemon in the first six independent selections, then use the probability of a non-lemon chocolate on each selection or the geometric model consistently." }, { "id": "cie-2022-m-52-q06c", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-52-q06d", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-52-q06e", "paperId": "cie-2022-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(e)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Given that none of the three chocolates is orange, find the probability that at least two are strawberry.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_52.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates found this part challenging and made little progress; common errors ignored the different ways to select one lemon and two strawberry chocolates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Condition on drawing all three from the ten non-orange chocolates. For the numerator count the cases with two strawberries and one lemon or three strawberries, using combinations for sampling without replacement." }, { "id": "cie-2022-m-62-q01a", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=4", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-m-62-q01b", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give a statistical reason why the sample estimates of rod length mean and variance may not be reliable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=4", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many candidates did not identify that the sample size was small and instead gave unrelated reasons such as poor measurement or insufficient population coverage.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Refer to the small sample size of 12, which may not represent the population well; avoid explanations about measurement accuracy or variability that are not stated in the question." }, { "id": "cie-2022-m-62-q02", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the two-sided 5% test of whether the spinner lands on blue with probability 1/5, using a suitable approximation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=5", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates found this question challenging; incorrect approximating parameters, missing or wrong continuity corrections, and unsupported in-context conclusions were common.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Under H₀ use the binomial model with n=300 and p=0.2, apply the normal approximation and continuity correction, compare the test statistic or tail area with the 5% two-sided critical value, and conclude in context." }, { "id": "cie-2022-m-62-q03", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Infer the confidence level from the given confidence interval for a population proportion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls this question challenging; many candidates could not progress without identifying the estimate p̂=0.4, and others used one of the interval endpoints as p̂.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Use the midpoint of the interval as the estimated proportion, recover the standard error for n=500, standardise the half-width to find the critical normal value, then convert it to a two-sided confidence level and round to the nearest integer." }, { "id": "cie-2022-m-62-q04a", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-m-62-q04b", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-m-62-q05a", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-62-q05b", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the difference in heights between two randomly chosen buildings is less than 1 m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as particularly challenging. Many found the mean difference and standardisation but included only one direction rather than both ordered height differences.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Model the difference of the two independent normal heights, then include both ordered directions: Building 1−Building 2<1 and Building 2−Building 1<1. Standardise with the difference distribution and combine the probabilities." }, { "id": "cie-2022-m-62-q06a", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Explain without calculation why the mean of the stated symmetric density is 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates gave an incorrect reason, using only the midpoint of the endpoints or interval width instead of referring to the symmetry of the density.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Point out that the quadratic density is symmetric about x=2 on its interval, so the mean lies on the axis of symmetry." }, { "id": "cie-2022-m-62-q06b", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-m-62-q06c", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(c): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-m-62-q06d", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the largest of three independent ball distances exceeds 2.5 m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates were unsure how to calculate this probability; those who found the one-roll tail often stopped at that value or multiplied it by three instead of using a complement over all three rolls.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_er.pdf", "sourceEvidencePages": [ 16, 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Find P(X≤2.5) from the density, use independence to obtain P(all three are at most 2.5), then take the complement." }, { "id": "cie-2022-m-62-q07ai", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a)(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-62-q07aii", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a)(ii): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-m-62-q07b", "paperId": "cie-2022-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_qp_62.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m22_ms_62.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-11-q01a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q01b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1(b): Expand through the fourth term and write the terms in ascending powers of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the binomial theorem for the first four terms, simplify each coefficient, and arrange the result in ascending powers of x." }, { "id": "cie-2021-w-11-q01c", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1(c): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-w-11-q02", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q03", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange using the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-w-11-q04a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-w-11-q04b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-w-11-q05a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-11-q05bi", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "5(b)(i): Use the graph to count where the curve and the corresponding line meet in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates answered the two graph-based parts correctly; many did not recognize that the straight lines had to be intersected with the curve. The report discusses the printed 2-mark parent part 5(b) jointly; this evidence covers both one-mark children 5(b)(i) and 5(b)(ii), which share a tied priority.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "For each equation, count where its straight line meets the cosine curve in the given interval, using the diagram's scale and endpoints." }, { "id": "cie-2021-w-11-q05bii", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "5(b)(ii): Use the graph to count where the curve and the corresponding line meet in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates answered the two graph-based parts correctly; many did not recognize that the straight lines had to be intersected with the curve. The report discusses the printed 2-mark parent part 5(b) jointly; this evidence covers both one-mark children 5(b)(i) and 5(b)(ii), which share a tied priority.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "For each equation, count where its straight line meets the cosine curve in the given interval, using the diagram's scale and endpoints." }, { "id": "cie-2021-w-11-q06a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of the plate formed by a straight side and two circular arcs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A number of candidates could not find the angle from the given geometry and therefore made no progress on the perimeter.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use the equal radii and the triangle geometry to find the central angle, then add both arc lengths and the straight side." }, { "id": "cie-2021-w-11-q06b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2021-w-11-q07a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-w-11-q07b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-w-11-q08a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q08b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q08c", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "8(c): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2021-w-11-q08d", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the restricted quadratic function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates gave the inverse fully correctly because the range restriction requires rejecting the positive square-root branch.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Set y equal to the completed-square form, solve for x, then choose the root consistent with the function's restricted domain." }, { "id": "cie-2021-w-11-q08e", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(e)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Find the quadratic rule after translating the graph by the stated vector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "A significant number did not know how to modify the function equation for the translation or made algebra errors.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Apply the horizontal and vertical translation to the original function, then expand into the requested quadratic form." }, { "id": "cie-2021-w-11-q09a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q09b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-11-q09c", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q09d", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(d): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-11-q10a", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the improper integral of the stated curve from 1 to infinity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says few candidates completed the integral; common problems were rewriting the curve in an integrable form and using the limits to evaluate the improper term.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as a power of 3x − 2, integrate, and evaluate the finite and infinite limits carefully." }, { "id": "cie-2021-w-11-q10b", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the volume formed by rotating the stated region about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Most candidates omitted the required square of y in the volume integral; those who squared it usually continued correctly or made only a coefficient error.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Square the curve expression in the volume-of-revolution integral, integrate between x = 1 and x = 2, and simplify the exact result." }, { "id": "cie-2021-w-11-q10c", "paperId": "cie-2021-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find where the normal at the stated point crosses the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_11.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report notes that some candidates could not rewrite the curve to differentiate and others stopped after finding the tangent gradient without finding the normal intercept.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to find its tangent gradient at x = 1, take the negative reciprocal, and use the point to find the normal's y-intercept." }, { "id": "cie-2021-w-12-q01", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "1: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange using the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-w-12-q02a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-12-q02b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Map a point on the transformed curve back to its corresponding point on the original graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says this part was more challenging; many candidates transformed the given point in the forward direction instead of finding its original point.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Reverse the vertical translation first, then undo the horizontal scaling in f(2x) to recover the original coordinates." }, { "id": "cie-2021-w-12-q03a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-12-q03b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-12-q04", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4: Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2021-w-12-q05a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find the exact angle parameter from the first, third and fifth terms of an arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes the trigonometric arithmetic-progression question as particularly challenging; many candidates did not use all three terms to form the required equations.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Use equal differences between consecutive terms, form the resulting trigonometric equation, and choose the value in the stated interval." }, { "id": "cie-2021-w-12-q05b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2021-w-12-q06", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "6: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2021-w-12-q07a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-w-12-q07b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the shaded area bounded by a circular arc and two straight lines.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "A significant number omitted this part; weaker attempts used an incorrect radius or approximated too early while subtracting the triangle from the sector.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Use the angle and radius from the diagram to find the sector area, then subtract the relevant triangle area." }, { "id": "cie-2021-w-12-q08a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "8(a): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-w-12-q08b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the parameter values when the coefficient condition has exactly one solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A significant number omitted this part or ignored that the condition had changed; successful candidates used a quadratic in a and its discriminant.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Form the coefficient equation in a, use the zero-discriminant condition for a single root, then determine k and a." }, { "id": "cie-2021-w-12-q09a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(a): Integrate the stated function to find the requested expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2021-w-12-q09b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the mound's volume when its radius is increasing at the stated rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A sizeable minority omitted this part after struggling with the chain rule and linking the two rates in part (a).", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the volume model with respect to time using the chain rule, substitute dr/dt = 0.1, and solve for the radius and volume." }, { "id": "cie-2021-w-12-q10a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-12-q10b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-12-q10c", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "State the range of the function given that its stationary point is unique.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was more challenging and many candidates did not understand how the unique-stationary-point information determined the range.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Use the stationary value and the function's domain and end behaviour to determine the full range rather than giving a single value." }, { "id": "cie-2021-w-12-q11a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2021-w-12-q11b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "11(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-w-12-q12a", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "12(a): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=19", "qpPages": [ 18, 18 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2021-w-12-q12b", "paperId": "cie-2021-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 12, "questionPart": "12(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of the equilateral triangle whose vertices lie on the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_12.pdf#page=21", "qpPages": [ 18, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part very challenging and says almost half of candidates omitted it; many found the radius but were unsure how to proceed.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 4, 5, 6 ], "tagConfidence": "reviewed", "answerGuide": "Find the circle's centre and radius, use the equilateral triangle's central angles to obtain its side, then apply the exact triangle-area formula." }, { "id": "cie-2021-w-13-q01", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1: Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-13-q02a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "2(a): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-w-13-q02b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "2(b): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2021-w-13-q03a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-13-q03b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Determine whether the polynomial function is increasing, decreasing or neither.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only stronger responses justified why the derivative was always positive; complete proofs using the quadratic discriminant were rarely seen.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate, complete the square in the derivative, and establish its sign for every real x before stating the conclusion." }, { "id": "cie-2021-w-13-q04a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(a): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2021-w-13-q04b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2021-w-13-q05a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-w-13-q05b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(b): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2021-w-13-q06a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch the inverse function on the supplied axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Although many drew a reflection, few reflected in y = x, and the line y = x was rarely included.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Reflect the given curve in the line y = x and show the reflected graph on the original axes." }, { "id": "cie-2021-w-13-q06b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find an expression for the inverse of the stated restricted rational function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the correct inverse with the negative alternative was very rarely seen; errors commonly arose when clearing the fraction and squaring.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Set y equal to the function, rearrange to a quadratic in x, and select the root consistent with the original domain." }, { "id": "cie-2021-w-13-q06c", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-13-q06d", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(d): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-13-q07a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-13-q07b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange using the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-w-13-q08a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(a): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-w-13-q08b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-13-q09a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2021-w-13-q09b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two gradients of a tangent from the given external point to the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the most omitted part; the direct tangent-angle method was rarely seen and most attempts led to harder algebra.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 7, 8, 9 ], "tagConfidence": "reviewed", "answerGuide": "Write a line through (10, 0) with gradient m, substitute it into the circle equation, and set the resulting quadratic's discriminant to zero." }, { "id": "cie-2021-w-13-q10a", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-13-q10b", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-13-q10c", "paperId": "cie-2021-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2021, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(c): Differentiate the curve and use its zero gradient at the extremum to establish the stated condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_13.pdf#page=15", "qpPages": [ 16, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-21-q01", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-21-q02a", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-21-q02b", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(b): Solve for the intersection of the stated graphs and give its coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the graph equations equal, use the correct branch of any modulus expression, solve for the intersection, and state both coordinates." }, { "id": "cie-2021-w-21-q02c", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(c): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-w-21-q03", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the constants in a logarithmic straight-line relationship from two plotted points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few correct solutions were seen; the main difficulty was identifying the straight-line gradient after taking logarithms.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms of aʸ = kx to obtain y ln(a) = ln(x) + ln(k), then use the line's gradient and intercept to find a and k." }, { "id": "cie-2021-w-21-q04a", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Show the logarithmic equation satisfied by the minimum point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few candidates completed the rearrangement correctly, particularly the step needed to isolate the exponential term.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set the derivative to zero, rearrange the exponential equation, and take logarithms." }, { "id": "cie-2021-w-21-q04b", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-21-q05a", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-21-q05b", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-w-21-q05c", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-w-21-q06a", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-21-q06b", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-21-q06c", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Factorise the difference of the two polynomials completely.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few correct solutions were seen; candidates needed to recognize the shared factor and complete the factorisation.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the constants found earlier, factor out the common linear factor, and continue until every factor is linear." }, { "id": "cie-2021-w-21-q06d", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(d): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-21-q07a", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-21-q07b", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the exact value of the trigonometric expression at the specified angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few correct solutions were seen; candidates needed the identity from part (a) and an exact result.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the triple-angle identity from part (a), then substitute the exact cosine value and simplify." }, { "id": "cie-2021-w-21-q07c", "paperId": "cie-2021-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate the stated trigonometric integral exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_21.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few correct solutions were seen and notes that some candidates made no attempt.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 10, 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (a) to rewrite the integrand, integrate term by term, and evaluate the definite limits exactly." }, { "id": "cie-2021-w-22-q01a", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-22-q01b", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-22-q02a", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-22-q02b", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-22-q02c", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-22-q03", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-22-q04", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "4: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-w-22-q05", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact normal gradient to the implicitly defined curve at the stated point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this question demanding; candidates often omitted terms in the product differentiation or gave the tangent gradient instead of the normal.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate implicitly using the product and chain rules, substitute the point, then take the negative reciprocal of the tangent gradient." }, { "id": "cie-2021-w-22-q06a", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "functions", "exponentials" ], "primaryTopicId": "functions", "summary": "Use a sketch of two graphs to show the exponential-logarithmic equation has exactly one root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct sketches were seen and many candidates did not attempt this part.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Sketch both functions on common axes, identify their single intersection and explain why this gives one solution." }, { "id": "cie-2021-w-22-q06b", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-22-q06c", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "functions", "exponentials", "proof" ], "primaryTopicId": "functions", "summary": "Show that convergence of the given iteration implies convergence to the equation's root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few correct solutions were seen; many candidates did not reduce the iterative formula to the stated fixed-point relation.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 13, 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Set consecutive iterates equal at the limit and rearrange the resulting fixed-point equation to the original root equation." }, { "id": "cie-2021-w-22-q06d", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-w-22-q07a", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-22-q07b", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2021-w-22-q07c", "paperId": "cie-2021-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_22.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_22.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2021-w-23-q01", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-23-q02a", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-23-q02b", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(b): Solve for the intersection of the stated graphs and give its coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the graph equations equal, use the correct branch of any modulus expression, solve for the intersection, and state both coordinates." }, { "id": "cie-2021-w-23-q02c", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(c): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-w-23-q03", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3: Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-23-q04a", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(a): Differentiate the curve and use its zero gradient at the extremum to establish the stated condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set the derivative to zero at the minimum point, then rearrange and take logarithms to obtain the stated condition." }, { "id": "cie-2021-w-23-q04b", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-23-q05a", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-23-q05b", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-w-23-q05c", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2021-w-23-q06a", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-23-q06b", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the factor 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3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(c): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_23.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-23-q06d", "paperId": "cie-2021-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2021, "session": "October/November", 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it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-31-q01", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-31-q02a", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2021-w-31-q02b", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the greatest and least possible values of the squared trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=7", "qpPages": [ 3, 3 ], 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"9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Evaluate the improper integral using u = √x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few candidates obtained the stated exact form after substituting and integrating.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 19, 20, 21 ], "tagConfidence": "reviewed", "answerGuide": "Substitute u = √x, change the limits, rewrite dx in terms of u, and evaluate the resulting exponential integral to infinity." }, { "id": "cie-2021-w-31-q05a", 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"sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-31-q07a", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "7(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, 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"paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "8(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-w-31-q08b", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "8(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-w-31-q08c", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration to determine the parameter to the requested accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Many candidates did not respond, and the report notes unfamiliarity with iteration and errors from using an inappropriate starting value.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 19, 20, 21 ], "tagConfidence": "reviewed", "answerGuide": "Choose a starting value inside the 9-to-11 interval from part (b), apply the fixed-point formula, and show each iterate to 4 decimal places." }, { "id": "cie-2021-w-31-q09a", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(a): Use gradients or direction vectors to establish the stated line relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2021-w-31-q09b", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-w-31-q09c", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact perpendicular distance from the origin to the stated vector line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few candidates used the required general-point and scalar-product strategy, so this part was rarely completed.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 19, 20, 21 ], "tagConfidence": "reviewed", "answerGuide": "Take a general point on the line, use the scalar product to locate the perpendicular foot, and find the length of its position vector." }, { "id": "cie-2021-w-31-q10a", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2021-w-31-q10b", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2021-w-31-q10c", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Find the real factors of the cubic using its complex roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates knew how to use the conjugate pair to form a quadratic factor with real coefficients.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 19, 20, 21 ], "tagConfidence": "reviewed", "answerGuide": "Use the conjugate root with the known non-real root to form a real quadratic factor, then divide the cubic and factor the remaining linear term." }, { "id": "cie-2021-w-31-q10di", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(d)(i)", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "10(d)(i): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-31-q10dii", "paperId": "cie-2021-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(d)(ii)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(d)(ii): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_31.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-w-32-q01", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-w-32-q02", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-w-32-q03a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove the conjugate of a sum equals the sum of the conjugates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this proof particularly challenging; many candidates omitted the conjugate or assumed the result instead of proving both sides.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Write each complex number in real and imaginary parts, calculate both conjugates explicitly, and show their real and imaginary parts agree." }, { "id": "cie-2021-w-32-q03b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-32-q04", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-32-q05a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-32-q05b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the greatest argument in the stated Argand region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates gave no response; only a few fully correct solutions were seen, often after a clear diagram.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Use the boundary point or tangent from the origin to the shaded locus, then calculate the relevant angle in degrees." }, { "id": "cie-2021-w-32-q06a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-32-q06b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-32-q07", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-32-q08a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the identity relating fourth powers of sine and cosine to sin²2θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this proof was particularly challenging; many candidates set the squared sum equal to zero instead of one.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Expand (cos²θ + sin²θ)², use cos²θ + sin²θ = 1, and rearrange the cross term with the double-angle identity." }, { "id": "cie-2021-w-32-q08b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation using the result from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Fully correct solutions were rare; candidates often lost the factor of 2 in the angle or reported fewer than all four solutions.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the identity, solve for sin²2θ or cos2θ, then find all four angles in the stated interval without losing branches." }, { "id": "cie-2021-w-32-q09a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "9(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-32-q09b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2021-w-32-q10a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-w-32-q10b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-w-32-q10c", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-w-32-q11a", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "11(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-32-q11b", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Derive the cubic equation for the second point where the curve has gradient 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates found this part particularly challenging; most did not use the derivative expression from part (a) to form an equation in t.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 22, 23, 24, 25 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the square-root tangent function, equate its derivative to 1, substitute t = tan a, and simplify to the cubic." }, { "id": "cie-2021-w-32-q11c", "paperId": "cie-2021-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "11(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_32.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-w-33-q01", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-33-q02a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-33-q02b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2021-w-33-q03", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report calls this question particularly challenging; candidates who scored zero often used logarithm rules where index laws were required.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 26, 27, 28, 29 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite 4^(x−2) as a multiple of 4^x, solve for 4^x, then take logarithms and round to three decimal places." }, { "id": "cie-2021-w-33-q04", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate trigonometric identity to express the requested value in terms of the given value, and select signs using the stated angle range." }, { "id": "cie-2021-w-33-q05", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange using the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2021-w-33-q06a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite using the trigonometric identities suited to the given expression, combine terms or denominators as needed, and simplify to the stated result." }, { "id": "cie-2021-w-33-q06b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find where the trigonometric expression reaches its least value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part particularly challenging; many candidates chose the wrong cosine argument for the minimum.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 26, 27, 28, 29 ], "tagConfidence": "reviewed", "answerGuide": "Use the amplitude-phase form from part (a) and set the shifted cosine argument to 180 degrees for its minimum." }, { "id": "cie-2021-w-33-q07a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Show the derivative formula for the curve defined implicitly by a logarithmic equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the chain rule within this implicit differentiation was challenging for most candidates.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 26, 27, 28, 29 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides implicitly, apply the chain rule to ln(x + y), collect the dy/dx terms and simplify." }, { "id": "cie-2021-w-33-q07b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(b): Use gradients or direction vectors to establish the stated line relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2021-w-33-q08a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector equation for the line through the two stated points on the pyramid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Several candidates thought the vector equation was just MN; the report explicitly says this part was challenging.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 26, 27, 28, 29 ], "tagConfidence": "reviewed", "answerGuide": "Write position vectors for M and N, form the direction vector MN, then give r as a point vector plus a scalar multiple of MN." }, { "id": "cie-2021-w-33-q08b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "8(b): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2021-w-33-q09a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2021-w-33-q09b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "9(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2021-w-33-q10a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "10(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-33-q10b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "differentiation", "integration", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the logistic differential equation and establish the equation for x at t = 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates confused the constants while separating and integrating; the report says the required form was not often reached.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 26, 27, 28, 29 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange to (20 − x)/x dx = 19 dt, integrate, apply x = 1 at t = 0, and rearrange at t = 1 to the requested exponential equation." }, { "id": "cie-2021-w-33-q10c", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "10(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2021-w-33-q10d", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(d)", "marks": 1, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(d): Integrate the stated function to find the requested expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2021-w-33-q11a", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=14", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2021-w-33-q11b", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "11(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=14", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-33-q11ci", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)(i)", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "11(c)(i): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2021-w-33-q11cii", "paperId": "cie-2021-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2021, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)(ii)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "11(c)(ii): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_33.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2021-w-41-q01a", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-41-q01b", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "1(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2021-w-41-q02a", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-41-q02b", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(b): Apply conservation of momentum to determine the requested post-collision quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, apply conservation of momentum with signed velocities before and after impact, then solve for the requested post-collision quantity." }, { "id": "cie-2021-w-41-q03", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2021-w-41-q04a", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(a): Draw a free-body diagram with the weight, normal reaction, applied force and friction in their correct directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw the particle and label its weight vertically down, the normal reaction perpendicular to the plane, the applied force along the plane, and friction opposing the impending motion." }, { "id": "cie-2021-w-41-q04b", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(b): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2021-w-41-q05a", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2021-w-41-q05b", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-41-q05c", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(c): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-w-41-q05d", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(d): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-w-41-q06a", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2021-w-41-q06b", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2021-w-41-q06c", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(c): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2021-w-41-q07", "paperId": "cie-2021-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_41.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2021-w-42-q01a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2021-w-42-q01b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2021-w-42-q02a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2021-w-42-q02b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the least braking force that makes the towing rope slack.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not use T = 0 and so made no progress; values from part (a) no longer applied.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 34, 35, 36 ], "tagConfidence": "reviewed", "answerGuide": "At the limiting case set the rope tension to zero, use the same deceleration for van and trailer, and resolve the forces on each." }, { "id": "cie-2021-w-42-q03a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the block's speed at the bottom of the smooth semicircular track.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates used constant-acceleration formulae for the curved track; the report says conservation of energy was required.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 34, 35, 36 ], "tagConfidence": "reviewed", "answerGuide": "Apply conservation of energy from A to B, converting the loss of gravitational potential energy into kinetic energy." }, { "id": "cie-2021-w-42-q03b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(b): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2021-w-42-q04a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-42-q04b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-42-q05a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-w-42-q05b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the engine's speed after 60 seconds on the horizontal track.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates did not recognize that an energy method was required and incorrectly tried to use a resistance force with Newton's second law.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 34, 35, 36 ], "tagConfidence": "reviewed", "answerGuide": "Use work done by power over time, subtract work against resistance, and equate the remaining work to the change in kinetic energy." }, { "id": "cie-2021-w-42-q06a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2021-w-42-q06b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the least horizontal force that keeps the block in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part particularly challenging and says very few candidates solved it successfully; most did not set the lower tension to zero.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 34, 35, 36 ], "tagConfidence": "reviewed", "answerGuide": "Set the lower-string tension to zero at the limiting case, resolve forces horizontally and vertically, and solve for X." }, { "id": "cie-2021-w-42-q07a", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(a): Resolve the forces on the particle and apply newton’s second law to determine its acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the weight and friction along the plane, calculate any normal reaction needed for F = μR, then apply Newton’s second law along the direction of motion." }, { "id": "cie-2021-w-42-q07b", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-42-q07c", "paperId": "cie-2021-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(c): Apply conservation of momentum to determine the requested post-collision quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_42.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, apply conservation of momentum with signed velocities before and after impact, then solve for the requested post-collision quantity." }, { "id": "cie-2021-w-43-q01a", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-43-q01b", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-43-q02a", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2(a): Draw a free-body diagram with the weight, normal reaction, applied force and friction in their correct directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw the particle and label its weight vertically down, the normal reaction perpendicular to the plane, the applied force along the plane, and friction opposing the impending motion." }, { "id": "cie-2021-w-43-q02b", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2(b): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2021-w-43-q03a", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the ball's greatest height after losing kinetic energy at impact.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates misread the stated energy loss and equated 8 J to the final potential energy; the complete correction method was rarely seen.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 37, 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Subtract the lost kinetic energy from the initial potential energy, then equate the remaining energy after the rebound to gravitational potential energy at the top." }, { "id": "cie-2021-w-43-q03b", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "3(b): Use the stated energy or momentum relationship to find the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2021-w-43-q04ai", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a)(i): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2021-w-43-q04aii", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a)(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-w-43-q04aiii", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)(iii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a)(iii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2021-w-43-q04b", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2021-w-43-q05a", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-43-q05b", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2021-w-43-q05c", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(c): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2021-w-43-q06a", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2021-w-43-q06b", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and both string tensions after release.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This question's later parts were particularly challenging; candidates often equated the two tensions or omitted the table particle and its friction.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 37, 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Write Newton's second-law equations for all three connected particles, include friction on the table mass, and solve the simultaneous equations." }, { "id": "cie-2021-w-43-q06c", "paperId": "cie-2021-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2021, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the time until the remaining connected particles come to rest after the heavier particle hits the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_43.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says parts (b) and (c) were particularly challenging; many used the wrong post-impact acceleration or stopped after finding the first phase.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 37, 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Recalculate acceleration for the two particles still moving, use their shared post-impact velocity, and apply constant-acceleration equations." }, { "id": "cie-2021-w-51-q01a", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-51-q01b", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-51-q02a", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-51-q02b", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Calculate the variance from the coded summary of the 40 observations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A significant number made no attempt; common errors omitted the square on the coded mean or substituted the original mean.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Use the coded-data variance formula with the given sums, square the coded mean, and convert back to the variance of x." }, { "id": "cie-2021-w-51-q03", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3: Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-w-51-q04a", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-51-q04b", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-51-q05a", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-51-q05b", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-51-q05c", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-51-q05d", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that Raman and Sanjay are in the same row group.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many found this part challenging and a significant proportion made no attempt; common errors omitted one group case or combined the counts incorrectly.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Count the groupings with both people together in either the group of five or the group of four, then divide by the total groupings from part (c)." }, { "id": "cie-2021-w-51-q06a", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2021-w-51-q06b", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2021-w-51-q06c", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "6(c): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2021-w-51-q06d", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(d): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2021-w-51-q07ai", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a)(i): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-w-51-q07aii", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that fewer than two of ten days exceed the stated social-media time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few fully correct solutions were seen and a significant number made no attempt; common errors reused the wrong probability or included two days.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Use a binomial distribution with n = 10 and the exceedance probability from part (i), then add the probabilities for zero and one day." }, { "id": "cie-2021-w-51-q07b", "paperId": "cie-2021-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the time threshold that Karli exceeds on 90% of days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_51.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A significant number omitted this part or used the wrong tail, often finding a value above the mean despite the wording.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 40, 41, 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Use the lower-tail 10th percentile of the normal distribution and standardise to solve for t." }, { "id": "cie-2021-w-52-q01a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-52-q01b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-w-52-q01c", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-52-q02a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-52-q02b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count six-person groups that do not contain both sisters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says the combinations question challenged many; candidates often omitted the case where neither sister was selected.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Count all six-person groups and subtract those containing Jane and Kate, or enumerate disjoint cases including neither sister." }, { "id": "cie-2021-w-52-q03a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-52-q03b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Complete the probability distribution for the number of yellow marbles selected.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=10", "qpPages": [ 4, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A high number made no attempt; common errors omitted the zero-yellow outcome or included impossible counts.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "List the possible counts 0, 1, 2 and 3, calculate each hypergeometric probability, and check that they sum to one." }, { "id": "cie-2021-w-52-q03c", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=10", "qpPages": [ 4, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-52-q04a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-52-q04b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements of TELESCOPE with exactly two letters between T and C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the arrangement condition challenged many; candidates often omitted the reversed order or gave calculations without explaining the cases.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Choose the positions for T and C at the required separation, account for both orders, then arrange the remaining letters while dividing for the repeated E." }, { "id": "cie-2021-w-52-q05a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-52-q05b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-52-q05c", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that in exactly one of four years the first wet day is 8 October.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this very challenging; many continued with a geometric model or reused the original daily probability.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Use the probability from part (b) as the success probability in a binomial distribution with n = 4 and r = 1." }, { "id": "cie-2021-w-52-q06a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-52-q06b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-w-52-q06c", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the task time is within 15 minutes of its mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found this challenging and some omitted it; common attempts calculated only one side or substituted 15 as the time itself.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Standardise the two-sided interval and subtract the lower-tail probability from the upper-tail probability, or use symmetry to find both tails." }, { "id": "cie-2021-w-52-q07a", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "7(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2021-w-52-q07b", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Read Q1 and Q3 from the cumulative-frequency graph at N/4 and 3N/4 respectively, then subtract Q1 from Q3." }, { "id": "cie-2021-w-52-q07c", "paperId": "cie-2021-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean and standard deviation from the cumulative-frequency data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_52.pdf#page=16", "qpPages": [ 10, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates found this task challenging and used cumulative totals or class boundaries instead of class frequencies and midpoints.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 44, 45, 46, 47 ], "tagConfidence": "reviewed", "answerGuide": "Convert cumulative frequencies to class frequencies, use class midpoints, and calculate the grouped-data mean and variance." }, { "id": "cie-2021-w-53-q01", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1: Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-53-q02a", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-53-q02b", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "2(b): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2021-w-53-q03a", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "3(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2021-w-53-q03b", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-53-q04a", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-53-q04b", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-53-q04c", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-w-53-q05a", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-53-q05b", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count codes that contain A, digit 5, or both.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was one of the most challenging questions; only a minority used the efficient subtraction method and many confused the overlapping cases.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 48, 49, 50, 51 ], "tagConfidence": "reviewed", "answerGuide": "Use inclusion-exclusion: count codes with A and codes with 5, then subtract codes with both; preserve the order of the code symbols." }, { "id": "cie-2021-w-53-q05c", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability of the specified DE code followed by a number from 4500 to 5000.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found this part particularly challenging; many omitted one of the possible hundreds digits or failed to convert a successful-code count into a probability.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 48, 49, 50, 51 ], "tagConfidence": "reviewed", "answerGuide": "Count valid ordered codes satisfying both conditions and divide by the total number of codes, accounting for all possible hundreds digits." }, { "id": "cie-2021-w-53-q06a", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-53-q06b", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-53-q06c", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-w-53-q06d", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the first all-three-darts success occurs on turn 9.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this part particularly challenging; candidates often reused p from the table instead of the probability of three hits.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 48, 49, 50, 51 ], "tagConfidence": "reviewed", "answerGuide": "Use the geometric distribution with success probability P(X = 3) from the table: (1 − p)^8p." }, { "id": "cie-2021-w-53-q07a", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-53-q07b", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-w-53-q07c", "paperId": "cie-2021-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2021, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_53.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-w-61-q01a", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-61-q01b", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-61-q02", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2: Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-w-61-q03", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-61-q04a", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the stated density over the interval specified by the probability event, applying the support limits." }, { "id": "cie-2021-w-61-q04b", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Integrate x times the probability density over its support to find the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E[X] = ∫ x f(x) dx over the density’s support and evaluate the limits." }, { "id": "cie-2021-w-61-q04c", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Set the cumulative probability to one half and solve for the median.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the integral of the density from the lower support bound to the median equal to 0.5, then rearrange to obtain the required equation." }, { "id": "cie-2021-w-61-q05ai", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a)(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-61-q05aii", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a)(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-w-61-q05b", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "5(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2021-w-61-q06a", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-w-61-q06b", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-61-q07a", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2021-w-61-q07b", "paperId": "cie-2021-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the second sample test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_61.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a few candidates completed this successfully; many skipped the acceptance region or incorrectly repeated the significance-test steps.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 52, 53, 54 ], "tagConfidence": "reviewed", "answerGuide": "Find the acceptance region for the null hypothesis using the stated normal model, then evaluate the chance of landing in that region under the alternative mean." }, { "id": "cie-2021-w-62-q01a", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-62-q01b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-62-q02a", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2021-w-62-q02b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-62-q03a", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-62-q03b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly two of three confidence intervals contain the true parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Few candidates understood what a confidence interval's coverage probability means; many used unrelated probabilities or tried to form more intervals.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 55, 56, 57 ], "tagConfidence": "reviewed", "answerGuide": "Treat the independent intervals as Bernoulli outcomes with their respective coverage probabilities 0.90, 0.95 and 0.99, and add the three ways exactly two can cover." }, { "id": "cie-2021-w-62-q04a", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-62-q04b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "4(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2021-w-62-q04c", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "statistical-testing", "proof" ], "primaryTopicId": "statistical-testing", "summary": "Explain whether the Central Limit Theorem was needed for the sample-mean test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates explained this fully; many did not distinguish the unknown population distribution from the sampling distribution of the mean.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 55, 56, 57 ], "tagConfidence": "reviewed", "answerGuide": "State that the population distribution is unknown and use the large sample size to justify the approximate normal distribution of the sample mean." }, { "id": "cie-2021-w-62-q05a", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-62-q05b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-w-62-q05c", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-62-q06a", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "6(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2021-w-62-q06b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the rejection region for the test of whether the machine is biased.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the question was particularly challenging; many gave unsupported answers, compared point probabilities rather than cumulative probabilities, or used the wrong distribution.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 55, 56, 57 ], "tagConfidence": "reviewed", "answerGuide": "Use the null binomial model to compare cumulative probabilities at 0 and at 0 or 1, then select the largest rejection region within the significance level." }, { "id": "cie-2021-w-62-q06c", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2021-w-62-q06d", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error when the true zero probability is 1/40.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part was generally not well attempted; many used an incorrect sample size or the wrong distribution.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 55, 56, 57 ], "tagConfidence": "reviewed", "answerGuide": "Use the acceptance region from part (b), then calculate the probability of that region under Binomial(30, 1/40)." }, { "id": "cie-2021-w-62-q06e", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(e)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(e): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-62-q07ai", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a)(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2021-w-62-q07aii", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a)(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2021-w-62-q07b", "paperId": "cie-2021-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a probability from the symmetric density function using P(Y < a) = 0.2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_62.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates generally found this very challenging; many did not know what method to use, despite the prompt to use a diagram and symmetry.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 55, 56, 57 ], "tagConfidence": "reviewed", "answerGuide": "Sketch a symmetric density on 0 to 5, mark a and its reflected point 5 − a, and use the equal areas implied by symmetry." }, { "id": "cie-2021-w-63-q01a", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-63-q01b", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2021-w-63-q02", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2: Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-w-63-q03", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-63-q04a", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the stated density over the interval specified by the probability event, applying the support limits." }, { "id": "cie-2021-w-63-q04b", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Integrate x times the probability density over its support to find the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E[X] = ∫ x f(x) dx over the density’s support and evaluate the limits." }, { "id": "cie-2021-w-63-q04c", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Set the cumulative probability to one half and solve for the median.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the integral of the density from the lower support bound to the median equal to 0.5, then rearrange to obtain the required equation." }, { "id": "cie-2021-w-63-q05ai", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a)(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2021-w-63-q05aii", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a)(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2021-w-63-q05b", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "5(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2021-w-63-q06a", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2021-w-63-q06b", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2021-w-63-q07a", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2021-w-63-q07b", "paperId": "cie-2021-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2021, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error for the second sample test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w21_ms_63.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a few candidates completed this successfully; many skipped the acceptance region or repeated the significance-test steps.", "difficultySourceUrl": "https://dynamicpapers.com/wp-content/uploads/2015/09/9709_w21_er.pdf", "sourceEvidencePages": [ 58, 59, 60 ], "tagConfidence": "reviewed", "answerGuide": "Find the acceptance region for the null hypothesis using the stated normal model, then evaluate the chance of landing in that region under the alternative mean." }, { "id": "cie-2023-m-12-q01", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Show that a line and a quadratic curve intersect for every value of a parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 1: the discriminant was sometimes written as a perfect square, but its correct interpretation was rarely seen; sign errors meant only a minority earned more than half the marks.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Equate the line and curve to form a quadratic in x, calculate and simplify its discriminant as a perfect square, then conclude that it is non-negative for every parameter value." }, { "id": "cie-2023-m-12-q02", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Apply a horizontal stretch, reflection in the y-axis and vertical stretch to a quadratic graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 1: reflection in the y-axis was the most challenging transformation; candidates also commonly squared 2x incorrectly or failed to multiply every term by 3.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Apply the transformations in order by replacing x with 2x, then with −x, and finally multiplying the whole resulting function by 3." }, { "id": "cie-2023-m-12-q03", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find where a point on a curve has its x- and y-coordinates increasing at the same rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 1: candidates did not always interpret the rate condition correctly, and many incorrect solutions assumed dy/dx = 0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Use dy/dx = (dy/dt)/(dx/dt) and set dy/dx = 1 for equal coordinate rates; differentiate the curve and solve for x." }, { "id": "cie-2023-m-12-q04a", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the tree circumference 20 years after the first measurement under an arithmetic progression model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 2: most candidates missed that the 20-year term is 5.02 m at n = 20 (or 5.00 m at n = 21), so many earned only the formula mark.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Set the first term to 5.00 and the common difference to 0.02; use the 20th term, or the 21st term if counting the first measurement as year zero." }, { "id": "cie-2023-m-12-q04b", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2023-m-12-q05", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5: Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-m-12-q06", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "6: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-m-12-q07a", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-m-12-q07b", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-m-12-q08a", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(a): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2023-m-12-q08b", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2023-m-12-q09a", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "9(a): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2023-m-12-q09b", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of a restricted quadratic function, choosing the branch allowed by its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 3: few candidates recognised that the stated domain required the negative square-root branch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange y = −3x² + 2, solve for x, choose the negative root using x ≤ −1, then exchange x and y." }, { "id": "cie-2023-m-12-q09c", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve an equation involving the two compositions of restricted functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 3: sign and algebra errors when squaring and combining the compositions often stopped candidates before the required quartic; solving it as a quadratic in x² was an accepted route.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Form f(g(x)) and g(f(x)), simplify carefully to a quartic in x, solve it as a quadratic in x², then retain only roots consistent with the function domains." }, { "id": "cie-2023-m-12-q10a", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "10(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-m-12-q10b", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-m-12-q10c", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(c): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-m-12-q10d", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(d)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(d): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-m-12-q11a", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=14", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-m-12-q11b", "paperId": "cie-2023-m-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "February/March", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 9, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the volume formed when the region between a sideways parabola and its chord is rotated about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_12.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 3: many candidates earned few marks after overlooking the y-axis of rotation, failing to square the functions, or using inconsistent limits; full working and subtraction of the two volumes were required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 1, 2, 3 ], "tagConfidence": "reviewed", "answerGuide": "Express both boundaries as x in terms of y, square each radius, integrate with the correct y-limits, and subtract the curve-to-axis volume from the line-to-axis volume." }, { "id": "cie-2023-m-22-q01", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-m-22-q02", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-m-22-q03a", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-m-22-q03b", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-m-22-q04a", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-m-22-q04b", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2023-m-22-q04c", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(c): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2023-m-22-q05a", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-m-22-q05b", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-m-22-q06a", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-m-22-q06b", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-m-22-q07a", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-m-22-q07b", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-m-22-q07c", "paperId": "cie-2023-m-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "February/March", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_22.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2023-m-32-q01", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-m-32-q02a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Shade the Argand region bounded by two argument rays from 1 + 2i and the line Re(z) = 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "March 2023 Principal Examiner Report, physical pp. 5–6: few candidates scored more than one mark; common errors included unequal axis scales, drawing a horizontal line instead of x = 3, and using full lines rather than correctly directed half-lines.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Plot 1 + 2i with equal scales, draw the two half-lines at the specified angles and the vertical boundary x = 3, then shade their intersection." }, { "id": "cie-2023-m-32-q02b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the least argument of a complex number in the shaded Argand region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 6: fewer than 5% of candidates scored any marks; progress depended on a correct sketch in part (a), followed by finding the boundary intersection and its distance below the x-axis.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Use the lower argument boundary from 1 + 2i and the line Re(z) = 3 to locate the limiting point, then calculate its argument from the origin." }, { "id": "cie-2023-m-32-q03", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine two unknown coefficients in a quartic from the remainder on division by a quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 6: candidates found the unknown constants difficult in polynomial division and often confused the divisor, quotient and remainder; the coefficient-comparison route was the clearest successful method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Write p(x) = (x² − x + 1)(Ax² + Bx + C) + 3x + 2, expand, equate coefficients with the given quartic, and solve for a and b." }, { "id": "cie-2023-m-32-q04", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Solve a complex equation involving z and its conjugate by giving roots in x + iy form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 6: this question proved challenging; sign, omitted-constant and multiplication errors meant few candidates reached the correct simplified equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Substitute z = x + iy and z* = x − iy, expand, collect real and imaginary terms separately, set both equal to zero and solve the resulting equations." }, { "id": "cie-2023-m-32-q05a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-m-32-q05b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-m-32-q06a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-m-32-q06b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-m-32-q07a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use sector and triangle areas to establish the given equation for the circle angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 7: candidates often used π instead of 2π for the major sector, omitted the 1/2 area factor, or did not simplify the triangle area with the double-angle identity.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Write the major-sector area as ½r²(2π − x), the shaded segment as the minor sector minus triangle, and equate the major sector to three times the shaded area." }, { "id": "cie-2023-m-32-q07b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2023-m-32-q07c", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(c): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2023-m-32-q08a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-m-32-q08b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Find the exact shaded area under y = x³ ln x between the curve, the x-axis and x = 1/2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 7: integration by parts was usually handled, but only a few noticed that the area must be positive and that the integral needed its modulus.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Integrate x³ ln x by parts, evaluate between 0 and 1/2 using the limiting value at zero, and take the positive area." }, { "id": "cie-2023-m-32-q09", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9", "marks": 7, "topicIds": [ "differentiation", "exponentials", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Solve a separable differential equation with exponential and squared-sine factors, using an initial condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 8: this question was extremely difficult for most; converting the exponential and sin²(2x) terms into integrable forms was the main obstacle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, rewrite e^(−3y) and sin²(2x) with a double-angle identity, integrate both sides, apply y = 0 at x = 0, and evaluate at x = 1/2." }, { "id": "cie-2023-m-32-q10a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-m-32-q10b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-m-32-q10c", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-m-32-q11a", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=17", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-m-32-q11b", "paperId": "cie-2023-m-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "February/March", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the partial fractions found in part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_32.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 8: integrating the term with a linear numerator over x² + 4 proved challenging for most; the report recommends splitting it into a logarithmic integral and an arctangent integral.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 5, 6, 7, 8 ], "tagConfidence": "reviewed", "answerGuide": "Integrate each partial-fraction term separately; split the linear numerator over x² + 4 into a multiple of the derivative of the denominator and a constant term, then apply the limits." }, { "id": "cie-2023-m-42-q01a", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q01b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q02a", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q02b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the collision height of two particles moving vertically in opposite directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 9: only a minority obtained a fully correct answer; candidates often equated the distances without using the initial 18 m separation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Write each particle’s displacement from the ground as a function of time, use their separation of 18 m to find the collision time, then substitute to get the height." }, { "id": "cie-2023-m-42-q03a", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q03b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q04a", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2023-m-42-q04b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-m-42-q04c", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(c): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2023-m-42-q05a", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitudes of the two strut forces supporting a block in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 10: only a significant minority recognised that the struts were in thrust, directed toward D; many treated them as tension and lost the required sign interpretation for a magnitude.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Draw and label the forces at D, resolve vertically and use symmetry to equate the two strut forces, then report their positive magnitude." }, { "id": "cie-2023-m-42-q05b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal force that makes the force in one strut zero.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 10: most candidates who attempted this part assumed the remaining strut was in tension; some also reversed the applied horizontal force or reported a negative magnitude.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Set the force in AD to zero, resolve vertically and horizontally at D, and use the stated magnitude convention to report F positively." }, { "id": "cie-2023-m-42-q06ai", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a)(i): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-m-42-q06aii", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a)(ii): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-m-42-q06b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Decide whether a block pulled up a rough incline will move, given the coefficient of friction and rope tension.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 10: many candidates found the relevant force values but concluded that the block moved down the plane, without checking the direction that friction would then need to act.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Compare the friction needed to prevent upward motion with the maximum available friction; if the required value is smaller, state that the block remains at rest." }, { "id": "cie-2023-m-42-q07a", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q07b", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=17", "qpPages": [ 10, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-m-42-q07c", "paperId": "cie-2023-m-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "February/March", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find where the first particle collides with the combined particle after the latter travels up and back down a slope.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_42.pdf#page=18", "qpPages": [ 10, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 11: very few candidates earned full marks; the most common error was omitting the combined particle’s return time from the slope.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 9, 10, 11 ], "tagConfidence": "reviewed", "answerGuide": "Find Q’s time to C, the combined particle’s full up-and-back time, and the separation at that point; then use relative speed to find the additional time and distance." }, { "id": "cie-2023-m-52-q01a", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "1(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2023-m-52-q01b", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-m-52-q01c", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-m-52-q02a", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-52-q02b", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-m-52-q02c", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-m-52-q03a", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-52-q03b", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-52-q03c", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the seventh resident is the second person not in favour of the leisure centre.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 14: most candidates found this part challenging and many omitted it; the event requires exactly one earlier resident and the seventh resident to be not in favour.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Choose which one of the first six residents is not in favour, multiply by the probability the other five are in favour, and include the seventh resident’s probability of not being in favour." }, { "id": "cie-2023-m-52-q04", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-52-q05", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether two events about the positions of four coloured marbles are independent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 14: candidates found this probability question challenging and few attempted it; many could find P(A) = 1/2 but made little progress on P(B).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate the equally likely marble orders to find P(A), P(B) and P(A ∩ B), then compare P(A ∩ B) with P(A)P(B)." }, { "id": "cie-2023-m-52-q06a", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-52-q06b", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that four cyclists all finish between the stated time limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 15: a significant number stopped after finding the probability for one cyclist and did not find the probability that all four met the condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Standardise both time limits to find one cyclist’s interval probability, keep sufficient precision, and raise that probability to the fourth power." }, { "id": "cie-2023-m-52-q06c", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Estimate the mean and standard deviation of a second normal distribution from two percentile statements.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 15: an unexpected number omitted this part; common errors paired 36 minutes with the wrong cumulative probability or used an inappropriate approach instead of two standardised equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Convert the 23rd and 90th percentiles to their z-values, form two equations for the mean and standard deviation, and solve them simultaneously." }, { "id": "cie-2023-m-52-q07a", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count arrangements of DELIVERED with the three Es together and the two Ds separated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 15: the counting condition was challenging for some; common solutions forgot to divide by 2 because the two Ds are indistinguishable.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Treat EEE as one block, count arrangements with Ds separated by subtracting cases with both Ds grouped, and divide consistently for the indistinguishable Ds." }, { "id": "cie-2023-m-52-q07b", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that a random arrangement of DELIVERED has exactly four letters between the two Ds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "March 2023 Principal Examiner Report, physical pp. 15–16: most candidates did not switch to the distinct-outcome counting needed for a probability; the repeated letters had to be treated as identifiable when forming the total number of outcomes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 12, 13, 14, 15, 16 ], "tagConfidence": "reviewed", "answerGuide": "Place the two Ds four positions apart, arrange the other seven letters with the repeated Es handled consistently, and divide by the total arrangements using the same repeated-letter convention." }, { "id": "cie-2023-m-52-q07c", "paperId": "cie-2023-m-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "February/March", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_52.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_52.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-62-q01a", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-m-62-q01b", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-m-62-q02a", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-m-62-q02bi", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b)(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-62-q02bii", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of one order in one of two hours and at least two orders in the other hour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 18: very few candidates recognised that the two Poisson probabilities had to be multiplied and the product doubled for either hour to satisfy the one-order condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Use the one-hour mean, calculate P(X = 1) and P(X ≥ 2), multiply them, then multiply by 2 for the two possible hour assignments." }, { "id": "cie-2023-m-62-q02c", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(c): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-m-62-q03a", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2023-m-62-q03bi", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(b)(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-m-62-q03bii", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b)(ii): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2023-m-62-q04a", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-62-q04b", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-62-q05a", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-m-62-q05b", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a box’s total packet mass exceeds 4080 g when boxes contain either large or small packets.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 18: this unusual mixture question challenged many; few earned full marks because they needed separate total-mass distributions for the two box types before combining probabilities in the 60:40 proportions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Find the total-mass normal distribution and exceedance probability for each box type separately, then weight those probabilities by 0.6 and 0.4 and add." }, { "id": "cie-2023-m-62-q06a", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "6(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-m-62-q06b", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Explain the conclusion from a one-tailed test using the observed z-statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 19: many candidates did not make the required valid comparison, and conclusions were not always stated cautiously in context.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Compare z = −2.02 with the 2.5% lower-tail critical value −1.96, then state the evidence-based conclusion about this year’s mean time in context." }, { "id": "cie-2023-m-62-q06c", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the range of significance levels at which a one-tailed test rejects for z = 2.14.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 19: candidates often used Φ(2.14) instead of the upper-tail probability 1 − Φ(2.14), and some did not know how to approach the part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the upper-tail probability 1 − Φ(2.14), convert it to a percentage, and state that the significance level must be at least that value." }, { "id": "cie-2023-m-62-q06d", "paperId": "cie-2023-m-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "February/March", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the earlier mean that makes a one-tailed test just reach the 5% rejection boundary.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_qp_62.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_ms_62.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "March 2023 Principal Examiner Report, physical p. 19: the most common error was inconsistent signs when standardising and equating the result to the lower-tail critical value −1.645.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_m23_er.pdf", "sourceEvidencePages": [ 17, 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Use standard error 3.9/√100 and set (24.8 − m)/(3.9/√100) = −1.645, then solve for m." }, { "id": "ial-2021-1-WMA11-01-q1a", "paperId": "ial-2021-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given algebraic curve and simplify the derivative.", "answerGuide": "Apply the power rule to every term, including negative and fractional powers, then collect the result in simplest form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20210304.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/74/d0/c7/b6/de/76/a0/c3/e8/f7/3d/85/b2/4b/d6/97.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA11_01_que_20210304.pdf", "markScheme": "catalogue-WMA11-ms.pdf", "examinerReport": "WMA11_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "a41cd8189bb98b0685b20dd1dbb94dc07ae041168ef3f6223036951092e794e7", "markScheme": 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time, and round to one decimal place.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20210304.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/74/d0/c7/b6/de/76/a0/c3/e8/f7/3d/85/b2/4b/d6/97.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA11_01_que_20210304.pdf", "markScheme": "catalogue-WMA11-ms.pdf", "examinerReport": "WMA11_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "a41cd8189bb98b0685b20dd1dbb94dc07ae041168ef3f6223036951092e794e7", "markScheme": "dcfd02cc5b006d80ba3d2d8c575b8355a37f7a6a5b5c2efd01e6b75250ffbf0c", "examinerReport": "fe702a2a4d6318b9a4b5e789c1afb340b8efae87d0e781b14d1767e84e80c3db" } }, { "id": "ial-2021-1-WMA11-01-q3a", "paperId": "ial-2021-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "trigonometry", "graphs" ], "primaryTopicId": "trigonometry", "summary": "Read the coordinates of the two marked points on the cosine curve.", "answerGuide": "Use the labelled maxima and minima of y=4cos(x°) to state both ordered pairs; include the degree-based x-values.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20210304.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/74/d0/c7/b6/de/76/a0/c3/e8/f7/3d/85/b2/4b/d6/97.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": 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"answerGuide": "Use the minimum value to find the vertical translation, then identify the first positive maximum and its y-coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20210304.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/74/d0/c7/b6/de/76/a0/c3/e8/f7/3d/85/b2/4b/d6/97.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA11_01_que_20210304.pdf", "markScheme": "catalogue-WMA11-ms.pdf", "examinerReport": "WMA11_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "a41cd8189bb98b0685b20dd1dbb94dc07ae041168ef3f6223036951092e794e7", "markScheme": "dcfd02cc5b006d80ba3d2d8c575b8355a37f7a6a5b5c2efd01e6b75250ffbf0c", "examinerReport": 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"coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the quadratic curve equation from two given points and its minimum point.", "answerGuide": "Use the minimum point to write the curve in vertex form with an unknown scale factor, then substitute the other known point to determine that factor and expand if required.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20210304.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/74/d0/c7/b6/de/76/a0/c3/e8/f7/3d/85/b2/4b/d6/97.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson lists this part among the paper's most challenging items. 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round to three decimal places.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20210304.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/74/d0/c7/b6/de/76/a0/c3/e8/f7/3d/85/b2/4b/d6/97.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA11_01_que_20210304.pdf", "markScheme": "catalogue-WMA11-ms.pdf", "examinerReport": "WMA11_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "a41cd8189bb98b0685b20dd1dbb94dc07ae041168ef3f6223036951092e794e7", "markScheme": "dcfd02cc5b006d80ba3d2d8c575b8355a37f7a6a5b5c2efd01e6b75250ffbf0c", "examinerReport": "fe702a2a4d6318b9a4b5e789c1afb340b8efae87d0e781b14d1767e84e80c3db" } }, { "id": "ial-2021-1-WMA11-01-q5b", "paperId": "ial-2021-1-WMA11-01", "board": "Edexcel IAL", 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"integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the energy over the two-hour interval with the trapezium rule.", "answerGuide": "Use the completed equal-width table, apply the trapezium rule to all ordinates, and round the result to two decimal places.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=17", "qpPages": [ 20, 21 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q8a", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the second term of the recursively defined sequence in terms of p.", "answerGuide": "Substitute the first term into the recurrence once and simplify.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q8b", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "sequences", "quadratics", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the stated sum of the first three terms to find the possible values of the second term.", "answerGuide": "Express the first three terms using p, form the given sum equation, solve for p, and substitute each valid value into the expression from part (a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q9a", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "State the centre and radius of the circle in terms of k.", "answerGuide": "Read the centre from the completed-square form and take the positive square root of the radius-squared expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q9b", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "quadratics" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the point-on-circle condition to find the possible values of k.", "answerGuide": "Substitute P(2,3) into the circle equation and show the stated quadratic; then solve it and retain the positive roots.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q9c", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Calculate the exact area of triangle OPT using the tangent at P.", "answerGuide": "Use k<2 to choose the appropriate circle, find the tangent gradient from the radius, determine its x-intercept T, and use the triangle area formula.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q10a", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the Sunday distance in week five under the arithmetic training plan.", "answerGuide": "Find the common difference from the first and final Sunday distances, then use the fifth term of the arithmetic sequence.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q10b", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the Sunday distance in week five when the distance rises by a fixed percentage.", "answerGuide": "Use the first and twelfth terms to find the common ratio, then evaluate the fifth term and round to one decimal place.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA12_01_que_20210304.pdf", "markScheme": "WMA12_01_msc_20210304.pdf", "examinerReport": "WMA12_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "af7e7b524d58ab5b8841a20233ee043cc781e8fb0420891aeb74a358dd39c31a", "markScheme": "ac804337e81d0a411c4d865ae6133bfe97dfd990bc4174713175a64b4bdfcd5a", "examinerReport": "0f2f6905302daa46c7c670da268539f6a387d9c94a5fa85857515b1a0ac548c5" } }, { "id": "ial-2021-1-WMA12-01-q10c", "paperId": "ial-2021-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the largest integer Wednesday distance allowed under each training plan.", "answerGuide": "Find the 12-Sunday total for each plan, add 12x, impose the 360 km limit, and take the greatest integer satisfying each inequality.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20210304.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_msc_20210304.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": 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integrate term by term, and include a constant of integration.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_msc_20210304.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA13_01_que_20210304.pdf", "markScheme": "WMA13_01_msc_20210304.pdf", "examinerReport": "WMA13_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "8ecbe0653acaa9b7ae6aa9b043e93ef695908a0324307244390cb10474e9e751", "markScheme": "7a2c0c0554a5a56bdea05d2601a5076219e2698145aa207d3ae91b6a101ae99a", "examinerReport": "151b39371626ca93c052073415bf25e9ed92e33b91403ff4916d54895ef544a7" } }, { "id": "ial-2021-1-WMA13-01-q2i", "paperId": "ial-2021-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "transformations", "graphs" ], "primaryTopicId": "transformations", "summary": "Sketch the horizontally and vertically transformed polynomial graph.", "answerGuide": "Apply the horizontal scale change and vertical stretch to the intercept and both turning points, then label the transformed coordinates.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_msc_20210304.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA13_01_que_20210304.pdf", "markScheme": "WMA13_01_msc_20210304.pdf", "examinerReport": "WMA13_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "8ecbe0653acaa9b7ae6aa9b043e93ef695908a0324307244390cb10474e9e751", "markScheme": "7a2c0c0554a5a56bdea05d2601a5076219e2698145aa207d3ae91b6a101ae99a", "examinerReport": "151b39371626ca93c052073415bf25e9ed92e33b91403ff4916d54895ef544a7" } }, { "id": "ial-2021-1-WMA13-01-q2ii", "paperId": "ial-2021-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2021, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "transformations", "graphs" ], "primaryTopicId": "transformations", "summary": "Sketch the reflection and vertical translation of the original graph.", "answerGuide": "Reflect the x-coordinates in the y-axis, translate the y-values down by one, and label the intercept and turning points.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_msc_20210304.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA13_01_que_20210304.pdf", "markScheme": "WMA13_01_msc_20210304.pdf", "examinerReport": "WMA13_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": 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the exact area of the bounded region under the given curve.", "answerGuide": "Use the antiderivative from part (a), identify the consecutive zeros delimiting the finite region, and evaluate the definite integral exactly.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=18", "qpPages": [ 14, 17 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q8", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "vectors", "geometry", "proof" ], "primaryTopicId": "vectors", "summary": "Prove that the two parametrically defined lines are skew for every b other than seven.", "answerGuide": "Set the coordinate expressions equal, solve two component equations for the parameters, and show the third component is compatible only when b=7; also rule out parallel directions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=21", "qpPages": [ 18, 19 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q9a", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 7, "topicIds": [ "integration", "trigonometry", "geometry" ], "primaryTopicId": "integration", "summary": "Show that the solid-of-revolution volume has the stated integral form.", "answerGuide": "Use the parametric equations to express radius and dx/dθ, set up π∫y²dx, determine the angular limit from x=3, and simplify using a double-angle identity.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=23", "qpPages": [ 20, 23 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q9b", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact volume by algebraic integration.", "answerGuide": "Integrate the cosine expression from part (a) over the stated bounds and simplify the exact result.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=23", "qpPages": [ 20, 23 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q10a", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Write the rational function of H in partial-fraction form.", "answerGuide": "Set up constants over the two linear factors, equate coefficients or substitute convenient H-values, and state the decomposition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q10b", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 7, "topicIds": [ "integration", "exponentials", "algebra" ], "primaryTopicId": "integration", "summary": "Solve the separable tank-depth differential equation using the initial condition.", "answerGuide": "Separate variables, integrate using the partial fractions, apply H(0)=13, and rearrange to the requested exponential expression.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q10c", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find when the tank depth reaches eight metres.", "answerGuide": "Substitute H=8 into the solution, isolate the exponential, take logarithms, and give the positive time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WMA14-01-q10d", "paperId": "ial-2021-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(d)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "State the eventual water depth predicted by the model.", "answerGuide": "Take the long-time limit of the solution; the decaying exponential term tends to zero, leaving the constant depth.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20210304.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/fe/fb/1e/4e/ad/96/c1/6e/2d/69/43/17/59/58/3b/49.pdf#page=25", "qpPages": [ 24, 28 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14_01_que_20210304.pdf", "markScheme": "catalogue-WMA14-ms.pdf", "examinerReport": "WMA14_01_pef_20210304.pdf" }, "sourceSha256": { "questionPaper": "4b169ad4728f32335613446ba2a55ca7b0bb69c89b675134fb530fa7997f99b7", "markScheme": "d5c64cb537fc3f90b6b7b0b27e25111b3c294002b517c99f973e586fa62b339f", "examinerReport": "071cbc60e57a5935760600fa69545fce0c604013a8715cc858c0188dd071cbbd" } }, { "id": "ial-2021-1-WDM11-01-q1", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use binary search to locate a named word in an ordered list, showing pivots and rejected sections.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=2", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At each step choose the middle entry, compare it with the target, reject the half that cannot contain it, and continue until the word is found or excluded." }, { "id": "ial-2021-1-WDM11-01-q2", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate the pizza production constraints and cost objective as a linear programme.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=3", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Let x and y be the small and large quantities; translate the minimum total, the large-to-small ratio and the maximum large share into integer-coefficient inequalities, then state the cost to minimize." }, { "id": "ial-2021-1-WDM11-01-q3a", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Pack the listed values into bins of capacity 5 using first fit.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Process the values in the given order and place each in the earliest bin with enough remaining capacity; open a new bin only when needed." }, { "id": "ial-2021-1-WDM11-01-q3b", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Carry out two bubble-sort passes and report each resulting list plus comparisons and swaps.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare adjacent entries from the left on each pass, swap out-of-order pairs, record the list after each pass, and count every comparison and swap." }, { "id": "ial-2021-1-WDM11-01-q3c", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Quick-sort the supplied intermediate list and make the pivots clear.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose and state a permitted pivot at each stage, partition into lower and higher values, then repeat on each sublist until sorted." }, { "id": "ial-2021-1-WDM11-01-q3d", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use first-fit decreasing to pack the sorted values into bins of capacity 5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sort the values into descending order first, then apply first fit and show the contents of each bin." }, { "id": "ial-2021-1-WDM11-01-q4a", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain how the classical and practical travelling salesperson problems differ.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=5", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the visit/return requirements of the classical problem and contrast them with the practical requirement that vertices may only need to be visited at least once." }, { "id": "ial-2021-1-WDM11-01-q4b", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use nearest neighbour from A to find a route upper bound.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=5", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Repeatedly travel to the nearest unvisited museum, return to A, and add the selected distances." }, { "id": "ial-2021-1-WDM11-01-q4c", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Compare a second stated route length with the first upper bound.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=5", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the smaller route length as the better upper bound and state why." }, { "id": "ial-2021-1-WDM11-01-q4d", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct a minimum spanning tree on the reduced network with Prim’s algorithm and use it for a route lower bound.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=5", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Starting at A, add the least eligible edge without creating a cycle until the reduced network is connected; total the tree and apply the stated lower-bound construction." }, { "id": "ial-2021-1-WDM11-01-q4e", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Compare a second lower bound with the bound from the reduced network.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=5", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The larger valid lower bound is better because it narrows the interval containing the optimum." }, { "id": "ial-2021-1-WDM11-01-q4f", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(f)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the smallest interval known to contain the optimal route length.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=5", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the greatest available lower bound and least available upper bound as the interval endpoints." }, { "id": "ial-2021-1-WDM11-01-q5a", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra’s algorithm to find and give the shortest route from A to H.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Update permanent and temporary labels from A, record predecessor vertices, and trace the final predecessor chain from H to find the route and length." }, { "id": "ial-2021-1-WDM11-01-q5b", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a shortest route from F to K that passes through A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the shortest path from F to A and from A to K, then concatenate the paths and add their lengths." }, { "id": "ial-2021-1-WDM11-01-q5c", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Compare the three pairings of the four relevant odd vertices, then identify the repeated roads and the inspection route length.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this route-inspection part caused substantial mark loss: many candidates paired only the two odd vertices, missing the additional start and finish vertices and the full set of pairings.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "List all three distinct pairings of the odd vertices after accounting for the required start and finish. Add the pairing totals, choose the least, state the duplicated arcs, and add their length to the network total.", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2021-1-WDM11-01-q5d", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State how many times the inspection route passes through F.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Trace the chosen route and duplicated arcs and count each visit to F." }, { "id": "ial-2021-1-WDM11-01-q5e", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the finishing vertex for the revised open inspection route.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Re-evaluate the odd-vertex condition for a route starting at D and finishing at a free vertex; state the resulting endpoint." }, { "id": "ial-2021-1-WDM11-01-q5f", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(f)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the difference between the lengths of the two inspection routes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=12", "qpPages": [ 6, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates gave the correct difference; the report notes a common error that subtracted the wrong repeated road lengths.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the correct completed route lengths from the earlier parts and subtract them in the order requested, preserving any follow-through allowed by the question.", "sourceEvidencePages": { "examinerReport": [ 8 ] } }, { "id": "ial-2021-1-WDM11-01-q6a", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the activity network by adding G, H and I with the fewest dummies.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=8", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Place each activity after all of its immediate predecessors and add only the dummy arcs needed to preserve the distinct dependency sets." }, { "id": "ial-2021-1-WDM11-01-q6b", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Add early and late event times to the network.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=8", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Make a forward pass taking maxima at merges, then a backward pass taking minima at splits from the required completion time." }, { "id": "ial-2021-1-WDM11-01-q6c", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Identify the critical activities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=8", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Select the activities with zero total float from the event-time calculations." }, { "id": "ial-2021-1-WDM11-01-q6d", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate a lower bound for the number of workers needed.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=8", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the total activity worker-days by the minimum project duration and round up to a whole worker." }, { "id": "ial-2021-1-WDM11-01-q6e", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Schedule the activities using the fewest workers while keeping the project completion time minimal.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=8", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes the scheduling diagram as poorly answered. Many candidates ignored the available floats or omitted activities, preventing them from meeting the worker constraint.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the activity precedence and float windows to place every activity within the critical-path duration. Check overlaps and confirm that the schedule uses the minimum number of workers.", "sourceEvidencePages": { "examinerReport": [ 9 ] } }, { "id": "ial-2021-1-WDM11-01-q7a", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Write the inequalities defining the feasible region from the graph and constraints.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=10", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read each boundary line and shaded side, use the positive-constant information, and include the non-negativity restrictions." }, { "id": "ial-2021-1-WDM11-01-q7b", "paperId": "ial-2021-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine the parameter values for which the stated objective has its given optimal value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20210304.pdf#page=10", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Decision/D1/MS/January%202021%20MS.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls the final question challenging and reports a very wide range of attempts. Weak algebra and failure to compare all feasible vertices were common, and full marks were rare.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the relevant vertices of the feasible region as expressions in the parameter, evaluate the objective at each candidate vertex, and solve the resulting algebraic conditions. Check every candidate against all vertices before accepting it.", "sourceEvidencePages": { "examinerReport": [ 9, 10 ] } }, { "id": "ial-2021-1-WME01-01-q1a", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the stone’s speed when it is 2 m above the ground.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v²=u²+2as with the vertical displacement from the projection point, taking care with the sign of the acceleration." }, { "id": "ial-2021-1-WME01-01-q1b", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time from projection until the stone first reaches the ground.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the vertical displacement equation with the initial height and acceleration −g; solve the quadratic and select the positive time." }, { "id": "ial-2021-1-WME01-01-q2a", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find P’s speed after the direct collision from the impulse on Q.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the impulse as Q’s momentum change to find Q’s final velocity, then apply conservation of momentum to determine P’s final velocity and speed." }, { "id": "ial-2021-1-WME01-01-q2b", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find Q’s speed after the direct collision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Q’s signed momentum change from the impulse and its direction to calculate the final velocity, then give its magnitude." }, { "id": "ial-2021-1-WME01-01-q3", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the smallest applied force T that keeps the parcel in equilibrium on the rough floor.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve both angled forces and weight vertically to find the normal reaction; use limiting friction opposing the impending motion and resolve horizontally to solve for T." }, { "id": "ial-2021-1-WME01-01-q4", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the non-uniform girder’s centre-of-mass distance from A using the two tipping conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about D for the first limiting case and about C for the second, equate the expressions for the weight, and solve for x." }, { "id": "ial-2021-1-WME01-01-q5", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find the magnitude and bearing of the resultant of two specified forces.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into north/east components using its bearing, add the components, find the magnitude with Pythagoras, and calculate the bearing from the component ratio." }, { "id": "ial-2021-1-WME01-01-q6a", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find Agatha’s position vector after t seconds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add t times her constant velocity vector to her initial position vector." }, { "id": "ial-2021-1-WME01-01-q6b", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show Brionie passes through P and find when this occurs.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find P from Agatha’s position at t=6, equate Brionie’s coordinate expressions to P, and verify both coordinates give the same time." }, { "id": "ial-2021-1-WME01-01-q6c", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the least distance between two skaters moving with constant velocities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates earned full marks in the final part. Many could use Pythagoras on the given expression but did not complete the minimization step.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Form the relative position vector as a function of time, square its magnitude, and minimize the resulting quadratic for t ≥ 0. Substitute the minimizing time into the separation distance.", "sourceEvidencePages": { "examinerReport": [ 3, 5 ] } }, { "id": "ial-2021-1-WME01-01-q6d", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the girls’ minimum separation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Square the magnitude of the relative position vector, minimize the quadratic for t≥0, then take the square root at the minimizing time." }, { "id": "ial-2021-1-WME01-01-q7a", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the speed when the parachute opens.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v=u+at for the initial free-fall interval from rest." }, { "id": "ial-2021-1-WME01-01-q7b", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the parachutist’s speed-time graph.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the initial linear increase, the constant deceleration segment, and the final constant-speed segment with correct time and speed labels." }, { "id": "ial-2021-1-WME01-01-q7c", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the duration T of the deceleration phase.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the 73.75 m displacement and the initial/final speeds in constant-acceleration equations, solving for the positive duration consistent with T<6." }, { "id": "ial-2021-1-WME01-01-q7d", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the helicopter’s height above the ground to the nearest metre.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the distances from free fall, deceleration and the final constant-speed segment; use the total time to find the last duration." }, { "id": "ial-2021-1-WME01-01-q8a", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension while the two-particle system accelerates down the inclined plane.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=15", "qpPages": [ 20, 24 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces on A and B along the string, include limiting friction on A in the correct direction, and solve the simultaneous equations for acceleration and tension." }, { "id": "ial-2021-1-WME01-01-q8b", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the range of starting distances that lets the particle stop before reaching the pulley after the hanging mass hits the ground.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=15", "qpPages": [ 20, 24 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates omitted this part, stopped after finding half the height, or failed to state the final inequality in the correct direction.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the acceleration and speed found while the system is connected, then model the particle's motion after the hanging mass stops. Compare the total distance travelled with d and solve the strict inequality.", "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-2021-1-WME01-01-q8c", "paperId": "ial-2021-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "State one additional physical factor that would make the model more realistic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20210304.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/e3/da/ac/1a/b3/b0/42/e5/a9/98/5f/ed/bf/bd/6a/71.pdf#page=15", "qpPages": [ 20, 24 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Name a relevant omitted effect, such as air resistance, pulley inertia or a non-light string, without repeating the excluded air-resistance example." }, { "id": "ial-2021-1-WME02-01-q1", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 7, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find both possible velocities after a particle receives a specified impulse.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=9", "qpPages": [ 2, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the impulse-momentum equation to relate the change in velocity vector to the impulse, then impose the given impulse magnitude to obtain a quadratic with two possible values." }, { "id": "ial-2021-1-WME02-01-q2", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal force required to keep the suspended triangular lamina in equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=10", "qpPages": [ 4, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the triangle’s centre of mass, find its perpendicular moment arms, and equate clockwise and anticlockwise moments about A." }, { "id": "ial-2021-1-WME02-01-q3", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the engine power using two constant-speed/acceleration observations on a car.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P=Fv and Newton’s second law on the horizontal road to relate power and resistance; resolve forces on the incline at constant speed, then solve for P." }, { "id": "ial-2021-1-WME02-01-q4", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the equilibrium orientation of a suspended rigid framework made from three rods.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find each rod’s centre of mass and use their lengths to weight the combined centre; take moments about the suspension point and solve for the angle." }, { "id": "ial-2021-1-WME02-01-q5a", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the particle’s velocity is parallel to i+j.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the i and j velocity components equal and solve for the non-negative time." }, { "id": "ial-2021-1-WME02-01-q5b", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration magnitude as the particle passes through A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the velocity components, substitute t=3, and calculate the vector magnitude." }, { "id": "ial-2021-1-WME02-01-q5c", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the position vector of A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each velocity component from t=0 to t=3 and use the initial position O." }, { "id": "ial-2021-1-WME02-01-q6a", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the resultant force exerted on the ladder by the ground when the man stands 2 m from A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical and horizontal force balance and take moments about a convenient point to find the ground reaction components, then combine them as a resultant." }, { "id": "ial-2021-1-WME02-01-q6b", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find how far the man can climb before the ladder is about to slip.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use limiting friction and the friction coefficient to determine the reaction components, then take moments about the foot of the ladder to solve for AD." }, { "id": "ial-2021-1-WME02-01-q6c", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "State how the rod model is used.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the ladder is treated as having its mass uniformly distributed, so its weight acts at its midpoint." }, { "id": "ial-2021-1-WME02-01-q7a", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the impact speed using energy conservation from A to B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate initial kinetic plus potential energy with final kinetic energy, using the 20 m vertical drop." }, { "id": "ial-2021-1-WME02-01-q7b", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the particle’s least speed during its flight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the constant horizontal velocity component from the displacement/time data; the least speed occurs when the vertical component is zero." }, { "id": "ial-2021-1-WME02-01-q7c", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find how long the projectile's speed exceeds 10 m/s during its flight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=15", "qpPages": [ 20, 23 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report calls this a consistently discriminating task and notes several blank responses. Many candidates solved a different question by finding when the speed was below the threshold.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve velocity into constant horizontal and changing vertical components. Set the squared speed greater than 100, solve the resulting quadratic in time, and restrict the solution to the five-second flight.", "sourceEvidencePages": { "examinerReport": [ 3, 5 ] } }, { "id": "ial-2021-1-WME02-01-q8a", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of restitution for the direct collision of A and B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=17", "qpPages": [ 24, 28 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the impulse to relate the before-and-after velocities, apply conservation of momentum, then divide the relative speed of separation by the relative speed of approach." }, { "id": "ial-2021-1-WME02-01-q8b", "paperId": "ial-2021-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the location x of the second collision after B rebounds from the wall.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20210304.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_msc_20210304.pdf#page=17", "qpPages": [ 24, 28 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first collision velocities, apply the wall’s coefficient of restitution to B, compare travel times for A and B, and solve the position equation for x." }, { "id": "ial-2021-1-WME03-01-q1a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the volume of the rotated solid from the reciprocal curve and bounded region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use V=π∫y²dx over the interval from x=1 to x=a and simplify the exact expression." }, { "id": "ial-2021-1-WME03-01-q1b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass x-coordinate of the solid of revolution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the first moment by integrating xπy², divide by the volume from part (a), and simplify." }, { "id": "ial-2021-1-WME03-01-q2a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the inverse-square gravitational force on a particle at height x above Earth’s surface.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the force as k/(x+R)², use the surface condition F=mg to determine k, and substitute it back." }, { "id": "ial-2021-1-WME03-01-q2b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the impact speed for a particle released from height R above Earth’s surface, given its speed at height R.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of energy between the point at distance 2R from the centre and the surface at R, accounting for gravitational potential change under the inverse-square field." }, { "id": "ial-2021-1-WME03-01-q3a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tensions in the upper and lower arms of a rotating fairground cabin.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the 3-4-5 geometry for the arm directions and resolve vertically and horizontally; use centripetal force for the horizontal resultant." }, { "id": "ial-2021-1-WME03-01-q3b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Explain how stretching both arms changes the sum of the tensions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the increased horizontal radius to compare the required centripetal resultant at unchanged angular speed; state the resulting change in the tension sum." }, { "id": "ial-2021-1-WME03-01-q4a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a solid formed by removing two cones from a larger cone.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=14", "qpPages": [ 12, 15 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat removed cones as negative masses, find their volume ratios and centroid distances from O, then take the signed first moment divided by the remaining mass." }, { "id": "ial-2021-1-WME03-01-q4b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the angle made by a line segment with the downward vertical when a hollowed cone hangs in equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=14", "qpPages": [ 12, 15 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says this part was challenging. Many candidates had the needed angles but failed to identify the final angle, often because their diagram was unclear.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of the remaining solid, draw a large clear vertical-plane diagram, and use the geometry of the suspended body to calculate the required angle.", "sourceEvidencePages": { "examinerReport": [ 3, 5 ] } }, { "id": "ial-2021-1-WME03-01-q5a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that both elastic strings have extension 2a in equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Hooke’s law to each string and balance the upward and downward forces together with the weight; solve the two extension equations." }, { "id": "ial-2021-1-WME03-01-q5b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the vertical motion after displacement is simple harmonic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the resultant restoring force in terms of displacement from equilibrium and show it is proportional to displacement and directed toward equilibrium." }, { "id": "ial-2021-1-WME03-01-q5c", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed at the stated height above A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of energy including gravitational potential and both elastic energies between release and the specified position." }, { "id": "ial-2021-1-WME03-01-q6a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle completes vertical circles when the string is assumed not to break.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use energy to find speeds at key points and apply the radial equation to verify the tension stays positive, especially at the top of the circle." }, { "id": "ial-2021-1-WME03-01-q6b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle still completes vertical circles when the string breaks above 7mg.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=18", "qpPages": [ 20, 23 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the maximum tension from the speed and radius at the lowest point and show it does not exceed the breaking threshold; check the top condition too." }, { "id": "ial-2021-1-WME03-01-q7a", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the spring extension at the turning point after an impulse sends a particle up a rough incline.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=21", "qpPages": [ 24, 28 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says full marks were not achieved by most candidates. Common costly errors were omitting an energy term or failing to form and solve the required three-term quadratic.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert the impulse to the initial speed, then equate initial kinetic energy to the gain in gravitational potential energy, elastic strain energy and work against friction. Solve the resulting quadratic and choose the physically valid extension.", "sourceEvidencePages": { "examinerReport": [ 3, 6, 7 ] } }, { "id": "ial-2021-1-WME03-01-q7b", "paperId": "ial-2021-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle will not remain at rest at A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20210304.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_msc_20210304.pdf#page=21", "qpPages": [ 24, 28 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the component of weight and elastic tension along the plane with the maximum available friction; show equilibrium cannot be maintained." }, { "id": "ial-2021-1-WST01-01-q1a", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of not being in event B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the regions outside B or use 1−P(B)." }, { "id": "ial-2021-1-WST01-01-q1b", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of A or C.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the disjoint regions in A∪C without double-counting any intersection." }, { "id": "ial-2021-1-WST01-01-q1c", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability of A given not B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A∩B′)/P(B′), reading both probabilities from the Venn diagram." }, { "id": "ial-2021-1-WST01-01-q2a", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Find the median age from the stem-and-leaf diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count the observations and identify the central ordered value." }, { "id": "ial-2021-1-WST01-01-q2b", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Find the interquartile range of the ages.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the lower and upper quartiles from the ordered data and subtract Q1 from Q3." }, { "id": "ial-2021-1-WST01-01-q2c", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Identify outliers using the 1.5×IQR rule.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the lower and upper fences from Q1−1.5IQR and Q3+1.5IQR, then identify any observations outside them." }, { "id": "ial-2021-1-WST01-01-q2d", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Draw a box plot of the ages on the supplied scale.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=6", "qpPages": [ 4, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Mark the quartiles, median and whisker endpoints correctly; plot any outlier separately." }, { "id": "ial-2021-1-WST01-01-q3a", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that a package weighs less than 10 kg.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardize 10 using the normal mean and standard deviation, then evaluate the lower-tail probability." }, { "id": "ial-2021-1-WST01-01-q3b", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the lightest weight in the heaviest 15% of packages.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the 85th percentile of the stated normal distribution and convert the z-score back to weight." }, { "id": "ial-2021-1-WST01-01-q3c", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the conditional probability that a package not handled by Jemima weighs more than the mean.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was much more challenging than the preceding parts. Candidates who used a probability ratio often struggled to define its numerator correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the condition into a ratio: the probability of a package above 18 kg and not in the heaviest 15%, divided by the probability of not being in that top group. Use the normal model consistently.", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2021-1-WST01-01-q3d", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that exactly two of four independently delivered packages are moved by Jemima.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the most difficult part of Question 3. Candidates often used the wrong binomial parameter or confused the number of packages with the tail percentage.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with four trials and the probability 0.15 that a package falls in the heaviest 15%; calculate the probability of exactly two and retain sufficient accuracy.", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2021-1-WST01-01-q4a", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Name the distribution of the equally likely spinner score.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the discrete uniform distribution on the four listed values." }, { "id": "ial-2021-1-WST01-01-q4bi", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected value of X.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use symmetry or the arithmetic mean of the four equally likely outcomes." }, { "id": "ial-2021-1-WST01-01-q4bii", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of X.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(X²) from the four outcomes and subtract [E(X)]²." }, { "id": "ial-2021-1-WST01-01-q4ci", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(Y) from its probability table.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sum yP(Y=y) over the five possible values." }, { "id": "ial-2021-1-WST01-01-q4cii", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Var(Y) from its probability table.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(Y²) and subtract [E(Y)]²." }, { "id": "ial-2021-1-WST01-01-q4d", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Choose a and b so W=aX+b matches Y’s mean and variance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Var(W)=a²Var(X) to find positive a, then use E(W)=aE(X)+b to solve for b." }, { "id": "ial-2021-1-WST01-01-q4e", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that W equals Y when the two spinners are spun once.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the independent outcome pairs satisfying aX+b=Y and sum their probabilities." }, { "id": "ial-2021-1-WST01-01-q5a", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe the correlation between performance score and salary.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the direction and strength shown by the scatter diagram." }, { "id": "ial-2021-1-WST01-01-q5bi", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the given value of Sxy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Sxy=Σxy−(ΣxΣy)/n with the supplied totals." }, { "id": "ial-2021-1-WST01-01-q5bii", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find Syy.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Σy²−(Σy)²/n." }, { "id": "ial-2021-1-WST01-01-q5c", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the product-moment correlation coefficient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate Sxx and use r=Sxy/√(SxxSyy), retaining the sign." }, { "id": "ial-2021-1-WST01-01-q5d", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Assess whether the sample is consistent with a linear relationship.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the correlation evidence and distinguish consistency with a linear pattern from proof of causation." }, { "id": "ial-2021-1-WST01-01-q5e", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the regression line of salary on performance score.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate b=Sxy/Sxx and a=ȳ−b x̄, then state both coefficients to three significant figures." }, { "id": "ial-2021-1-WST01-01-q5f", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(f)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the regression slope in context.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain the predicted change in salary, in thousands of pounds, for each one-point increase in performance score." }, { "id": "ial-2021-1-WST01-01-q5g", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(g)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Plot the regression line on the scatter diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate two points on the line, plot them, and draw the straight line through them." }, { "id": "ial-2021-1-WST01-01-q5h", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(h)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the underperforming manager on the scatter diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Circle the observation with a notably low performance score relative to the salary trend, as indicated by the graph." }, { "id": "ial-2021-1-WST01-01-q5i", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the salary for a manager with performance score 30.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x=30 into the fitted regression line and report the result to three significant figures." }, { "id": "ial-2021-1-WST01-01-q6a", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show the probability of winning on one roll of the disc is 1/5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=11", "qpPages": [ 18, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The centre must lie at least 1 cm from every edge, leaving a safe region 3 cm by 1 cm inside each 5-by-3 rectangle; divide its area by 15." }, { "id": "ial-2021-1-WST01-01-q6b", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the standard deviation of prizes won per student in the first group.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=11", "qpPages": [ 18, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sample’s frequency-independent variance formula from Σx and Σx², then take the square root." }, { "id": "ial-2021-1-WST01-01-q6c", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 7, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the combined mean and standard deviation for the two student groups.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=11", "qpPages": [ 18, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine group totals to obtain the overall mean; combine within-group sums of squares and between-group mean offsets to find the overall variance, then take its square root." }, { "id": "ial-2021-1-WST01-01-q6d", "paperId": "ial-2021-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use the corner-hit estimate to estimate π.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_msc_20210304.pdf#page=11", "qpPages": [ 18, 20 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the corner-hit probability as the appropriate circular-area ratio, rearrange for π, and substitute 0.2216." }, { "id": "ial-2021-1-WST02-01-q1a", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State a suitable model for the number of oysters that do not survive.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with n=30 and probability of non-survival 0.05." }, { "id": "ial-2021-1-WST02-01-q1b", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State an assumption required by the binomial model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that each oyster has the same survival probability or that outcomes are independent." }, { "id": "ial-2021-1-WST02-01-q1c", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probabilities for exactly 24 survivors and at least 3 non-survivors.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial distribution with p(non-survival)=0.05, converting 24 survivors to 6 failures for the first event." }, { "id": "ial-2021-1-WST02-01-q1d", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use an approximation to find the largest n with a specified upper-tail probability above 0.8.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Approximate the binomial count by a normal distribution with continuity correction, solve the tail inequality, and choose the largest integer n meeting it." }, { "id": "ial-2021-1-WST02-01-q1e", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Test at 5% whether the non-survival proportion is greater than 5%.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=5", "qpPages": [ 2, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-tailed binomial hypotheses, use the sample count of four failures from 25, find the exact tail probability or critical region, and conclude in context." }, { "id": "ial-2021-1-WST02-01-q2a", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability a novice walks at least 3.5 m before falling.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate 1−F(3.5) from the given cumulative distribution function." }, { "id": "ial-2021-1-WST02-01-q2b", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability more than one of 30 novices walks at least 3.5 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with the probability from part (a) and calculate 1−P(X≤1)." }, { "id": "ial-2021-1-WST02-01-q2c", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use algebraic integration to find Var(W), given E(W)=1.6.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate w²f(w) over the distribution’s support to obtain E(W²), then subtract [E(W)]²." }, { "id": "ial-2021-1-WST02-01-q3a", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability a 100 ml sample does not contain exactly four water fleas.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson probability with mean 7 for exactly four, then subtract from one." }, { "id": "ial-2021-1-WST02-01-q3b", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability exactly one of five samples contains exactly four fleas.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with five trials and success probability P(Poisson(7)=4)." }, { "id": "ial-2021-1-WST02-01-q3ci", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)(i)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the equation produced by a normal approximation for a sample of n ml.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson mean and variance to 0.07n, apply the continuity correction for more than three fleas, standardize, and rearrange to the stated equation." }, { "id": "ial-2021-1-WST02-01-q3cii", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Solve for the sample volume n.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the equation from part (i), retain a positive physically meaningful volume, and round as required." }, { "id": "ial-2021-1-WST02-01-q3d", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Test at 1% whether the average flea count per 100 ml has increased.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-tailed Poisson hypotheses, evaluate the upper-tail probability at 15 under mean 7, compare with 0.01, and conclude in context." }, { "id": "ial-2021-1-WST02-01-q4a", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the normalization condition relating k and a for the given density.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate k(a−x)² from 0 to a, set the total probability to one, and simplify to ka³=3." }, { "id": "ial-2021-1-WST02-01-q4b", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use E(X)=1.5 to show a=6.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x times the density over its support to obtain E(X), substitute ka³=3, and solve for the positive a." }, { "id": "ial-2021-1-WST02-01-q4c", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Verify the median is 1.2 to one decimal place.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find m satisfying the integral of the density from 0 to m equals one half, substitute a=6, and show the root rounds to 1.2." }, { "id": "ial-2021-1-WST02-01-q5a", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability the randomly cut first piece is longer than 1.8 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=9", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model W as uniform on [0,3] and calculate the length of the interval above 1.8 divided by 3." }, { "id": "ial-2021-1-WST02-01-q5b", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the squared hypotenuse length in terms of W.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=9", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Pythagoras to the two piece lengths W and 3−W and simplify." }, { "id": "ial-2021-1-WST02-01-q5c", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(X²) for the hypotenuse length.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=9", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the expression from part (b) and the required moments of a uniform variable on [0,3] to evaluate the expectation." }, { "id": "ial-2021-1-WST02-01-q5d", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability X² exceeds 5.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=9", "qpPages": [ 18, 21 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve 2W²−6W+9>5 over 0≤W≤3, identify the qualifying interval lengths, and divide by 3." }, { "id": "ial-2021-1-WST02-01-q6a", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain why the owner samples club members rather than asking everyone.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=10", "qpPages": [ 22, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Give a practical reason such as lower cost, less time or easier administration." }, { "id": "ial-2021-1-WST02-01-q6b", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Suggest a sampling frame for the youth-club members.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=10", "qpPages": [ 22, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Name a complete list or register containing all current members." }, { "id": "ial-2021-1-WST02-01-q6c", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the sampling units.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=10", "qpPages": [ 22, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the individual youth-club members are the units." }, { "id": "ial-2021-1-WST02-01-q6d", "paperId": "ial-2021-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 7, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the observed probabilities to show the probability that the sample mean is 50.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20210304.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_msc_20210304.pdf#page=10", "qpPages": [ 22, 24 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This was one of the more demanding parts. The report explains that the shorter solution uses the probabilities summing to one; a more complicated alternative leads to a quartic.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Write p + q + r = 1, use the given event probabilities to obtain equations for p and qr, then solve the quadratic for q and r and apply q > r to choose the correct ordering.", "sourceEvidencePages": { "examinerReport": [ 3, 6, 7 ] } }, { "id": "ial-2021-1-WST03-01-q1a", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Complete the random sample using the stated column of random numbers.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=2", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Continue down the specified column, skip values outside 1–60 and duplicates, and stop when ten valid player numbers have been selected." }, { "id": "ial-2021-1-WST03-01-q1b", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show how to use random numbers to select a proportional stratified sample of players.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=2", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was rarely answered correctly, even by candidates who had found a valid simple random sample in the previous part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the population into girls and boys, select four distinct valid labels from the girls' range and six from the boys' range using the random-number table, and show the accepted labels clearly.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2021-1-WST03-01-q1c", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Name a group that may be missed by the age-ordered simple random sample and explain why.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=2", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=4", "qpPages": [ 2, 3 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify an age-related group concentrated toward one end of the ordered list and explain that a sample of ten may omit them." }, { "id": "ial-2021-1-WST03-01-q2a", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate Spearman’s rank correlation coefficient for memory and maths scores.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=5", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rank both variables, calculate d and d² for each student, sum d², and substitute in Spearman’s formula." }, { "id": "ial-2021-1-WST03-01-q2b", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Test at 5% whether greater mathematical ability is associated with better short-term memory.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=5", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-tailed hypotheses for Spearman correlation, use the sample coefficient and critical value, and give a contextual conclusion." }, { "id": "ial-2021-1-WST03-01-q2c", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Assess whether a positive correlation supports the claim that memory improvement causes stronger mathematics performance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=4", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=5", "qpPages": [ 4, 5 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this causal distinction was rarely made. Many candidates treated positive correlation as proof of the teacher's causal argument.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Explain that correlation establishes an association only. It does not prove that improved memory causes higher test scores; other variables or the direction of influence may explain the relationship.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2021-1-WST03-01-q3a", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Complete the chi-squared goodness-of-fit test for uniform choice among four clubs.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find expected frequencies, use the given test statistic and three degrees of freedom, compare with the 5% critical value, and conclude." }, { "id": "ial-2021-1-WST03-01-q3b", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State suitable hypotheses for association between school and club choice.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State independence as the null and an association as the alternative." }, { "id": "ial-2021-1-WST03-01-q3c", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Find the expected frequency for School C choosing Computers.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the School C row total by the Computers column total and divide by the grand total." }, { "id": "ial-2021-1-WST03-01-q3d", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Explain why the adjusted chi-squared test has four degrees of freedom after grouping categories.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this explanation was more challenging: many candidates mentioned a small expected count but did not identify the pooled categories or resulting table size.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the expected cell below five, explain that the relevant club categories were pooled to form a 3 by 3 table, and use (rows − 1)(columns − 1) = 4.", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2021-1-WST03-01-q3e", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Complete the association test at 5% significance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=6", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=6", "qpPages": [ 6, 9 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated statistic with four degrees of freedom, compare against the 5% chi-squared critical value, and conclude in context." }, { "id": "ial-2021-1-WST03-01-q4a", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Calculate a 98% confidence interval for the mean score in town A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=10", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use x̄±zσ/√n with z for a 98% two-sided interval and the known standard deviation." }, { "id": "ial-2021-1-WST03-01-q4b", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test the newspaper’s claim that town B’s mean score is greater than town A’s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=10", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-tailed hypotheses, standardize the difference in sample means using the known common standard deviation and both sample sizes, compare with the 5% critical value, and conclude." }, { "id": "ial-2021-1-WST03-01-q4c", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Find the largest national mean consistent with both town samples failing to show the mean is lower.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=10", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=7", "qpPages": [ 10, 13 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says correct solutions were rare. Candidates often calculated separate confidence intervals but did not set up inequalities or choose the stricter threshold.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "For each town, find the one-sided critical upper limit using the known standard deviation and sample size. The largest value satisfying both conditions is the smaller of the two limits.", "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2021-1-WST03-01-q5a", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 10, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the grouped pine-cone lengths fit the stated normal model at 10%.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=14", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate expected frequencies for the five classes, check adequacy, form the chi-squared statistic, use the correct degrees of freedom after parameter estimation, and conclude." }, { "id": "ial-2021-1-WST03-01-q5b", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Calculate unbiased estimates of the mean and variance from the supplied totals.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=14", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use x̄=Σx/n and the unbiased sample variance formula based on Σx²−(Σx)²/n." }, { "id": "ial-2021-1-WST03-01-q5c", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Complete the friend’s normality chi-squared test at 10% without pooling classes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=14", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine degrees of freedom from the five classes and estimated parameters, compare 3.50 with the critical value, and state the conclusion." }, { "id": "ial-2021-1-WST03-01-q5d", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Estimate the proportion of pine cones longer than 7 cm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=14", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardize 7 using the sample estimates, calculate the upper-tail normal probability, and round to two significant figures." }, { "id": "ial-2021-1-WST03-01-q6a", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability a randomly chosen yellow tile is longer than a red tile.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=18", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=9", "qpPages": [ 18, 20 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the normal difference Y−R using independent variables, calculate its mean and variance, and find P(Y−R>0)." }, { "id": "ial-2021-1-WST03-01-q6b", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability the total length of three red tiles is less than four yellow-tile lengths.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=18", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=9", "qpPages": [ 18, 20 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal distributions of the independent sums, form the difference 3R−4Y, and evaluate its lower-tail probability." }, { "id": "ial-2021-1-WST03-01-q6c", "paperId": "ial-2021-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2021, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 7, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Choose coefficients with a fixed mean so that the variance of a linear combination is minimized.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20210304.pdf#page=18", "msUrl": "https://pmt.physicsandmathstutor.com/download/Maths/A-level/Papers/Edexcel-IAL/Statistics/S3/MS/January%202021%20MS.pdf#page=9", "qpPages": [ 18, 20 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found the mean and variance expressions but stopped before optimizing. The report says the most successful methods expressed the variance in one variable and minimized it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20210304.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the mean constraint to express one coefficient in terms of the other, substitute into the variance formula for independent variables, and minimize the resulting quadratic.", "sourceEvidencePages": { "examinerReport": [ 3, 4, 5 ] } }, { "id": "cie-2022-s-11-q01a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-11-q01b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-11-q02", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2022-s-11-q03a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Express two binomial coefficients in terms of a parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this part challenged many candidates, who expanded unnecessarily or made index and coefficient errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 1 ], "tagConfidence": "reviewed", "answerGuide": "Use the general binomial term to identify only the requested powers, then equate their coefficients carefully." }, { "id": "cie-2022-s-11-q03b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(b): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-s-11-q04a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-s-11-q04b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the proved identity to solve a trigonometric equation over an open interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few correct solutions were seen; many divided by a trigonometric factor and lost valid solutions or included an endpoint outside the interval.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the identity, rearrange without dividing by a factor that may be zero, and check every solution against the open interval." }, { "id": "cie-2022-s-11-q05a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(a): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2022-s-11-q05b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact perimeter of a sector segment when its perpendicular chord length is given.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a very small minority noticed that the changed chord length implies a different central angle; using the earlier angle limits the available credit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Use the perpendicular geometry to determine the new angle, then add the two straight boundary lengths and the corresponding arc length." }, { "id": "cie-2022-s-11-q06a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-11-q06b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "6(b): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2022-s-11-q06c", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "6(c): Compose the two defined functions in the stated order.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute in the defined function rules in the correct order, then simplify the resulting expression." }, { "id": "cie-2022-s-11-q07a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-11-q07b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "7(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-s-11-q08a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the ordered transformations that convert the basic sine graph to a given transformed graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Almost all candidates made errors, especially in the horizontal scale, direction or translation vector.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Read the horizontal stretch, vertical stretch and translation from the transformed argument and amplitude, then state a valid sequence in the required order." }, { "id": "cie-2022-s-11-q08b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Solve a transformed sine equation exactly over the stated full-turn interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=16", "qpPages": [ 15, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was particularly challenging for some candidates, who rearranged in the wrong order or gave only one solution.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Isolate the sine term while reversing the horizontal transformation, solve the reference-angle equation, then list every exact solution in the interval." }, { "id": "cie-2022-s-11-q09a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-11-q09b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter values for which a line cuts a circle at two distinct points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=17", "qpPages": [ 17, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Only a minority reached a fully correct inequality; sign, expansion and discriminant errors were common in the final algebra.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the circle to form a quadratic in x, require a positive discriminant, and solve the resulting strict inequality." }, { "id": "cie-2022-s-11-q10a", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=18", "qpPages": [ 18, 18 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-s-11-q10b", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Determine the curve from derivative information and a stationary-point condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=18", "qpPages": [ 18, 18 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this one of the paper's most challenging questions; only a small number of fully correct solutions were seen, with many missing constants or not integrating twice.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the second derivative twice, include a constant at each stage, and use the given stationary-point data to determine the constants." }, { "id": "cie-2022-s-11-q10c", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(c): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=18", "qpPages": [ 18, 19 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-s-11-q10d", "paperId": "cie-2022-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(d)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(d): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_11.pdf#page=19", "qpPages": [ 18, 19 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-12-q01", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-s-12-q02", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-s-12-q03", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3: Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2022-s-12-q04a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-12-q04b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2022-s-12-q05a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine two unknowns from conditions that a line and a curve meet at two specified points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Both parts of Question 5 challenged many candidates; in this part many used a discriminant instead of substituting the two given points.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Substitute each stated point into the equality between the curve and line to form two simultaneous equations, then solve for the unknowns." }, { "id": "cie-2022-s-12-q05b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find parameter values for which a line is tangent to a curve under an alternative condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says both parts challenged many; candidates who missed the phrase ‘given instead’ reused the previous conditions, while correct solutions required a repeated-root condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Use only the new conditions, equate the line and curve, set the discriminant to zero, and solve the resulting quadratic in the parameter." }, { "id": "cie-2022-s-12-q06", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "6: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-s-12-q07a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-s-12-q07b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Calculate the shaded area in a sector diagram to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says the triangle-area step in this part was more challenging for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Find the sector area and the relevant triangle area using the diagram, subtract them in the correct order, and round only the final result." }, { "id": "cie-2022-s-12-q08a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-12-q08b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-s-12-q09a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-12-q09b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of a curve and classify its nature.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=17", "qpPages": [ 15, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says some candidates found solving the resulting equation challenging and forgot to find both coordinates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate and set the first derivative to zero, solve for x, substitute into the curve for y, then use the second derivative to classify the point." }, { "id": "cie-2022-s-12-q10a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the domain of the inverse of a rational function from its graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=18", "qpPages": [ 16, 16 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was one of the most challenging parts, with many candidates omitting it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Read the range of the given function from the graph, including any excluded value, and state that set as the inverse function's domain." }, { "id": "cie-2022-s-12-q10b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=18", "qpPages": [ 16, 16 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-12-q10c", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=18", "qpPages": [ 16, 16 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-12-q10d", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(d)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Explain why a specified function has no inverse function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=18", "qpPages": [ 17, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report notes a reasonable number omitted this part and many misidentified the issue as a negative square root instead of a failure of one-to-one behaviour.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "State that the function is not one-to-one on its domain, using the graph's horizontal-line test or the repeated outputs from a quadratic rule." }, { "id": "cie-2022-s-12-q10e", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(e)", "marks": 6, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Use a stated decomposition to find the area bounded by a tangent and the coordinate axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=19", "qpPages": [ 17, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says nearly one fifth omitted this part; many stopped after the preliminary identity or found the area under the curve instead of the tangent.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Use the provided decomposition to integrate the derivative or identify the tangent, find its intercepts, then calculate the enclosed triangle area." }, { "id": "cie-2022-s-12-q11a", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-12-q11b", "paperId": "cie-2022-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "trigonometry", "algebra", "proof" ], "primaryTopicId": "trigonometry", "summary": "Show that a trigonometric equation has no solutions when the parameter exceeds the stated threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_12.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_12.pdf#page=21", "qpPages": [ 19, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the paper's most challenging part; many omitted it and very few made a good attempt at the required quadratic-form argument.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Set u = cos²x, use 0 ≤ u ≤ 1, and show the resulting quadratic equation cannot have a valid root when the parameter is above the threshold." }, { "id": "cie-2022-s-13-q01", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-s-13-q02a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q02b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q02c", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q03a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-13-q03b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "3(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2022-s-13-q04a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q04b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the single transformation that maps one quadratic graph to another.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a minority gave a fully correct stretch description, including its scale factor and direction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 8 ], "tagConfidence": "reviewed", "answerGuide": "Compare the coefficients or completed-square forms to identify the invariant axis, scale factor and stretch direction." }, { "id": "cie-2022-s-13-q05a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q05b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-13-q06a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q06b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "6(b): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2022-s-13-q06c", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(c): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q06d", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(d): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q07a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-13-q07b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-13-q08a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-s-13-q08b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-s-13-q09a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "9(a): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2022-s-13-q09b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "9(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=12", "qpPages": [ 14, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2022-s-13-q10a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-13-q10b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-13-q11a", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Apply the tangency condition to determine the line and curve parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=13", "qpPages": [ 18, 18 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-s-13-q11b", "paperId": "cie-2022-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find where the normal at a tangent point meets the curve again.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_13.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_13.pdf#page=14", "qpPages": [ 19, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only a few candidates factorised the resulting quadratic with algebraic coefficients to obtain the second point.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 10 ], "tagConfidence": "reviewed", "answerGuide": "Find the tangent and normal gradients, substitute the normal equation into the curve, factor the intersection equation and select the second point." }, { "id": "cie-2022-s-21-q01a", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Show the gradient relation for a straight line formed by taking logarithms of an exponential model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct responses were seen; the report says candidates needed to take logarithms and identify the gradient before reaching the stated result.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 11 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms on both sides, rewrite as a linear function of x, and identify its gradient from the coefficient of x." }, { "id": "cie-2022-s-21-q01b", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Determine the integer parameter in the logarithmic model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Again, very few correct responses were seen; some candidates who had a method did not read the condition that the parameter is an integer.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 11 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the given data into the logarithmic relation, solve for the parameter and state the required integer." }, { "id": "cie-2022-s-21-q02a", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-21-q02b", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-21-q03a", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-s-21-q03b", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find the exact shaded area associated with the trigonometric curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says coefficient errors were common and few candidates obtained a fully correct solution; some differentiated instead of integrating.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 11 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the required trigonometric expression using a double-angle identity, substitute the bounds in radians and simplify exactly." }, { "id": "cie-2022-s-21-q04", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-21-q05a", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-s-21-q05b", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the larger intersection root satisfies a specified iterative equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Few candidates identified that the larger root must be associated with the right-hand branch of the modulus graph.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the larger root's defining equation, select the correct branch and substitute to verify the iterative relation." }, { "id": "cie-2022-s-21-q05c", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-21-q05d", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-s-21-q06a", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-s-21-q06b", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Use a given derivative result to prove uniqueness and find an exact stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few candidates supplied the required uniqueness argument; many also failed to give an exact solution or the y-coordinate.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Exclude zero factors explicitly, solve the remaining derivative equation exactly, and substitute the x-coordinate into the curve." }, { "id": "cie-2022-s-21-q07a", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-21-q07b", "paperId": "cie-2022-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the exact integral after decomposing a rational function into partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says very few correct responses were seen; candidates often omitted one partial-fraction term and made little progress after that.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the quotient from part (a) to write the complete decomposition, integrate every term and apply the limits exactly." }, { "id": "cie-2022-s-22-q01", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-s-22-q02a", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-s-22-q02b", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the modulus equation and select the valid root using the graph from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates used their graph to select the final answer, and most gave an invalid root in addition to the correct one.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 13 ], "tagConfidence": "reviewed", "answerGuide": "Solve the equation separately on each branch of the modulus graph, then keep only the intersection consistent with the graph." }, { "id": "cie-2022-s-22-q03", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-s-22-q04a", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(a): Apply the stated numerical procedure to approximate the required root or value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_22.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-s-22-q04b", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-22-q05a", "paperId": "cie-2022-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", 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"questionPart": "8(a)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_23.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-23-q08b", "paperId": "cie-2022-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the result from part (a) to solve a trigonometric equation in the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_23.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_23.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct solutions were seen; better candidates substituted the doubled angle into their earlier result, but many failed to complete the linked method accurately.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the amplitude-phase result from part (a) with the doubled angle, solve within the stated interval and give all required solutions." }, { "id": "cie-2022-s-31-q01", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-31-q02a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-s-31-q02b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "State the interval in which a binomial expansion is valid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates did not attempt this part, while others gave only part of the required set of values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Apply the binomial convergence condition to the expansion parameter and express the full interval for x." }, { "id": "cie-2022-s-31-q03", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-31-q04", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-31-q05a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-s-31-q05b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Factorise the cubic completely after determining its unknown coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates recognised that the repeated linear factor in the polynomial and its derivative implied a squared factor in the cubic.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Use the repeated-factor implication or divide by the repeated factor, then factorise the remaining quadratic completely." }, { "id": "cie-2022-s-31-q06a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an integral using a trigonometric substitution and its associated differential.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates attempted the required substitution; many who did failed to use the trigonometric identity needed to simplify the integrand.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Set x = tan θ, use dx = sec²θ dθ, simplify with 1 + tan²θ = sec²θ and transform the limits before integrating." }, { "id": "cie-2022-s-31-q06b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-31-q07a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-s-31-q07b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-s-31-q07c", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(c): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-s-31-q08a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-s-31-q08b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact tangent of the angle between two tangents to a curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=14", "qpPages": [ 15, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not know where to start, and among those who found both gradients very few knew how to determine the angle between the lines.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Find both tangent gradients at the intercept points, then use the tangent-of-angle formula for two line gradients." }, { "id": "cie-2022-s-31-q09a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-s-31-q09b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-s-31-q09c", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "vectors", "coordinate-geometry", "algebra" ], "primaryTopicId": "vectors", "summary": "Find the perpendicular distance from the origin to a line specified by two points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=15", "qpPages": [ 17, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this part was more challenging; successful candidates began by finding the foot of the perpendicular and then used the resulting geometry.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 21 ], "tagConfidence": "reviewed", "answerGuide": "Find a general point on the line, use a zero dot product with the line direction to locate the perpendicular foot, and calculate its distance from the origin." }, { "id": "cie-2022-s-31-q10a", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-s-31-q10b", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=16", "qpPages": [ 18, 19 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-s-31-q10c", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Justify why a specified tangent iteration converges to the intended root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=16", "qpPages": [ 19, 19 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The majority did not understand the key relation needed in this explanation, which the report identifies explicitly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 21 ], "tagConfidence": "reviewed", "answerGuide": "Show the relationship between the tangent values at supplementary angles and rearrange it into the stated fixed-point iteration." }, { "id": "cie-2022-s-31-q10d", "paperId": "cie-2022-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "10(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_31.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-s-32-q01", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-s-32-q02", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-32-q03", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-s-32-q04", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point of a trigonometric curve involving powers of sine and cosine.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this differentiation question was quite challenging; correct use of both product and chain rules was uncommon and algebraic simplification caused further errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate with product and chain rules, set the derivative to zero, use a double-angle or single-function identity, and solve in radians." }, { "id": "cie-2022-s-32-q05a", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Sketch the two graphs used to establish a unique root of an equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the sketch required for this numerical-methods question challenged many; common errors included omitting the asymptote or drawing the quadratic incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Sketch y = ln x with its vertical asymptote and the given quadratic with its intercepts, then show the intersection that represents the root." }, { "id": "cie-2022-s-32-q05b", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-32-q05c", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-s-32-q06a", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 7, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve a separable differential equation involving exponential terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report records a significant number who could not separate the exponential expression correctly, alongside repeated sign and integration errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Separate x and y factors first, integrate both sides, apply the initial condition and simplify the logarithmic/exponential relation." }, { "id": "cie-2022-s-32-q06b", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-s-32-q07a", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation to establish the derivative of a curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A significant minority were unfamiliar with implicit differentiation and could make no progress on this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 24 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate every term with respect to x, apply the product and chain rules to terms containing y, and collect dy/dx." }, { "id": "cie-2022-s-32-q07b", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-s-32-q08a", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-32-q08b", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-32-q09a", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the intersection of two vector lines to establish a relation between their parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this unfamiliar vector task challenged many candidates, particularly those without a clear elimination strategy.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 24 ], "tagConfidence": "reviewed", "answerGuide": "Write a general point on each line, equate corresponding components and eliminate the line parameters to obtain the required relation." }, { "id": "cie-2022-s-32-q09b", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2022-s-32-q09c", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=16", "qpPages": [ 14, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-s-32-q10a", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-s-32-q10b", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-s-32-q10c", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Sketch the locus of complex numbers at a fixed distance from a given complex centre.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=18", "qpPages": [ 17, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies the complex-number question's required sketch as a challenge for many candidates.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 25 ], "tagConfidence": "reviewed", "answerGuide": "Draw a circle centred at the given complex number with radius 2, using equal scales and indicating the centre and radius." }, { "id": "cie-2022-s-32-q10d", "paperId": "cie-2022-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(d)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(d): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_32.pdf#page=18", "qpPages": [ 16, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-33-q01", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2022-s-33-q02", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-33-q03a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "3(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-33-q03b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Solve the logarithmic equation and state the valid solution to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A considerable number found this part difficult; errors included solving for the wrong variable, retaining a root that should be rejected and rounding incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 27 ], "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic in the substituted logarithmic variable, convert back to x, reject the extraneous root and round to two decimal places." }, { "id": "cie-2022-s-33-q04a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-s-33-q04b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-s-33-q05a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-s-33-q05b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-s-33-q05c", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(c): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-s-33-q06a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-33-q06b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=16", "qpPages": [ 10, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-s-33-q07a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-33-q07b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=17", "qpPages": [ 12, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-s-33-q08a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve a separable differential equation for a time-varying population.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=18", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes this as a challenging differential-equation question and notes coefficient and sign errors in both separated terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 28 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate the population power and the cosine term, then use the initial population and time to determine the integration constant." }, { "id": "cie-2022-s-33-q08b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-s-33-q08c", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the maximum population value from the solution of a differential equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=19", "qpPages": [ 15, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says candidates found the algebra difficult and that a correct maximum was rarely obtained.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 28 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the parameter from part (b) into the population formula, determine where the oscillating term is maximal, and evaluate the resulting expression." }, { "id": "cie-2022-s-33-q09a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-s-33-q09b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2022-s-33-q09c", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=21", "qpPages": [ 16, 17 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-s-33-q10a", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "10(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=22", "qpPages": [ 18, 18 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-s-33-q10b", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "10(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=22", "qpPages": [ 18, 19 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-s-33-q10c", "paperId": "cie-2022-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "10(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_33.pdf#page=23", "qpPages": [ 18, 19 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-s-41-q01a", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q01b", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2022-s-41-q01c", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(c): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q02", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2: Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q03", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2022-s-41-q04", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4: Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2022-s-41-q05a", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q05b", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q05c", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the engine power at a stated point from speed and resistance information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report lists this part among the more demanding questions and says few attempted it; common errors came from using the wrong force to calculate power.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 30 ], "tagConfidence": "reviewed", "answerGuide": "Find the correct driving force at the point, calculate power as force times speed, then use the power to determine the required value." }, { "id": "cie-2022-s-41-q06a", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q06b", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "kinematics", "differentiation", "integration" ], "primaryTopicId": "kinematics", "summary": "Determine a parameter from an acceleration condition and calculate total distance over a two-second motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was rather challenging, with some omissions and relatively few fully correct solutions; candidates often used incorrect limits or stopped after finding a turning time.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 30 ], "tagConfidence": "reviewed", "answerGuide": "Find the time when velocity is zero, differentiate to obtain acceleration and solve for the parameter, then split the distance integral at the turning point." }, { "id": "cie-2022-s-41-q07a", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-41-q07b", "paperId": "cie-2022-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed of the joined particles after a second collision on a smooth incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_41.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part very challenging and says many candidates omitted it; only very few correctly found both pre-collision velocities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 31 ], "tagConfidence": "reviewed", "answerGuide": "Track each particle's motion between collisions with signed acceleration, determine their velocities just before impact, then apply conservation of momentum." }, { "id": "cie-2022-s-42-q01a", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-42-q01b", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-s-42-q02", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2: Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2022-s-42-q03a", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-42-q03b", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-42-q04a", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Set up the positions of two particles when one begins moving after a time delay.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 4 was the most challenging question for most candidates; many misread the four-second delay or used the wrong particle as the time reference.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Choose a single time origin, express each displacement using the correct elapsed time for that particle, and include the four-second offset." }, { "id": "cie-2022-s-42-q04b", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find when the two delayed particles are the same distance from the origin.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies Question 4 as the paper's most challenging question; errors in the time definition made the resulting equation difficult to solve.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Use the displacement expressions and the correct shared time variable, equate the required distances and solve the resulting equation." }, { "id": "cie-2022-s-42-q04c", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "kinematics", "data-analysis" ], "primaryTopicId": "kinematics", "summary": "Sketch and label the two displacement-time graphs for the delayed motions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates produced fully correct graphs; common errors included a straight-line sketch for the quadratic path and missing key axis labels.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Draw the linear displacement for one particle and the piecewise quadratic displacement for the other, marking the origin, crossing time and relevant intervals." }, { "id": "cie-2022-s-42-q05", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5: Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2022-s-42-q06a", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Calculate work against resistance for a car travelling uphill at constant power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies Question 6 as among the most challenging and says many incorrectly used constant-acceleration equations despite the changing driving force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Use work-energy: work by the engine equals kinetic-energy gain, potential-energy gain and work against resistance; solve for the resistive work." }, { "id": "cie-2022-s-42-q06b", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-42-q07a", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2022-s-42-q07b", "paperId": "cie-2022-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance from a velocity function that changes sign during the interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_42.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_42.pdf#page=17", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "This part was among the most challenging; most candidates integrated over the negative-velocity interval without changing the sign to obtain distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 34 ], "tagConfidence": "reviewed", "answerGuide": "Find the zeroes of velocity, integrate on each interval, take the absolute values of the signed displacements and add them." }, { "id": "cie-2022-s-43-q01a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q01b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q02a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q02b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the duration for which the projectile's speed is at least the given threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part was more challenging; candidates often found the time below the threshold or misread the ‘at least’ condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 35 ], "tagConfidence": "reviewed", "answerGuide": "Use constant-acceleration equations to find the two threshold-speed times, then subtract them to obtain the full duration at or above the threshold." }, { "id": "cie-2022-s-43-q03a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q03b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q03c", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(c): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q04a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2022-s-43-q04b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the greatest applied force that keeps a suspended block in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says a significant number did not attempt this more challenging part; it required identifying which string becomes slack at the limiting force.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 36 ], "tagConfidence": "reviewed", "answerGuide": "At the limiting case set the relevant string tension to zero, replace the original force in the resolved equations and solve for the new force." }, { "id": "cie-2022-s-43-q05a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-s-43-q05b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-s-43-q05c", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(c): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-s-43-q06a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-s-43-q06b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the friction coefficient when two connected particles are at rest under a changed tension.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=9", "qpPages": [ 11, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report identifies this as one of the most challenging parts; candidates often reused the earlier tension, omitted a force or chose the wrong friction direction.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Recalculate the tension for the equilibrium case, resolve along and perpendicular to each plane, test the friction direction and use F = μR." }, { "id": "cie-2022-s-43-q07a", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=10", "qpPages": [ 12, 12 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-43-q07b", "paperId": "cie-2022-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find distance travelled over a time interval from a velocity function with a turning point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_43.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_43.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies this as one of the most challenging parts and notes that candidates often found displacement instead of splitting the distance integral at the zero of velocity.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Split the integral at t = 5, integrate on each side, and add the magnitudes of the two displacements." }, { "id": "cie-2022-s-51-q01a", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-51-q01b", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-51-q02a", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-51-q02b", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-51-q03a", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-51-q03b", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-51-q03c", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-51-q03d", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(d): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-51-q04a", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-51-q04b", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-51-q04c", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-51-q04d", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a geometric-distribution probability from repeated four-coin trials.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part challenged some candidates, who inserted a binomial coefficient or failed to combine the separate geometric outcomes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Identify the per-trial success probability for the specified event, use the geometric model and sum the required disjoint cases." }, { "id": "cie-2022-s-51-q05a", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-s-51-q05b", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-51-q05c", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Estimate the expected number of leaves more than one standard deviation from the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the majority found this part challenging; many calculated only one tail or counted leaves within one standard deviation instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 40 ], "tagConfidence": "reviewed", "answerGuide": "Use the two tails beyond z = −1 and z = 1, add their probabilities, multiply by the number of leaves and keep table precision until the final integer." }, { "id": "cie-2022-s-51-q06a", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-51-q06b", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-51-q06c", "paperId": "cie-2022-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_51.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2022-s-52-q01", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Recover the sample size from coded-sum information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this coding question challenged some candidates; very few wrote the expansion needed to connect the coded total with the unknown n.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Expand the coded summation into Σx − 200n, equate it to the supplied total and solve for n." }, { "id": "cie-2022-s-52-q02a", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-52-q02b", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-52-q03a", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2022-s-52-q03b", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "3(b): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2022-s-52-q03c", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-52-q04a", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a normally distributed bag weight lies between two bounds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Question 4's context challenged many candidates; this part caused errors from treating the interval as symmetric or using the variance as the standard deviation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Standardise each endpoint using σ = √variance, find both cumulative probabilities and subtract to obtain the interval probability." }, { "id": "cie-2022-s-52-q04b", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation of a normal distribution from a tail probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the context in Question 4 challenged many candidates; this part required translating the tail probability into a z-value and solving for the unknown spread.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Convert the stated tail probability to a standard normal z-value, equate the standardisation formula to it and solve for σ." }, { "id": "cie-2022-s-52-q05a", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-52-q05b", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-52-q06a", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-52-q06b", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-52-q06c", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count four-letter selections where the counts of two repeated letters differ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates found this part challenging or omitted it; frequent errors came from treating repeated letters as distinct or leaving out valid count scenarios.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 43 ], "tagConfidence": "reviewed", "answerGuide": "List the allowed combinations of C and O counts that differ, account for remaining letters, and add the counts for all valid cases." }, { "id": "cie-2022-s-52-q06d", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count divisions of the repeated-letter word into three groups when the Cs must be separated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most demanding part of the paper; most candidates extended an earlier method without accounting for how the two Os can be distributed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 44 ], "tagConfidence": "reviewed", "answerGuide": "Separate the possible placements of the two Os across groups, count each valid scenario with repeated letters treated as indistinguishable, and add the cases." }, { "id": "cie-2022-s-52-q07a", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-52-q07b", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2022-s-52-q07c", "paperId": "cie-2022-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that at least one child receives an orange sweet when sweets are chosen without replacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_52.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_52.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says a significant number omitted this part, and successful solutions used the complement of drawing no orange sweets.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 44 ], "tagConfidence": "reviewed", "answerGuide": "Use 1 minus the probability of selecting three sweets from the non-orange set, or count the valid orange-count scenarios without replacement." }, { "id": "cie-2022-s-53-q01a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "1(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2022-s-53-q01b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2022-s-53-q02a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-53-q02b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2022-s-53-q02c", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2022-s-53-q03a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-53-q03b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-53-q04a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-53-q04b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-53-q04c", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-53-q05a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a probability for apple weights lying between two stated thresholds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies finding the correct normal-probability area as the more challenging part; many subtracted the wrong cumulative area and did not check whether the result was plausible.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 46 ], "tagConfidence": "reviewed", "answerGuide": "Use the z-values for both thresholds, convert the lower-tail value using symmetry where needed, and take the difference of the correct cumulative probabilities." }, { "id": "cie-2022-s-53-q05b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-s-53-q05c", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(c): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-s-53-q06a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-53-q06b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2022-s-53-q06c", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-53-q07a", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-53-q07b", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-53-q07c", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-53-q07d", "paperId": "cie-2022-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(d): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_53.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2022-s-61-q01a", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 6, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-61-q01b", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2022-s-61-q02", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Test whether a die player can predict outcomes using a binomial model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates found this hypothesis test challenging; few formed the correct binomial tail or made a valid comparison with the significance level.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 48, 49 ], "tagConfidence": "reviewed", "answerGuide": "State hypotheses for the success probability, calculate the appropriate binomial tail, compare it with 0.10 and give a contextual conclusion with uncertainty." }, { "id": "cie-2022-s-61-q03", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-61-q04a", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-61-q04b", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a joint probability involving a Poisson variable and a binomial variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was particularly challenging; many could not identify the required joint outcomes and others tried an unsuitable normal model.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 49 ], "tagConfidence": "reviewed", "answerGuide": "List the joint outcomes that satisfy the event, calculate each independent Poisson-binomial probability and add them." }, { "id": "cie-2022-s-61-q05a", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-61-q05b", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-s-61-q05c", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-61-q06a", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2022-s-61-q06b", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-s-61-q06c", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(c): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-61-q06d", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Calculate a probability from the area under the density graph in the final part of the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates were successful and a significant number made no attempt; diagrams often helped identify the required region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 49 ], "tagConfidence": "reviewed", "answerGuide": "Sketch the density, identify the requested interval, and calculate the corresponding area using the uniform density or the appropriate geometric decomposition." }, { "id": "cie-2022-s-61-q07a", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-61-q07b", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-61-q07c", "paperId": "cie-2022-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_61.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-62-q01a", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-62-q01b", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-62-q02", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "2: Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2022-s-62-q03a", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-s-62-q03b", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-s-62-q04a", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "4(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2022-s-62-q04b", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region for a Poisson test at the stated significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many found this part challenging; it was necessary to compare two cumulative probabilities, and many gave only a probability instead of the critical region.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 52 ], "tagConfidence": "reviewed", "answerGuide": "Calculate P(X ≤ 3) and P(X ≤ 4) under the null model, compare both with the significance level, and state the largest valid critical region." }, { "id": "cie-2022-s-62-q04c", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "4(c): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2022-s-62-q04d", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(d): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2022-s-62-q04e", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(e)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(e): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-s-62-q05a", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-s-62-q05b", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-s-62-q05c", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find a probability involving the median of a continuous density function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates found this part challenging; approaches that tried to locate the median first were less successful than using the median's half-area property.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_er.pdf", "sourceEvidencePages": [ 52 ], "tagConfidence": "reviewed", "answerGuide": "Use the fact that half the probability lies below the median, then integrate or subtract the appropriate density area to obtain the requested probability." }, { "id": "cie-2022-s-62-q06a", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-62-q06b", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-62-q07", "paperId": "cie-2022-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_62.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-63-q01", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-s-63-q02a", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-63-q02b", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-63-q02c", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2022-s-63-q03a", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-63-q03b", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "3(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2022-s-63-q03c", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-63-q04a", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-63-q04b", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-63-q05a", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-63-q05b", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-s-63-q05c", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(c): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-s-63-q06a", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-s-63-q06b", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=10", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-s-63-q07ai", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a)(i): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-s-63-q07aii", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a)(ii): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-s-63-q07aiii", "paperId": "cie-2022-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level 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"qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s22_ms_63.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-11-q01", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-11-q02a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-w-11-q02b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "2(b): Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2022-w-11-q03", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-11-q04", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "4: Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-w-11-q05a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of the shaded circular segment from the diagram and arc length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates assumed the chord equalled the radius, an error the report says restricted marks in this part and affected the follow-up.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Determine the radius and central angle from the given perimeter and arc length, find the chord, then add it to the arc length." }, { "id": "cie-2022-w-11-q05b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "5(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2022-w-11-q06a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-w-11-q06b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-11-q07a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "7(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-w-11-q07b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "7(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-w-11-q07c", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "7(c): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-w-11-q08a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(a): Use the derivative sign to find the least integer boundary for an increasing function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor the derivative, determine where it is positive, and choose the smallest integer n that makes the whole domain x > n lie in that interval." }, { "id": "cie-2022-w-11-q08b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-11-q08c", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-11-q09a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-11-q09b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-11-q09c", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Explain whether the stated quadratic function has an inverse on its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates often missed that the minimum point prevents one-to-one behaviour, so correct explanations were relatively rare.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Use the minimum point and restricted domain to decide whether the function is one-to-one, then state the implication for an inverse." }, { "id": "cie-2022-w-11-q09d", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(d)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the transformations connecting the two function graphs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Fully correct descriptions were rarely seen; the report emphasizes that accurate transformation language was needed.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Compare the completed-square forms and state the horizontal and vertical transformations in a clear order." }, { "id": "cie-2022-w-11-q10a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(a): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=14", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-11-q10b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the angle between tangents to two curves at the stated points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report describes this as very challenging for all but the strongest candidates and outlines several tangent-geometry methods.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate each curve to get the tangent gradients, then find the angle between the two lines using their inclinations or the tangent-angle formula." }, { "id": "cie-2022-w-11-q11a", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "11(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-11-q11b", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "11(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=16", "qpPages": [ 18, 19 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-11-q11c", "paperId": "cie-2022-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the required angle in the final circle-geometry part.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_11.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says almost all candidates found this final angle question very challenging; only a small number showed a workable method.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Use the circle and tangent angle relations to obtain the intermediate angles, then combine them to find the requested angle." }, { "id": "cie-2022-w-12-q01a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "1(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-12-q01b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "1(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-12-q02", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2022-w-12-q03a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-12-q03b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-12-q04", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-w-12-q05a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-12-q05b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the transformed function rule after the specified graph transformations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates missed this part and often did not understand the required function notation.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Apply the horizontal stretch and translation to the input and output of the original function, preserving the order stated in the question." }, { "id": "cie-2022-w-12-q06a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-12-q06b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-12-q06c", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch the quadratic using the completed-square information from the earlier part.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates omitted the sketch or gave the wrong orientation or intercept interpretation; the report recommends practice on this type of task.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Read the vertex and orientation from the completed-square form, then draw and label the corresponding curve rather than plotting arbitrary points." }, { "id": "cie-2022-w-12-q07a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the stated trigonometric identity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a small number of candidates produced a convincing proof after reaching the common-denominator step.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Start from one side, combine the fractions with a common denominator, and use the Pythagorean identity to simplify to the other side." }, { "id": "cie-2022-w-12-q07b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-12-q08a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(a): Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2022-w-12-q08b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-12-q08c", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Use the stationary-point result to state the range of x satisfying the required condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "About a third of candidates omitted this part, and many who attempted it chose the wrong side of the critical value.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Use the derivative-zero boundary found in part (b), then choose the side of that value that makes the required derivative condition true." }, { "id": "cie-2022-w-12-q09a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-12-q09b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-12-q09c", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-12-q09d", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(d): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-12-q09e", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(e)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Explain the final property of the composite or inverse-function construction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most challenging part on the paper and says only the very best candidates gave a satisfactory explanation.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Use the function's stated domain and range restrictions to justify the requested conclusion rather than relying on a graph-test label alone." }, { "id": "cie-2022-w-12-q10a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(a): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2022-w-12-q10b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=15", "qpPages": [ 16, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2022-w-12-q10c", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(c): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=15", "qpPages": [ 16, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2022-w-12-q11a", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-12-q11b", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "11(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=17", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-12-q11c", "paperId": "cie-2022-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(c): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_12.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_12.pdf#page=18", "qpPages": [ 19, 19 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q01", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "1: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-13-q02a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-13-q02b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "2(b): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2022-w-13-q02c", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(c): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-13-q03a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "3(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-w-13-q03b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "3(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-w-13-q03c", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(c): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2022-w-13-q04", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4: Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2022-w-13-q05a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the scaled ellipse information from the stated graph transformation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This part challenged many candidates; the report notes frequent incorrect scale factors and reversed division.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 9 ], "tagConfidence": "reviewed", "answerGuide": "Apply the stated stretch to the ellipse dimensions and keep the scale direction and factor consistent." }, { "id": "cie-2022-w-13-q05b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q05c", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(c): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q05d", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(d): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q06", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-13-q07a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-w-13-q07b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-13-q08a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8(a): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2022-w-13-q08b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-13-q09a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(a): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-w-13-q09b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2022-w-13-q10a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=12", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q10b", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the area bounded by the curve and the geometric boundary shown.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found this more challenging than the preceding part, often choosing the wrong integral because a straight line also appeared in the diagram.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 10 ], "tagConfidence": "reviewed", "answerGuide": "Identify which curve expression is required, use the intersection limits from the circle, and integrate the correct function over those limits." }, { "id": "cie-2022-w-13-q10c", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Find an exact perimeter expression for the shaded region in the circle diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=13", "qpPages": [ 14, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this part was very challenging; common errors included using the gradient as an angle and converting exact work to decimals.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 10 ], "tagConfidence": "reviewed", "answerGuide": "Use the circle and intersection geometry to find the straight boundary lengths and the arc angle, then add the straight lengths to the exact arc length in radians." }, { "id": "cie-2022-w-13-q11a", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q11bi", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)(i)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b)(i): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=14", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-13-q11bii", "paperId": "cie-2022-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2022, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)(ii)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b)(ii): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_13.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-21-q01", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2022-w-21-q02", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation using logarithms to the stated accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few correct solutions were seen; the report identifies errors in logarithm manipulation and rearrangement.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides, apply the power rule correctly, collect the terms in x and solve the resulting linear equation." }, { "id": "cie-2022-w-21-q03", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Find the exact compound-angle value from a given trigonometric ratio.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many treated this as a calculator exercise and received no credit without the identity and compound-angle method.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 12, 13 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity to find the exact tangent value in the stated quadrant, then apply the compound-angle formula." }, { "id": "cie-2022-w-21-q04a", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-21-q04b", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-21-q05", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-w-21-q06a", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-21-q06b", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-w-21-q07a", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-21-q07b", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Show that the stationary point is a maximum using a sign change.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few correct responses were seen; many used the curve equation instead of its gradient function.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 13 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the interval endpoints into the derivative function, show the sign changes, and state what that implies about the stationary point." }, { "id": "cie-2022-w-21-q07c", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Apply the stated numerical procedure to approximate the required root or value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-21-q08a", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the trigonometric expression in amplitude-phase form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few correct solutions were seen; many candidates missed that the expression involved double angles before forming the required single-angle form.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Use double-angle identities first, collect the sine and cosine terms, then express the result as a single shifted cosine." }, { "id": "cie-2022-w-21-q08b", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the amplitude-phase form to solve the linked trigonometric equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Of the few correct results in part (a), few candidates completed this linked trigonometric solution correctly.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the result from part (a), solve the cosine equation on the given interval, and check both branches." }, { "id": "cie-2022-w-21-q08c", "paperId": "cie-2022-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_21.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-22-q01", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-22-q02", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-22-q03", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-w-22-q04a", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-w-22-q04b", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-w-22-q05a", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Show the equation satisfied by the stationary point of the logarithmic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few candidates completed this proof; the report gives the needed rearrangement as the key step.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 16 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve implicitly or explicitly as given, set the derivative to zero, and rearrange to the stated equation." }, { "id": "cie-2022-w-22-q05b", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-22-q05c", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-22-q06", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 9, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "6: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-22-q07a", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-w-22-q07b", "paperId": "cie-2022-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Use the gradient function to establish the nature of the stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_22.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says fewer correct methods and solutions were seen in this part than in the earlier stationary-point work.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the gradient function at the two supplied x-values and show the sign change that establishes a maximum." }, { "id": "cie-2022-w-23-q01", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2022-w-23-q02", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation by logarithmic manipulation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes few correct solutions because candidates struggled with logarithm manipulation and algebraic rearrangement.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms on both sides, apply the power rule and solve the resulting linear equation carefully." }, { "id": "cie-2022-w-23-q03", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Find the exact compound-angle value from the given secant information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A significant number used a calculator instead of the required identities and compound-angle method, losing all credit.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 18, 19 ], "tagConfidence": "reviewed", "answerGuide": "Derive the tangent value from the identity and quadrant, then apply the compound-angle formula to obtain an exact result." }, { "id": "cie-2022-w-23-q04a", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-23-q04b", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-23-q05", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-w-23-q06a", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-23-q06b", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-w-23-q07a", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-23-q07b", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Use a derivative sign change to identify the maximum point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few correct responses were seen; many substituted into the curve instead of its gradient function.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the gradient function at the supplied endpoints, demonstrate a sign change, then state the maximum conclusion." }, { "id": "cie-2022-w-23-q07c", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Apply the stated numerical procedure to approximate the required root or value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-23-q08a", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Express the doubled-angle trigonometric expression in amplitude-phase form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few correct solutions were seen; many candidates missed how the double angles fitted the requested form.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Apply double-angle identities, combine the sine and cosine terms, and write the result with the specified shifted double angle." }, { "id": "cie-2022-w-23-q08b", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the linked trigonometric equation using the result from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few correct responses to this part followed the already uncommon correct result in part (a).", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the amplitude-phase result, solve the resulting cosine equation, and follow the required sequence of angle steps." }, { "id": "cie-2022-w-23-q08c", "paperId": "cie-2022-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2022, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_23.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-31-q01a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-31-q01b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2022-w-31-q02", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2: Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-31-q03", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-w-31-q04", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-31-q05a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2022-w-31-q05b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Interpret the complex-number geometry and determine the required argument quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few candidates showed understanding of what this part required; numerous responses were blank or guesses.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Use the diagram to identify the relevant point and angle, then select the correct argument in the stated range." }, { "id": "cie-2022-w-31-q06a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-w-31-q06b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-31-q07a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-w-31-q07b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2022-w-31-q07c", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-31-q08a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-31-q08b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Continue the differential-equation solution to the requested value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this continuation was a challenge and many scripts left it blank, often after an incomplete first part.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate both sides with the needed trigonometric forms, apply the initial condition and present the requested exact or decimal form." }, { "id": "cie-2022-w-31-q09a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2022-w-31-q09b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-31-q10a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-31-q10b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use binomial expansions to obtain the requested coefficient or term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=15", "qpPages": [ 16, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This part proved challenging, with many candidates not attempting the binomial expansion and others making expansion errors.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Expand the specified factors only as far as needed, retain the correct coefficients, and multiply terms to collect the requested power." }, { "id": "cie-2022-w-31-q11a", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-w-31-q11b", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=16", "qpPages": [ 18, 19 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-w-31-q11c", "paperId": "cie-2022-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 4, "topicIds": [ "vectors", "coordinate-geometry", "algebra" ], "primaryTopicId": "vectors", "summary": "Complete the final vector-intersection calculation for the line construction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_31.pdf#page=17", "qpPages": [ 18, 19 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates offered no solution to this part and only a small number made progress.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Form vector equations for the lines with distinct parameters, equate components and solve the simultaneous equations." }, { "id": "cie-2022-w-32-q01", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-w-32-q02a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2022-w-32-q02b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Use the factorization to establish the sign of the cubic expression over the required domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a minority gained full marks; many tried to solve a cubic with the quadratic formula or assumed the quadratic factor had real roots.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 25 ], "tagConfidence": "reviewed", "answerGuide": "Factor out the known linear term, analyze the quadratic factor and combine the signs of the factors over the stated interval." }, { "id": "cie-2022-w-32-q03", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-32-q04a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-32-q04b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-32-q05a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-32-q05b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(b): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-32-q05c", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-32-q05d", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(d)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(d): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-32-q06a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "6(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-w-32-q06b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "vectors", "trigonometry", "coordinate-geometry" ], "primaryTopicId": "vectors", "summary": "Find the exact triangle area using the angle relation from the vector work.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates could not obtain an exact value; only a minority recalled how to evaluate the required inverse-trigonometric expression exactly.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 26 ], "tagConfidence": "reviewed", "answerGuide": "Use the exact sine of the included angle with the two side lengths, and retain exact values rather than substituting a decimal angle." }, { "id": "cie-2022-w-32-q07a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2022-w-32-q07b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "differentiation", "integration", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Separate and integrate the differential equation using the result from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=16", "qpPages": [ 10, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the integral involving tan squared over sin squared was challenging, with relatively few candidates recognizing its standard form.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 26 ], "tagConfidence": "reviewed", "answerGuide": "Separate the variables, use the derivative result and standard trigonometric integrals, then apply the given condition to determine the constant." }, { "id": "cie-2022-w-32-q08a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=17", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-w-32-q08b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=17", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-32-q09a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "9(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=18", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-w-32-q09b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(b): Integrate the stated function to find the requested expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2022-w-32-q09c", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "9(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=19", "qpPages": [ 14, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-32-q10a", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-32-q10b", "paperId": "cie-2022-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the remaining partial-fraction term in exact form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_32.pdf#page=20", "qpPages": [ 16, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The second integral proved more challenging; many candidates used the wrong inverse-tangent form or treated the rational term as a logarithm.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 27 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the quadratic denominator into the standard arctangent form, preserve the scale factor, and combine the logarithmic and inverse-tangent terms." }, { "id": "cie-2022-w-33-q01", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2022-w-33-q02", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2: Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-w-33-q03", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-33-q04", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric equations and obtain the required derivative relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "About half of candidates mishandled the chain rule in dy/dt, commonly omitting the factor arising from the inner angle.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 29 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to the parameter, apply the chain rule to dy/dx and simplify with the trigonometric identities shown by the question." }, { "id": "cie-2022-w-33-q05a", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "5(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2022-w-33-q05b", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Locate the complex-number point that maximizes the argument in the stated region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part challenged candidates even when their diagram was correct; many made no attempt because they could not locate the maximizing point.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 29 ], "tagConfidence": "reviewed", "answerGuide": "Use the Argand diagram to identify the boundary point, form the relevant right triangle and calculate the extremal argument geometrically." }, { "id": "cie-2022-w-33-q06", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the quadratic equation with complex coefficients and give roots in Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the equation looked unfamiliar; all but a few exceptional candidates using real and imaginary parts made no further progress.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 29, 30 ], "tagConfidence": "reviewed", "answerGuide": "Use the complex quadratic formula, simplify the discriminant, and rationalize the complex denominator while showing each algebraic step." }, { "id": "cie-2022-w-33-q07a", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2022-w-33-q07b", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2022-w-33-q08a", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "8(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-33-q08b", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2022-w-33-q08c", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "8(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2022-w-33-q09a", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-w-33-q09b", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2022-w-33-q09c", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection point of the two vector lines using simultaneous equations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only about a third of candidates earned all marks in the vector section, with many carrying earlier vector errors into the final line-intersection work.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 30 ], "tagConfidence": "reviewed", "answerGuide": "Write both line equations with different parameters, equate corresponding components and solve for the point of intersection." }, { "id": "cie-2022-w-33-q10a", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-33-q10b", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "differentiation", "integration", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the variable-separable differential equation and apply the initial condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this very challenging for many candidates; a significant number did not attempt it and common errors lost later marks.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 31 ], "tagConfidence": "reviewed", "answerGuide": "Separate the variables, integrate both sides, evaluate the arbitrary constant from the initial values and rearrange for the dependent variable." }, { "id": "cie-2022-w-33-q10c", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the solution expression to determine its limiting behaviour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only about one in ten candidates produced the correct expression before this part, and the report notes continued rearrangement difficulties.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 31 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange the expression to make the dependent variable the subject, then evaluate the limit as time increases." }, { "id": "cie-2022-w-33-q11a", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=15", "qpPages": [ 16, 16 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-33-q11b", "paperId": "cie-2022-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2022, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_33.pdf#page=15", "qpPages": [ 16, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2022-w-41-q01", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "1: Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2022-w-41-q02", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the collision data to find the post-collision speeds and kinetic-energy change.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Only a minority used momentum correctly and many interpreted the stated speed difference in a way incompatible with the collision direction.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 32 ], "tagConfidence": "reviewed", "answerGuide": "Apply conservation of momentum with signed velocities, use the stated speed difference to solve, then compare initial and final kinetic energies." }, { "id": "cie-2022-w-41-q03a", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-41-q03b", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-41-q03c", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(c): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-41-q04a", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 7, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find the tension in the connected-particle system on a rough inclined plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a small number of fully correct responses were seen; common methods omitted a force component or used Newton's second law perpendicular to the plane.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and perpendicular to the plane, include the tension component in the normal reaction and use friction proportional to the reaction." }, { "id": "cie-2022-w-41-q04b", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-41-q05a", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2022-w-41-q05b", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled while velocity changes sign.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report explains that candidates often integrated over the full interval as displacement instead of splitting at the two velocity zeros.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Integrate velocity over the three intervals separated by its zeros, take the magnitude of each displacement and add them." }, { "id": "cie-2022-w-41-q06a", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "6(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2022-w-41-q06b", "paperId": "cie-2022-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed when the pulley particles reach the same horizontal level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_41.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this more challenging and notes relatively few fully correct solutions; many omitted it or did not find the equal-height displacement.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 33 ], "tagConfidence": "reviewed", "answerGuide": "Use the string geometry to find the distance at equal heights, then equate the total loss of gravitational potential energy to the total gain in kinetic energy." }, { "id": "cie-2022-w-42-q01", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q02a", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2022-w-42-q02b", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2(b): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2022-w-42-q03", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2022-w-42-q04a", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-w-42-q04b", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the final speed using the constant-power information.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "This part was very challenging for many; most tried constant acceleration even though the constant power makes acceleration variable.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 35 ], "tagConfidence": "reviewed", "answerGuide": "Use work done by the engine as power multiplied by time, calculate the kinetic-energy change and equate the two energy quantities." }, { "id": "cie-2022-w-42-q05a", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q05b", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=13", "qpPages": [ 6, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q06a", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q06b", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q06c", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the time from the B–C collision until A collides with the combined particle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=15", "qpPages": [ 8, 9 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This part was very challenging for the vast majority; many did not attempt it or could not devise a starting strategy.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 35 ], "tagConfidence": "reviewed", "answerGuide": "Use the particles' velocities after the first collision and after B and C coalesce, determine their separation at the B–C impact, then equate their subsequent displacements to find when they meet." }, { "id": "cie-2022-w-42-q07a", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=18", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q07b", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=19", "qpPages": [ 10, 10 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-42-q07c", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the time when the particle comes to rest using the velocity expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=20", "qpPages": [ 10, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Candidates who had the correct velocity expression often succeeded, but many errors came from omitting the integration constant in the earlier part.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 36 ], "tagConfidence": "reviewed", "answerGuide": "Set the velocity function from part (b) equal to zero and solve for the positive time, retaining the required accuracy." }, { "id": "cie-2022-w-42-q07d", "paperId": "cie-2022-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the total distance over the two phases of the particle's motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_42.pdf#page=21", "qpPages": [ 10, 11 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this part fairly challenging and says many candidates lacked a suitable second-phase expression to integrate.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 36 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity expression separately before and after the velocity reaches zero, use the appropriate limits and add the magnitudes." }, { "id": "cie-2022-w-43-q01a", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-43-q01b", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-43-q02a", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-w-43-q02b", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-w-43-q03", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Resolve the forces on the ring at the contact point to determine the required force quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says success depended on identifying the reaction direction; it was often missing or drawn incorrectly, making the force resolution fail.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 37, 38 ], "tagConfidence": "reviewed", "answerGuide": "Draw the normal reaction perpendicular to the tangent and away from the centre, then resolve the tension, weight and reaction consistently." }, { "id": "cie-2022-w-43-q04a", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the initial speed and acceleration from distances travelled during specified seconds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the question was frequently misread; many used displacement from the start instead of the difference across the named second.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 38 ], "tagConfidence": "reviewed", "answerGuide": "Express each stated distance in a given second as the difference of displacements at its endpoints, form two equations and solve for the initial conditions." }, { "id": "cie-2022-w-43-q04b", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-43-q05a", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the collision time and separation for particles with time-dependent acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number used constant-acceleration formulas instead of integration, and common errors included adding displacements or using only one particle's displacement.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 38 ], "tagConfidence": "reviewed", "answerGuide": "Integrate each acceleration to obtain velocity and displacement, use equal velocities to find the collision time, and subtract positions for the separation." }, { "id": "cie-2022-w-43-q05b", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-43-q06ai", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a)(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-w-43-q06aii", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a)(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2022-w-43-q06b", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-43-q07a", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2022-w-43-q07b", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the motion after collision, including the time or distance condition in the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates found this challenging and often ignored the velocity change at impact or introduced acceleration during the constant-speed phase.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Use constant acceleration before impact, conserve momentum at the collision, then apply the stated constant-speed condition to the next phase." }, { "id": "cie-2022-w-43-q07c", "paperId": "cie-2022-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2022, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(c): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_43.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2022-w-51-q01", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-51-q02a", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-51-q02b", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-51-q03a", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-51-q03b", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2022-w-51-q03c", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare the two basketball teams' height distributions in context.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates found this comparison challenging and needed contextual comparisons of spread and central tendency.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Compare both central tendency and spread using the stem-and-leaf data, then state which team is generally taller and/or more variable." }, { "id": "cie-2022-w-51-q04a", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-51-q04b", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-w-51-q04c", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that all three adults satisfy the stated blood-pressure condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many solutions stopped after finding the individual probability and did not continue to the probability for all three adults.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Use symmetry of the normal distribution to find one adult's probability, then raise that probability to the third power for independent adults." }, { "id": "cie-2022-w-51-q05a", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-51-q05b", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-51-q05c", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether the two events in the dice game are independent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many found this part challenging and successful methods explicitly compared P(A)P(B) with P(A∩B).", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Compare P(A∩B) with P(A)P(B), using the values obtained from the stated game outcomes, and state the conclusion." }, { "id": "cie-2022-w-51-q05d", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-51-q06a", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-51-q06b", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the ways to divide all 15 club members into groups of 3, 5 and 7.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates found this part challenging and often treated each group as if it were selected again from all 15 members.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Choose the first group from all 15, the second from the remaining 12 and assign the remaining members to the last group; the group sizes distinguish the groups." }, { "id": "cie-2022-w-51-q06c", "paperId": "cie-2022-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the back-row arrangements with Abel and Betty together but Freya and Gino apart.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_51.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was the most challenging question and only a small number obtained the correct final answer.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Treat Abel and Betty as one block, arrange it with the other five people, subtract cases where Freya and Gino are also adjacent, and account for both orders within each pair." }, { "id": "cie-2022-w-52-q01a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-52-q01b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the required conditional probability for the travel-to-school scenario.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this conditional probability caused more difficulty; candidates often gave an unsupported numerator or confused the complement in the denominator.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 44 ], "tagConfidence": "reviewed", "answerGuide": "Use the total probability from part (a), identify the correct joint event and divide by the conditioning probability." }, { "id": "cie-2022-w-52-q02a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2022-w-52-q02b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-w-52-q03a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-52-q03b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use the geometric model to find the probability associated with the specified waiting time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes repeated errors in reading the success criterion, including an incorrect exponent in the geometric probability.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 44 ], "tagConfidence": "reviewed", "answerGuide": "Calculate the success probability from one three-dice trial, then use the geometric distribution with the correct number of failures before success." }, { "id": "cie-2022-w-52-q04a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-52-q04b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-52-q05a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-52-q05b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for the biased-coin experiment.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The conditional-probability part was challenging for many, with frequent confusion between the intersection and the numerator event.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 45 ], "tagConfidence": "reviewed", "answerGuide": "Identify the intersection event, use the probability from part (a) where specified, and divide by the conditioning-event probability." }, { "id": "cie-2022-w-52-q05c", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-52-q06a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-52-q06b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-52-q06c", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Justify whether a normal approximation is appropriate for the binomial model in part (b).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few correct solutions were presented and over a quarter omitted this part; the report warns against assuming an approximation is valid without checking it.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 45 ], "tagConfidence": "reviewed", "answerGuide": "Check the binomial-to-normal conditions using the sample size and success and failure probabilities, then state why the approximation is suitable here." }, { "id": "cie-2022-w-52-q07a", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-52-q07b", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability of the letter arrangement with the Ls together and six letters between the As.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says most candidates found this part challenging and commonly forgot that the Ls also had to remain together.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 46 ], "tagConfidence": "reviewed", "answerGuide": "Place the two As at the required separation, count valid positions for the adjacent L block, then divide by the total number of distinct arrangements of ALLIGATOR." }, { "id": "cie-2022-w-52-q07c", "paperId": "cie-2022-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_52.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_52.pdf#page=18", "qpPages": [ 13, 13 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-53-q01", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1: Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-53-q02", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2: Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-53-q03a", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "3(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2022-w-53-q03b", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2022-w-53-q03c", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-53-q04a", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-53-q04b", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-53-q04c", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-53-q04d", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(d): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-53-q05a", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2022-w-53-q05b", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-53-q05c", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(c): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=13", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2022-w-53-q06a", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-53-q06b", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2022-w-53-q06c", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count the selections satisfying the condition comparing the numbers of two repeated letters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the paper's most challenging question; only a small number found the correct final answer.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 49 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate the allowed count cases, count each selection with repeated letters treated as indistinguishable, and divide by the total number of equally likely selections if the question asks for probability." }, { "id": "cie-2022-w-53-q07a", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-53-q07b", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-53-q07c", "paperId": "cie-2022-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2022, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_53.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2022-w-61-q01", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-61-q02a", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "2(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-61-q02b", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "2(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-61-q03a", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-61-q03b", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-w-61-q04a", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-61-q04b", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-61-q05a", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-61-q05b", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-61-q06a", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-61-q06b", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-61-q06c", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(c): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-61-q06d", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(d): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-61-q07a", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2022-w-61-q07b", "paperId": "cie-2022-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_61.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2022-w-62-q01a", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-62-q01b", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2022-w-62-q02", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-62-q03a", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-w-62-q03b", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-w-62-q04a", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2022-w-62-q04b", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use the Poisson model to carry out the stated hypothesis-test decision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report explicitly calls this part challenging and notes that many candidates omitted one of the two cumulative probabilities.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 55 ], "tagConfidence": "reviewed", "answerGuide": "Calculate both cumulative probabilities needed for the decision, compare with the significance level or critical region and state a contextual conclusion." }, { "id": "cie-2022-w-62-q04c", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Complete the Poisson test decision from the probability comparison.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates compared a point probability instead of the required cumulative probability, and the report stresses the contextual conclusion.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 55 ], "tagConfidence": "reviewed", "answerGuide": "Compare the correct cumulative probability with the significance level, then give a non-definite conclusion in the context of the claim." }, { "id": "cie-2022-w-62-q04d", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Continue the Poisson calculation for the stated follow-up probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that some candidates used an incorrect Poisson mean or continued with an unsuitable normal distribution.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 55 ], "tagConfidence": "reviewed", "answerGuide": "Use the correct Poisson parameter for the scenario and calculate the requested tail or cumulative probability without switching to a normal model." }, { "id": "cie-2022-w-62-q05a", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-62-q05b", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-62-q06a", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-62-q06b", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-62-q07a", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the density diagram and its symmetry to express the requested probability in terms of p.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not know how to start and seemed unfamiliar with finding probabilities from a pdf graph.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 56 ], "tagConfidence": "reviewed", "answerGuide": "Mark the known areas on the diagram and use symmetry about the given axis to relate the target interval to P(X<0)=p." }, { "id": "cie-2022-w-62-q07bi", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(b)(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-62-q07bii", "paperId": "cie-2022-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use a zero of the density graph to find a second relation between its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_62.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says most candidates ignored the indicated graph method and repeated integration attempts instead of using a zero of the density to form the second equation.", "difficultySourceUrl": "https://pastpapers.co/caie/A-Level/Mathematics-9709/2022-Oct-Nov/9709_w22_er.pdf", "sourceEvidencePages": [ 56 ], "tagConfidence": "reviewed", "answerGuide": "Substitute an endpoint where the graph meets the x-axis into the density, combine the resulting equation with part (b)(i), and solve for both constants." }, { "id": "cie-2022-w-63-q01", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-63-q02a", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "2(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-63-q02b", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "2(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2022-w-63-q03a", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-63-q03b", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2022-w-63-q04a", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-63-q04b", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2022-w-63-q05a", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-63-q05b", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2022-w-63-q06a", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-63-q06b", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-63-q06c", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(c): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2022-w-63-q06d", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(d): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2022-w-63-q07a", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2022-w-63-q07b", "paperId": "cie-2022-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2022, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w22_ms_63.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2023-s-11-q01", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "1: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-11-q02a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-s-11-q02b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-s-11-q02c", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "2(c): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-s-11-q03", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3: Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-11-q04", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "4: Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2023-s-11-q05", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra", "differentiation" ], "primaryTopicId": "algebra", "summary": "Find the positive parameter and contact point for a line tangent to the curve y = −1/(2x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes this question as challenging; many candidates could not establish the required tangent condition and made algebraic or sign errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Equate the line and curve and use the repeated-root condition, or equate gradients; reject parameter values that do not fit the stated context." }, { "id": "cie-2023-s-11-q06a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "6(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2023-s-11-q06b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "6(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2023-s-11-q07a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-11-q07b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-11-q07c", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-11-q08a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use function values, including a composite value, to determine two constants in the definitions of f and g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says finding the second constant was harder for candidates unsure how to form a composite function, and notes a common omission in substituting into f.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Use f(7) first to determine its parameter, evaluate f(5) before applying g, then solve the resulting linear equation for the other constant." }, { "id": "cie-2023-s-11-q08b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the domain of the inverse function from the range of f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "Only a small number gave the correct range for f inverse; many confused the range with the original domain or gave an excluded-value condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Find the range of f from its definition and domain, then state that range as the domain of the inverse." }, { "id": "cie-2023-s-11-q08c", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(c): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=15", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-11-q09a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the instantaneous rate at which the water depth changes when the depth is 10 cm, given volume as a function of depth and time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says most candidates did not recognise that the rate question required differentiation; some substituted the depth directly into the volume formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the volume relation with respect to time using the chain rule, then substitute the given depth and rate data." }, { "id": "cie-2023-s-11-q09b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Use the given rate of change to determine the water volume at a later instant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=16", "qpPages": [ 13, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says only the strongest candidates successfully used the earlier rate result with the chain rule; many left the part blank or substituted the rate directly for the depth.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Combine dV/dt = (dV/dh)(dh/dt), solve for the depth at the stated rate, then substitute that depth into the volume formula." }, { "id": "cie-2023-s-11-q10a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(a): Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2023-s-11-q10b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the area of the triangle bounded by the tangent and normal at a point on the curve and the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=18", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority produced fully correct solutions, and the report explicitly says the problem-solving aspects were more challenging; it notes missing diagrams, chain-rule errors and unnecessary side-length work.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate to obtain the tangent gradient, use its negative reciprocal for the normal, find both x-intercepts and use the known vertical height of the point to calculate the triangle area." }, { "id": "cie-2023-s-11-q11a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=18", "qpPages": [ 16, 16 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-11-q11b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-11-q11c", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-11-q11d", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(d)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(d): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-11-q12a", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 12, "questionPart": "12(a)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "12(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-s-11-q12b", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 12, "questionPart": "12(b)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "12(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-s-11-q12c", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 12, "questionPart": "12(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Verify that the two circles intersect at points whose y-coordinate is 11.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=20", "qpPages": [ 19, 19 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most candidates did not attempt this verification and that many who tried equated the circle equations instead of using the common y-value efficiently.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Substitute y = 11 into each circle equation and show that both give the same x-coordinate or x-squared value." }, { "id": "cie-2023-s-11-q12d", "paperId": "cie-2023-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 12, "questionPart": "12(d)", "marks": 3, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "12(d): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_11.pdf#page=21", "qpPages": [ 18, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-s-12-q01", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "1: Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2023-s-12-q02", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "2: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-s-12-q03a", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-12-q03b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the parameter range for which the quadratic has no real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 9, "difficultyEvidence": "The report says only a few used the link with completing the square; simplifying the discriminant inequality proved challenging for many.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 5 ], "tagConfidence": "reviewed", "answerGuide": "Use the completed-square condition or require the discriminant to be negative, then solve the resulting strict inequality." }, { "id": "cie-2023-s-12-q04", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-12-q05", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "5: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-s-12-q06a", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Show the stated arc-length relationship for a sector using the isosceles triangle in the diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "The report says this was where the paper began to become more challenging; candidates often used the wrong triangle angle or could not reach the required expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Bisect the isosceles triangle, identify the right-triangle angle and express the radius in terms of the given chord-side length before using arc length." }, { "id": "cie-2023-s-12-q06b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2023-s-12-q07ai", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a)(i): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-12-q07aii", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Verify which of the earlier trigonometric solutions satisfy the modified equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says 30% omitted this verification and many overcomplicated the selection of valid solutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Substitute each solution from part (i) into the modified equation and retain only those that satisfy it." }, { "id": "cie-2023-s-12-q07b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-12-q07c", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the resulting trigonometric equation and give all solutions in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Around one fifth omitted the part; the report says many who attempted it cancelled a factor incorrectly or returned only two of the required solutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Factor the expression without dividing by a potentially zero factor, solve each factor over the full interval and check that four valid values are included." }, { "id": "cie-2023-s-12-q08a", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "8(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-12-q08b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-12-q08c", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Complete a graph sketch and use it to decide whether the function has an inverse.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says this part was challenging: many omitted the flattening behaviour of the graph or failed to connect the sketch to the inverse-function test.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Complete the periodic graph on the specified interval and apply the horizontal-line test to decide whether the function is one-to-one." }, { "id": "cie-2023-s-12-q08d", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(d)", "marks": 6, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe, in the required order, three transformations taking the restricted sine graph to f.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=15", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report states that a significant number found the transformations topic a significant challenge, despite many fully correct solutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Read the amplitude, horizontal scale, vertical shift and restricted domain from f; state each transformation and apply them in the correct order." }, { "id": "cie-2023-s-12-q09a", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "9(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2023-s-12-q09b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "sequences", "summary": "Prove the stated relationship between the nth terms of the two possible geometric progressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=17", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls this part much more challenging and says about 20% omitted it; candidates often did not use the results from part (a) or could not complete the proof.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Express both nth terms using the two possible first terms and common ratios, divide one by the other, and simplify the resulting power expression." }, { "id": "cie-2023-s-12-q10a", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=18", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-s-12-q10b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=19", "qpPages": [ 14, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-s-12-q10c", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equations of the two tangents to a circle parallel to the normal found earlier.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=20", "qpPages": [ 15, 15 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this part as much more challenging: 43% omitted it and many repeated work from the preceding part without finding the required parallel tangents.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Use the known normal gradient for both tangents, substitute y = mx + c into the circle, and impose a repeated root to find the two offsets." }, { "id": "cie-2023-s-12-q11a", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=21", "qpPages": [ 16, 16 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-12-q11b", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=21", "qpPages": [ 16, 16 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-12-q11c", "paperId": "cie-2023-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation at the point where the tangent makes the specified angle with the positive x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_12.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_12.pdf#page=22", "qpPages": [ 17, 17 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says many were unaware that a line's gradient is the tangent of its angle with the horizontal, so they omitted the part or set the derivative equal to zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Set dy/dx equal to tan of the given angle, solve for x, find y, then use the negative reciprocal gradient to form the normal." }, { "id": "cie-2023-s-13-q01", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1: Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-13-q02", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-13-q03a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(a): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-s-13-q03b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3(b): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-s-13-q04a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-13-q04b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-13-q05a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-s-13-q05b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-s-13-q06a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(a): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2023-s-13-q06b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(b): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2023-s-13-q07a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the range of the defined function on its restricted domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says only about half of candidates found and expressed the range correctly, despite the function and domain being given.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 9 ], "tagConfidence": "reviewed", "answerGuide": "Identify the limiting values of the function over the stated domain and express the range with correct strict or inclusive endpoints." }, { "id": "cie-2023-s-13-q07b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "7(b): Apply the function definitions to find the requested expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2023-s-13-q07c", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(c): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-13-q08a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "8(a): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2023-s-13-q08b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the least integer n for which the arithmetic progression's partial sum falls below the stated bound.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few showed how to solve the quadratic reached in the algebra, while candidates who preserved the inequality were able to select the correct integer n.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 10 ], "tagConfidence": "reviewed", "answerGuide": "Form the partial-sum inequality, solve the resulting quadratic, retain the inequality direction and choose the least integer satisfying it." }, { "id": "cie-2023-s-13-q09a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-s-13-q09b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-s-13-q09c", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(c): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-s-13-q10a", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-13-q10b", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-13-q10c", "paperId": "cie-2023-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(c): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_13.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-s-21-q01", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-s-21-q02", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-s-21-q03a", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-s-21-q03b", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use the specified iterative formula and starting value to find the parameter to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A surprisingly large number did not attempt the iteration part, although most candidates who tried it were successful; some stopped before meeting the accuracy requirement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Iterate from the given initial value, show each value to the required precision and round only the final result." }, { "id": "cie-2023-s-21-q04a", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-s-21-q04b", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide the cubic by its known linear factor and factorise the polynomial completely.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says a significant number gave only the quotient and did not factorise it further, missing the stated demand for complete factorisation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Use algebraic long division or synthetic division, then factor the quadratic quotient and state all factors." }, { "id": "cie-2023-s-21-q04c", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-s-21-q05a", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric curve to find dy/dx at a general parameter value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many did not recognise y as a product, and candidates who did often omitted the factor from differentiating the trigonometric term.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Apply the product rule to y and divide dy/dt by dx/dt, simplifying the result in terms of the parameter." }, { "id": "cie-2023-s-21-q05b", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates of the maximum point of the parametric curve over the given interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says most derivatives were not in a form that led to the required tangent equation, so very few correct solutions were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Set dy/dx to zero using the derivative from part (a), solve the resulting trigonometric equation within the interval, then calculate both coordinates." }, { "id": "cie-2023-s-21-q06", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate the definite trigonometric integral exactly using an appropriate double-angle identity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says few candidates used the required double-angle identity to obtain an integrable expression, and many gained only a recognition mark.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the squared cosine term with a double-angle identity, simplify the product into powers of cos 4x and integrate each term over the limits." }, { "id": "cie-2023-s-21-q07a", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-21-q07b", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-21-q07c", "paperId": "cie-2023-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Use the preceding trigonometric result to find the maximum of an expression and the smallest positive angle where it occurs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_21.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_21.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Completely correct solutions were very rare; candidates often failed to use the result from part (a), which was needed to make further progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 13 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the expression using the amplitude-phase form from part (a), identify when its cosine factor reaches its maximum, and select the smallest positive angle in range." }, { "id": "cie-2023-s-22-q01", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation involving secant and tangent over a full turn.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few responses were completely correct and stresses that the equation must first be reduced to one trigonometric function.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Use tan²θ = sec²θ − 1 to form a quadratic in sec θ or cos θ, solve it, then list all angles in the interval." }, { "id": "cie-2023-s-22-q02", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Determine two constants from a straight-line graph of ln y against x for an exponential model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few correct responses were seen; candidates often failed to connect the transformed gradient with the parameter difference or solve the resulting equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 14, 15 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain a linear relation in x, match intercept and gradient to the two data points, then solve simultaneously for the constants." }, { "id": "cie-2023-s-22-q03", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-s-22-q04a", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-22-q04b", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-22-q04c", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-s-22-q05a", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-22-q05b", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of the maximum point of an exponential-polynomial curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many did not factorise the derivative equation after setting it to zero, despite the exponential factor being non-zero; exact coordinates were required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 15 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate by the product rule, set the derivative to zero, divide by the non-zero exponential term, factor the remaining quadratic and substitute the root into the curve." }, { 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form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-22-q06b", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-22-q06c", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-22-q07a", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-22-q07b", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-s-22-q07c", "paperId": "cie-2023-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Show that the stationary-point equation has no further real solutions after using the given factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_22.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_22.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a small number of correct responses were seen; candidates needed to factor the cubic and establish that the remaining quadratic has no real roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 16 ], "tagConfidence": "reviewed", "answerGuide": "Divide the cubic by the known linear factor and use the discriminant of the remaining quadratic to show it has no real roots." }, { "id": "cie-2023-s-23-q01", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation involving secant and tangent over a full turn.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few responses were completely correct and stresses that the equation must first be reduced to one trigonometric function.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Use tan²θ = sec²θ − 1 to form a quadratic in sec θ or cos θ, solve it, then list all angles in the interval." }, { "id": "cie-2023-s-23-q02", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Determine two constants from a straight-line graph of ln y against x for an exponential model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few correct responses were seen; candidates often failed to connect the transformed gradient with the parameter difference or solve the resulting equations.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 17, 18 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain a linear relation in x, match intercept and gradient to the two data points, then solve simultaneously for the constants." }, { "id": "cie-2023-s-23-q03", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3: Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-s-23-q04a", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-23-q04b", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-23-q04c", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-s-23-q05a", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-s-23-q05b", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of the maximum point of an exponential-polynomial curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many did not factorise the derivative equation after setting it to zero, despite the exponential factor being non-zero; exact coordinates were required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate by the product rule, set the derivative to zero, divide by the non-zero exponential term, factor the remaining quadratic and substitute the root into the curve." }, { "id": "cie-2023-s-23-q06a", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-23-q06b", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-23-q06c", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-23-q07a", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-23-q07b", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-s-23-q07c", "paperId": "cie-2023-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Show that the stationary-point equation has no further real solutions after using the given factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_23.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_23.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a small number of correct responses were seen; candidates needed to factor the cubic and establish that the remaining quadratic has no real roots.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Divide the cubic by the known linear factor and use the discriminant of the remaining quadratic to show it has no real roots." }, { "id": "cie-2023-s-31-q01", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-31-q02a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-31-q02b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve a linear inequality involving an absolute-value expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Most candidates squared the inequality and obtained an incorrect critical value; only a minority used the intended interval-based approach.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Use the sketch from part (a) to split the absolute-value expression into intervals, solve each linear inequality and combine the valid ranges." }, { "id": "cie-2023-s-31-q03", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-s-31-q04a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-31-q04b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-31-q05a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-31-q05b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-s-31-q06a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "6(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-s-31-q06b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-31-q06c", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "vectors", "coordinate-geometry" ], "primaryTopicId": "vectors", "summary": "Find the exact area of a parallelogram from its side vectors and included angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the straightforward area relation was rarely used and the majority of candidates gave no response to this part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 21 ], "tagConfidence": "reviewed", "answerGuide": "Use the area formula for a parallelogram as twice the area of a triangle, with the side lengths and sine of the angle obtained in earlier parts." }, { "id": "cie-2023-s-31-q07", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 8, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve the separable differential equation with trigonometric and rational factors using the supplied initial condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says complete integration was unusual; only a minority obtained both required antiderivatives correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Separate the variables, use the double-angle identity in the y-integral and an appropriate secant integral in x, then apply the initial condition to determine the constant." }, { "id": "cie-2023-s-31-q08a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-31-q08b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of the partial-fraction decomposition and give the requested exact form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=14", "qpPages": [ 13, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes that few answers were fully correct after integration, with common sign and coefficient errors and unsuccessful integration of the repeated linear factor.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Integrate each partial-fraction term, use a logarithm for the repeated linear factor and substitute the limits carefully before simplifying." }, { "id": "cie-2023-s-31-q09a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Use integration by parts and the limits to establish the stated equation for the parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says using the limits to rearrange the integration-by-parts result proved more challenging and several candidates did not attempt the rearrangement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Apply integration by parts to the exponential-linear integrand, evaluate the definite integral at both limits and isolate the parameter in the given form." }, { "id": "cie-2023-s-31-q09b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Verify by calculation that the root lies between the two stated bounds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=15", "qpPages": [ 15, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "A large minority gave no response; the report expects numerical evidence for the sign change or corresponding comparison.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the defining equation at each endpoint and show that the results have opposite signs, or use the equivalent definite-integral test." }, { "id": "cie-2023-s-31-q09c", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Apply the specified iteration to find the parameter to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=15", "qpPages": [ 15, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 7, "difficultyEvidence": "A large minority gave no response; the report also notes accuracy and conclusion errors among those who iterated.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Iterate from the stated start value, show successive estimates to the requested precision and round the final parameter correctly." }, { "id": "cie-2023-s-31-q10a", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "10(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-31-q10b", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "10(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=17", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-31-q10c", "paperId": "cie-2023-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 7, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the earlier polynomial roots to find all complex numbers whose squares are roots of that polynomial.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_31.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_31.pdf#page=18", "qpPages": [ 17, 17 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 8, "difficultyEvidence": "The report says very few recognised the link to the earlier parts and most offered no response; many attempted an unhelpful direct square of the given complex root.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Use all roots established in parts (a) and (b), solve z² equal to each root, and include both square roots for every resulting value." }, { "id": "cie-2023-s-32-q01", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2023-s-32-q02", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "2: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-s-32-q03a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-32-q03b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-32-q04", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the equation involving cosine and a double-angle cosine over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a minority obtained a correct single-cosine equation; the report lists several common incorrect half-angle and double-angle substitutions.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 24 ], "tagConfidence": "reviewed", "answerGuide": "Convert cos 2x to 2cos²x − 1, reduce to a quadratic in cos x, solve and retain all solutions in the interval in radians." }, { "id": "cie-2023-s-32-q05a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2023-s-32-q05b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2023-s-32-q06a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-32-q06b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Give an algebraic argument that the root of the trigonometric equation is unique in the specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not understand that an algebraic argument was required and instead tried to iterate numerically or confused inverse tangent and cotangent.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 24 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange the fixed-point relation algebraically into the stated equivalent form and use the interval restrictions to justify the unique root." }, { "id": "cie-2023-s-32-q06c", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-s-32-q07a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-32-q07b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-s-32-q08a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 7, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-32-q08b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-32-q09a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-32-q09b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral after decomposing a rational function with a repeated linear factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=17", "qpPages": [ 15, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report calls this part more challenging and describes sign, coefficient and repeated-factor integration errors; some candidates reverse-engineered the given result.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 25 ], "tagConfidence": "reviewed", "answerGuide": "Integrate each partial fraction, including the reciprocal-square term, preserve the correct limit order and show enough working to verify the given exact result." }, { "id": "cie-2023-s-32-q10a", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=18", "qpPages": [ 16, 16 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-s-32-q10b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the exact area bounded by the curve and the x-axis using the curve's intersection points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=19", "qpPages": [ 17, 17 ], 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"proof", "summary": "11(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-32-q11b", "paperId": "cie-2023-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "vectors", "coordinate-geometry" ], "primaryTopicId": "vectors", "summary": "Find the position vector of the foot of the perpendicular from a point to a line in three dimensions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_32.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_32.pdf#page=21", "qpPages": [ 19, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates did not know how to start and a large number left the part blank; the report says the required scalar product needed to use the correct vectors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 26 ], "tagConfidence": "reviewed", "answerGuide": "Write a general point on the line, form the vector from the given point and impose perpendicularity with the line's direction vector to solve for the parameter." }, { "id": "cie-2023-s-33-q01", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-s-33-q02", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-s-33-q03", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3: Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-s-33-q04", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-33-q05a", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-33-q05b", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-s-33-q06a", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-33-q06b", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-s-33-q07a", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-s-33-q07b", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level 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"cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8", "marks": 8, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the separable differential equation and use its initial condition to determine the requested value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says a number of candidates found the question challenging and could only separate variables and integrate the 1/x term; some did not recognise the arctan integral.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 29 ], "tagConfidence": "reviewed", "answerGuide": "Separate variables, split the rational y-integrand into standard terms, integrate the arctan component in radians and apply the initial condition before solving for x." }, { "id": "cie-2023-s-33-q09a", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2023-s-33-q09b", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "vectors", "coordinate-geometry" ], "primaryTopicId": "vectors", "summary": "Find a specified position vector using the perpendicular geometry of two lines in three dimensions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=18", "qpPages": [ 15, 15 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Candidates found this part more difficult and many did not attempt it; the report notes that the simplest vector-perpendicular method uses both required vectors and avoids assuming an intersection.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 29 ], "tagConfidence": "reviewed", "answerGuide": "Choose a general point on the relevant line, form the displacement vector from the given point, use a zero dot product with the direction vector and then construct the requested position vector." }, { "id": "cie-2023-s-33-q10a", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-33-q10b", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "10(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=21", "qpPages": [ 16, 17 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-s-33-q11a", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=24", "qpPages": [ 18, 18 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2023-s-33-q11b", "paperId": "cie-2023-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the cube of a complex number in simplified modulus-argument form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_33.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_33.pdf#page=26", "qpPages": [ 19, 19 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority attempted this part and some used two inconsistent values from the earlier part; the report identifies the need to handle the signs of both real and imaginary parts.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 30 ], "tagConfidence": "reviewed", "answerGuide": "Use the correct root from part (a), write it in polar form, apply De Moivre's theorem to cube it and reduce the argument to the required interval." }, { "id": "cie-2023-s-41-q01a", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-41-q01b", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-41-q02a", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2023-s-41-q02b", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-41-q03", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the displacement when a particle with variable acceleration is next instantaneously at rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates struggled to solve the equation for the next rest time, especially after squaring it, and some failed to give the corresponding displacement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 36 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate displacement to obtain velocity, set velocity to zero, solve the resulting equation without introducing extraneous roots, then substitute the valid time into displacement." }, { "id": "cie-2023-s-41-q04a", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2023-s-41-q04b", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2023-s-41-q04c", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Given the greatest speed after the initial motion, find the total stopping time and average speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few candidates scored full marks and many left this part blank; a common error treated the given greatest speed as applying before the specified time instead of after it.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 36 ], "tagConfidence": "reviewed", "answerGuide": "Use the signed area under the velocity-time graph for the later interval to find its duration, then calculate total distance and divide by the total time." }, { "id": "cie-2023-s-41-q05a", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(a): Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2023-s-41-q05b", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the direction and exact magnitude of the resultant of two forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A large number did not give the exact resultant magnitude, leaving a negative radical expression or an approximate value despite the exact-answer requirement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 36 ], "tagConfidence": "reviewed", "answerGuide": "Resolve the two force components, combine them to find the resultant magnitude and use the component signs to state its direction." }, { "id": "cie-2023-s-41-q06a", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine whether two connected particles move on rough planes when released from rest.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates struggled to begin this non-standard question and their force equations were often difficult to follow; the report gives two valid net-force approaches.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 36, 37 ], "tagConfidence": "reviewed", "answerGuide": "Find the maximum friction first, then compare the driving component along the string with friction, or assume motion and check that the resulting acceleration is positive." }, { "id": "cie-2023-s-41-q06b", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-41-q07a", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-41-q07b", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-s-41-q07c", "paperId": "cie-2023-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to find a car's speed at the top of a hill when engine power and resistance vary as stated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_41.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_41.pdf#page=13", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many candidates struggled; the report says constant-acceleration methods are invalid because power and resistance are not both constant and describes errors in engine work and energy changes.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Calculate engine work from power and elapsed time, include the work of the changing resistance, gain in gravitational potential energy and change in kinetic energy, then solve for the final speed." }, { "id": "cie-2023-s-42-q01", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-42-q02a", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the initial speed of one particle when two particles moving towards each other come to rest after collision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A significant number failed to represent the initial opposite direction with a negative velocity, producing a negative speed that some changed without justification.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 38 ], "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, assign opposite signs to the initial velocities and apply conservation of momentum before solving for the speed." }, { "id": "cie-2023-s-42-q02b", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2023-s-42-q03a", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve forces in two directions and determine the unknown applied force and angle in the equilibrium setup.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was a challenge for many; although most formed two equations, candidates often substituted an approximate angle where exact trigonometric ratios were required.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally and vertically, substitute the exact sine and cosine values implied by the given tangent ratio, then solve the simultaneous equations." }, { "id": "cie-2023-s-42-q03b", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-42-q04a", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-s-42-q04b", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use a work-energy equation to find the athlete's speed after running along a sloped road section.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes that many lost credit by approximating too early or treating forces as energy terms; attempts using constant acceleration were rarely credited.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Equate the change in kinetic energy to work by the driving and resistive forces plus the gravitational potential-energy change, retaining exact terms until the end." }, { "id": "cie-2023-s-42-q05", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5: Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-s-42-q06a", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-42-q06b", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2023-s-42-q06c", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find when the particle is instantaneously at rest and its displacement then, given velocity as a fractional-power function of time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates struggled with the equation involving fractional powers and some incorrectly solved acceleration equal to zero instead of velocity equal to zero.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Set the velocity expression to zero and solve for the positive time; integrate velocity to obtain displacement and substitute that time." }, { "id": "cie-2023-s-42-q06d", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find the speed when the particle returns to the origin from its fractional-power velocity model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "A surprisingly large number struggled with the fractional-power equation, used constant-acceleration formulas despite the changing acceleration, or reported a negative speed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 40 ], "tagConfidence": "reviewed", "answerGuide": "Integrate velocity to obtain displacement, set displacement to zero for the later return time, substitute into velocity and give its magnitude." }, { "id": "cie-2023-s-42-q07ai", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a)(i): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-42-q07aii", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed at C after motion over a rough section followed by a smooth section of a plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=16", "qpPages": [ 11, 11 ], "msPages": [ 16, 19 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "A significant number did not distinguish the smooth BC section from the rough AB section and reused the earlier tension or acceleration; the report flags this part's section-specific reasoning.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 40 ], "tagConfidence": "reviewed", "answerGuide": "Use the rough-section acceleration only up to B, then set friction to zero on BC and apply the appropriate motion or energy equation for the remaining distance." }, { "id": "cie-2023-s-42-q07b", "paperId": "cie-2023-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_42.pdf#page=20", "qpPages": [ 10, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-43-q01", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-43-q02", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2: Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-43-q03", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve the forces on a ring held in equilibrium by a string and a perpendicular applied force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this force-resolution question was more challenging than the first two; some candidates omitted a tension component or treated one string as two.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 41, 42 ], "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and perpendicular to the applied force, use the single-string tension consistently and solve for the required force quantities." }, { "id": "cie-2023-s-43-q04a", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-s-43-q04b", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-43-q05a", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2023-s-43-q05b", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-43-q05c", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(c): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=13", "qpPages": [ 6, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2023-s-43-q06a", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2023-s-43-q06b", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-s-43-q06c", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Use Newton's second law to find the crate mass in an elevator with a specified cable tension.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few responses were fully correct; errors commonly omitted a force, included an extra force or used an incorrect sign.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Draw the force balance during the specified motion, use the elevator acceleration with a consistent sign convention and solve for the crate mass." }, { "id": "cie-2023-s-43-q06d", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest force exerted by the elevator floor on the crate and state its direction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few correct responses were seen and many candidates did not know how to begin; common errors used the wrong acceleration or included elevator mass.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 43 ], "tagConfidence": "reviewed", "answerGuide": "Use Newton's second law for the crate during the accelerating phase, identify the larger normal reaction and state that it acts upward." }, { "id": "cie-2023-s-43-q07a", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to find a child's speed at a point on a curved slide after resistance does work.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls the work-energy question somewhat challenging; candidates often used the wrong sign or omitted the work done against resistance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 43 ], "tagConfidence": "reviewed", "answerGuide": "Balance the loss in gravitational potential energy and the specified work against resistance with the gain in kinetic energy; do not assume constant acceleration." }, { "id": "cie-2023-s-43-q07b", "paperId": "cie-2023-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy method to find the coefficient of friction on the final part of a slide.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_43.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_43.pdf#page=16", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part remained challenging; many omitted the distance in friction work or formed a dimensionally incorrect energy equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 43 ], "tagConfidence": "reviewed", "answerGuide": "Write the full energy balance including friction force multiplied by distance, find the normal reaction, substitute F = μR and simplify μ as a fraction." }, { "id": "cie-2023-s-51-q01a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-51-q01b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-s-51-q02a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-51-q02b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Count committees with no more than two brothers under the stated selection rules.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many found the restrictions in this part more difficult; candidates often omitted one of the brother-count scenarios or selected the remaining members from the wrong group.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Separate the cases of zero, one and two brothers, multiply the corresponding combinations for each case and add the results." }, { "id": "cie-2023-s-51-q03a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-51-q03b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-51-q03c", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for arrangements of a word with repeated letters and adjacency restrictions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number found this conditional-probability part demanding; weaker solutions divided by the total arrangements rather than conditioning on the stated adjacency event.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 45 ], "tagConfidence": "reviewed", "answerGuide": "Count arrangements satisfying both the three-letter block and the condition, count all arrangements satisfying the condition, and divide the first count by the second." }, { "id": "cie-2023-s-51-q04a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-s-51-q04b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-s-51-q05a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "5(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2023-s-51-q05b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-s-51-q05c", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-s-51-q06a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-51-q06b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-51-q06c", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-51-q06d", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a suitable approximation to find the probability of fewer than 20 successes across 96 trials.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Fully correct solutions were rare; a significant number omitted the part or incorrectly tried to use a binomial distribution directly rather than an appropriate approximation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 46 ], "tagConfidence": "reviewed", "answerGuide": "Use the success probability from the earlier parts to form the binomial mean and variance, apply a normal approximation with continuity correction, and evaluate the lower-tail probability." }, { "id": "cie-2023-s-51-q07a", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the first elephant gift arrives before the sixth purchase.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many found this part challenging and some used a binomial model or counted only success on the fifth purchase rather than the cumulative event.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 47 ], "tagConfidence": "reviewed", "answerGuide": "Sum the geometric probabilities for a first elephant on purchases one through five, or use one minus the probability of no elephant in those five trials." }, { "id": "cie-2023-s-51-q07b", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-51-q07c", "paperId": "cie-2023-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability that five purchases contain exactly one of each animal type.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_51.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_51.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Correct solutions were rarely seen; many found the required single-occurrence probabilities but did not count the possible arrangements of the five distinct animals.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 47 ], "tagConfidence": "reviewed", "answerGuide": "Multiply the probability of one of each animal by the number of ways to arrange the five distinct outcomes." }, { "id": "cie-2023-s-52-q01a", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-52-q01b", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-52-q02a", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-52-q02b", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-52-q02c", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-52-q03a", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-s-52-q03b", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "3(b): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2023-s-52-q03c", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-s-52-q04a", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-52-q04b", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-52-q04c", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for a score from two spinners, given that the score is not two.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this conditional-probability part challenging and says a surprisingly high number omitted it; successful solutions used a complete outcome table.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 50 ], "tagConfidence": "reviewed", "answerGuide": "List all spinner-pair outcomes, identify those allowed by the condition and count the subset with a positive score." }, { "id": "cie-2023-s-52-q04d", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a score greater than two occurs at least three times in nine repeated spins.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number omitted the part and many found the context challenging; the report also flags misreading the success criterion as exactly three instead of at least three.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 50 ], "tagConfidence": "reviewed", "answerGuide": "Use the single-spin probability from the outcome table as the binomial success probability, then sum the probabilities for three or more successes in nine trials." }, { "id": "cie-2023-s-52-q05a", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected number of birds whose normally distributed lengths lie within the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many solutions stopped after finding the probability and did not calculate the expected count; the report also notes a common mistaken assumption that the interval probability was symmetric.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 50 ], "tagConfidence": "reviewed", "answerGuide": "Standardise both interval endpoints, subtract the two cumulative probabilities to get the interval probability, then multiply by the sample size and give an integer count." }, { "id": "cie-2023-s-52-q05b", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-52-q05c", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the probability that fewer than 80 of 120 birds exceed the given length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=15", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many candidates did not attempt this part; errors included a wrong probability area and using the wrong continuity-correction boundary.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 50, 51 ], "tagConfidence": "reviewed", "answerGuide": "Form the binomial mean and variance, apply a continuity correction at 79.5 for fewer than 80, standardise and use the correct lower-tail area." }, { "id": "cie-2023-s-52-q06a", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2023-s-52-q06b", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-52-q06c", "paperId": "cie-2023-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Count arrangements of the selected team in which no cyclists stand next to one another.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_52.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_52.pdf#page=17", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many found this part challenging and a surprising number left it unanswered; the report describes a common incomplete subtraction that removed only the cases with all cyclists together.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 51 ], "tagConfidence": "reviewed", "answerGuide": "Arrange the non-cyclists first and count valid gaps for the cyclists, or subtract all adjacency cases while distinguishing pairs together from all three together." }, { "id": "cie-2023-s-53-q01a", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the expected number of repeated two-coin trials until a pair of heads occurs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a challenging starter; many did not recognise the geometric distribution and commonly gave a value based on the chance of a single head instead of two heads.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 52 ], "tagConfidence": "reviewed", "answerGuide": "Use the geometric mean 1/p with p equal to the probability that both coins show heads on one trial." }, { "id": "cie-2023-s-53-q01b", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-53-q01c", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-53-q02", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the probability that the vote count lies between two inclusive limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number misunderstood the inclusive endpoints or applied the continuity correction in the wrong direction, despite finding the mean and variance correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 52, 53 ], "tagConfidence": "reviewed", "answerGuide": "Use the binomial mean and variance, convert the inclusive integer interval to half-integer boundaries, standardise and find the probability between the two z-values." }, { "id": "cie-2023-s-53-q03a", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-53-q03b", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-53-q03c", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-s-53-q04a", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-s-53-q04b", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare the race-time data sets with conclusions stated in context.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this comparison challenged even the strongest candidates; many compared numerical summaries without explaining what the values meant in the race context.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 53 ], "tagConfidence": "reviewed", "answerGuide": "Compare medians for typical speed and interquartile ranges for consistency, explicitly stating which team was generally faster and which had less spread." }, { "id": "cie-2023-s-53-q04c", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-53-q05a", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-53-q05b", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-53-q06a", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-s-53-q06b", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-53-q06c", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected number of days on which the normally distributed daily quantity lies within 1.3 standard deviations of its mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report calls this challenging; candidates often failed to simplify the standardised limits to ±1.3 or omitted the final expected-count calculation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 54 ], "tagConfidence": "reviewed", "answerGuide": "Use symmetry to find P(−1.3 < Z < 1.3), multiply by 365 and give the required integer number of days." }, { "id": "cie-2023-s-53-q07a", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-53-q07b", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Count arrangements with As at both ends and exactly three letters between the two Cs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=15", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report describes this part as more challenging and notes that candidates often overcomplicated the placement cases or gave up after listing many unnecessary scenarios.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 54 ], "tagConfidence": "reviewed", "answerGuide": "Fix the As at the ends, count arrangements of the six remaining letters and multiply by the four possible positions of the five-letter block containing the two Cs." }, { "id": "cie-2023-s-53-q07c", "paperId": "cie-2023-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_53.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_53.pdf#page=16", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-61-q01a", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-s-61-q01b", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-s-61-q02a", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2023-s-61-q02b", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis", "integration" ], "primaryTopicId": "data-analysis", "summary": "Find the radius of a semicircular probability-density graph whose total area is one.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few gave good attempts; many omitted the part or used an unnecessarily difficult integration method instead of the semicircle area formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Set one half of πr² equal to one and solve for the positive radius." }, { "id": "cie-2023-s-61-q02ci", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(c)(i): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-s-61-q02cii", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(c)(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-61-q03", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "3: Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-s-61-q04a", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-61-q04b", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Explain why a sample of journey times from only three months is unsuitable for estimating the annual mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only a few candidates answered well; stating the first three months alone was insufficient without explaining that they may not represent the whole year or its seasonal variation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Explain that the selected months may have systematically different conditions from the rest of the year, so the sample is not representative." }, { "id": "cie-2023-s-61-q04c", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that exactly three of several confidence intervals contain the true population mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=9", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a few recognised the required calculation and a large number gave no response; the report says some supplied only part of the needed expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Treat each interval as an independent success with the stated coverage probability and use the binomial probability for exactly three successes." }, { "id": "cie-2023-s-61-q05a", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-s-61-q05b", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-61-q06", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the unbiased sample-variance formula to determine the positive parameter in the population model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes responses ranging from correct to no attempt and notes frequent algebra errors when expanding the squared parameter expression.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the sample summaries into the unbiased variance equation, expand the resulting quadratic carefully and retain the positive parameter value." }, { "id": "cie-2023-s-61-q07a", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-s-61-q07b", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region for a Poisson hypothesis test and the probability of a Type I error.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says some candidates did not know how to approach it, used the wrong mean or found individual point probabilities instead of cumulative probabilities.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 58 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the relevant cumulative Poisson probabilities at successive integer cut-offs, identify the largest region within the significance level, and sum its null probabilities." }, { "id": "cie-2023-s-61-q07c", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-61-q07d", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State the test conclusion with a reason after observing three accidents.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "This part was well attempted by only a few; many gave a conclusion without the required comparison or without cautious contextual wording.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 58 ], "tagConfidence": "reviewed", "answerGuide": "Compare the observed count with the critical region or its probability, then state a non-definite conclusion about the mean accident rate in context." }, { "id": "cie-2023-s-61-q07e", "paperId": "cie-2023-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(e)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(e): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_61.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-62-q01", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-62-q02a", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-s-62-q02b", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "proof" ], "primaryTopicId": "distributions", "summary": "Use the mean and variance formulas to explain why the binomial-to-Poisson approximation requires a small success probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few used the requested formulas to deduce that 1 − p is close to one; many gave memorised approximation conditions instead.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 59 ], "tagConfidence": "reviewed", "answerGuide": "Compare np with np(1 − p), cancel np and explain that closeness requires 1 − p near one, hence p near zero." }, { "id": "cie-2023-s-62-q02c", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(c): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-s-62-q03a", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-62-q03b", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "3(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-s-62-q04a", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-s-62-q04b", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-62-q04c", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-s-62-q04d", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-62-q05a", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that two independent normal scores differ by more than two.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few achieved full marks; candidates often found only one of the two possible directional differences or multiplied the two tail probabilities rather than adding them.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 60 ], "tagConfidence": "reviewed", "answerGuide": "Form the distribution of X − Y, calculate both tails beyond 2 and −2, then add the disjoint probabilities." }, { "id": "cie-2023-s-62-q05bi", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b)(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-62-q05bii", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Comment on whether independence of the two score variables is a reasonable assumption.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part was not as well attempted; many incorrectly asserted independence, and some gave no response.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 60 ], "tagConfidence": "reviewed", "answerGuide": "Explain that a candidate's performance in one component is likely related to performance in the other, so independence is doubtful." }, { "id": "cie-2023-s-62-q06a", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Model the count with a binomial distribution and calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify n and p, express the event with the required binomial probabilities, and add or complement the relevant terms." }, { "id": "cie-2023-s-62-q06b", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find the probability of a Type II error given the stated probability of that error in a binomial test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=11", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A large number struggled to set up the correct equation and mixed the success probability with its complement.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 61 ], "tagConfidence": "reviewed", "answerGuide": "Write the Type II error as the relevant binomial probability using the test's acceptance outcome, then solve the equation for p with clear p and 1 − p roles." }, { "id": "cie-2023-s-62-q07ai", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a)(i): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-s-62-q07aii", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a)(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-62-q07b", "paperId": "cie-2023-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "data-analysis", "integration" ], "primaryTopicId": "data-analysis", "summary": "Find the endpoint of a triangular probability density when its median is one.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_62.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_62.pdf#page=12", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few reached the correct parameter; the report says expressing the density in terms of the unknown endpoint and integrating proved difficult, while a similar-triangle ratio was available.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 61 ], "tagConfidence": "reviewed", "answerGuide": "Set the cumulative area up to one equal to one half, express the triangle's height in terms of its endpoint and solve the resulting equation." }, { "id": "cie-2023-s-63-q01", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2023-s-63-q02a", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-63-q02b", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-63-q03a", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the two possible sample sizes implied by the width of an approximate confidence interval for a proportion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Candidates had difficulty handling x/100 and 1 − x/100 and some found proportions but did not convert them into the required possible values of x.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 62 ], "tagConfidence": "reviewed", "answerGuide": "Write the confidence-interval width equation in terms of p = x/100, solve for p and convert both valid proportions back to integer values of x." }, { "id": "cie-2023-s-63-q03b", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-s-63-q04", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4: Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-s-63-q05a", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-63-q05b", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "5(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-s-63-q05c", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "State when the Central Limit Theorem would not be needed for the test in the preceding part.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only about a third answered this correctly, according to the report.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 63 ], "tagConfidence": "reviewed", "answerGuide": "State that the original population distribution is normal, so the sample mean has a normal distribution without relying on the Central Limit Theorem." }, { "id": "cie-2023-s-63-q06a", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-s-63-q06b", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-s-63-q07a", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "7(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-s-63-q07b", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "probability", "integration" ], "primaryTopicId": "probability", "summary": "Find a probability from the area under a semicircular density curve between the stated values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many unsuccessful attempts used integration for this part; the report says it became challenging because the density formula first had to be found, although some correct integration steps earned marks.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_er.pdf", "sourceEvidencePages": [ 63 ], "tagConfidence": "reviewed", "answerGuide": "Use the angle and sector geometry to identify the required area, subtract the triangle from the sector, and divide by the total area where needed; integration is an alternative after deriving the density." }, { "id": "cie-2023-s-63-q08a", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "8(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-s-63-q08b", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "8(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-s-63-q08c", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "8(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-s-63-q08d", "paperId": "cie-2023-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "8(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s23_ms_63.pdf#page=14", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "ial-2021-10-WDM11-01-q1a", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain what a path means in a network.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Define a path using connected edges and the rule that no vertex is repeated." }, { "id": "ial-2021-10-WDM11-01-q1b", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the shortest route from A to J using Dijkstra’s algorithm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Run Dijkstra’s labelling process from A, retain each predecessor, and state the resulting route and total length." }, { "id": "ial-2021-10-WDM11-01-q1c", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the shortest route from J to A that passes through G.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the shortest J-to-G and G-to-A legs, join them at G, and add their lengths." }, { "id": "ial-2021-10-WDM11-01-q2", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate the triangle-angle constraints as a linear-programming problem in x and y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=3", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use x+y+z=180° to eliminate z, translate each angle condition to an inequality, then state the simplified objective to maximise." }, { "id": "ial-2021-10-WDM11-01-q3a", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct a minimum spanning tree using Prim’s algorithm starting at A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the least eligible edge at each step without creating a cycle, recording the selected edges in order." }, { "id": "ial-2021-10-WDM11-01-q3b", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the weight of the minimum spanning tree.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the weights of the seven selected tree edges." }, { "id": "ial-2021-10-WDM11-01-q3c", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the tree weight to give an initial upper bound for a closed visiting route.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Double the minimum spanning-tree weight to obtain a route that traverses every tree edge in both directions." }, { "id": "ial-2021-10-WDM11-01-q3d", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the two nearest-neighbour tours starting at A and calculate their costs.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At each step choose a nearest unvisited city; resolve the tie at A both ways, return to A, and total both tours." }, { "id": "ial-2021-10-WDM11-01-q3e", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Choose the best upper bound from the routes already found.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take the smaller of the initial bound and the nearest-neighbour tour costs." }, { "id": "ial-2021-10-WDM11-01-q3f", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound after deleting A and its incident edges.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find a minimum spanning tree on the remaining cities, then add the two smallest edges incident to A." }, { "id": "ial-2021-10-WDM11-01-q3g", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(g)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the smallest interval guaranteed to contain the optimal tour cost.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the lower bound from part (f) and the best upper bound from part (e), with the correct inclusive endpoints." }, { "id": "ial-2021-10-WDM11-01-q4a", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the activity precedence table from the network diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each activity, list every activity that must finish immediately before it; leave the first activities without predecessors." }, { "id": "ial-2021-10-WDM11-01-q4b", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Read the missing event times and activity labels from the network and cascade chart.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the event-time diagram and cascade chart to identify v, w, x, y and z." }, { "id": "ial-2021-10-WDM11-01-q4c", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate a lower bound for the number of workers needed to finish in minimum time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the total activity duration by the minimum project duration and round up to the next whole worker." }, { "id": "ial-2021-10-WDM11-01-q4d", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct a minimum-time schedule using as few workers as possible.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Schedule each activity within its precedence and time-window constraints, then show a feasible allocation using four workers." }, { "id": "ial-2021-10-WDM11-01-q4e", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Revise the project schedule after activity F takes five extra hours; give the new completion time and critical path.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part separated candidates well; many omitted either critical path ACFK or the revised duration of 27.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Update the affected event times, identify the new longest path and read off the revised completion time.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2021-10-WDM11-01-q5a", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the shortest closed inspection route that traverses every cycle lane, stating the repeated lanes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the four odd vertices, compare the three possible pairings using shortest paths, repeat the least-cost pairing, and add it to the network weight." }, { "id": "ial-2021-10-WDM11-01-q5b", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Count the number of times vertex C appears in the inspection route.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count C in the base network traversal and include any repeated path that passes through C." }, { "id": "ial-2021-10-WDM11-01-q5c", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Choose the best finishing vertex for the cycle-lane inspection route and give its minimum length.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this part discriminated well; many omitted why the C-to-D path through G was the least-weight path that could be repeated.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare open-route choices by pairing odd vertices; repeat the least-weight connecting path that excludes A and add its length to the route total.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2021-10-WDM11-01-q6a", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Plot the constraints and identify the feasible region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=9", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw each boundary line, shade the correct half-plane for every inequality, and label their common intersection R." }, { "id": "ial-2021-10-WDM11-01-q6bi", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use an objective line to identify the optimum when k=0.8.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=9", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw a line of gradient −0.8 and slide it parallel across the feasible region to the last point of contact, labelling that vertex V." }, { "id": "ial-2021-10-WDM11-01-q6bii", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate the coordinates of V and the corresponding objective value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=9", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the two boundary equations meeting at V, then substitute its coordinates into P=0.8x+y." }, { "id": "ial-2021-10-WDM11-01-q6c", 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candidates attempted this part correctly, and many did not show a clear method.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Evaluate the objective function at the adjacent feasible vertices and solve the resulting inequalities for the parameter.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2021-10-WDM11-01-q7a", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the lower bound of four bins to restrict the possible bin size n.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the total of all items with three and four bins, then apply the integer condition on n." }, { "id": "ial-2021-10-WDM11-01-q7b", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Quick-sort the item list into descending order, showing each pass and pivot.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the required middle pivot at each pass, partition values above and below it, and continue until the list is ordered." }, { "id": "ial-2021-10-WDM11-01-q7c", "paperId": "ial-2021-10-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2021, "session": "October", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Infer the bin capacity from the supplied first-fit and first-fit-decreasing allocations, with a full justification.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20211020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20220113.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says many struggled to justify the capacity: the decreasing allocation gives n at most 72, while the first-fit bin totals 72.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the failed fit of 13 after 32 and 28 to bound n above, then use the first-fit bin containing 14, 20, 23 and 15 to show n is at least 72.", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2021-10-WMA11-01-q1", "paperId": "ial-2021-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the given algebraic expression, leaving each term simplified.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20211008.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the summands as powers of x, integrate each term with the power rule, and simplify the result.", "sourceFiles": { "questionPaper": "WMA11-01-QP.pdf", "markScheme": "WMA11-01-MS.pdf", "examinerReport": "WMA11-01-ER.pdf" }, "sourceSha256": { "questionPaper": "72C32C76D8A62B99818C69BD360A93D68BBE510AA0120B2DE53AD6DF7B6E2207", "markScheme": "2717CBF3427A5EA0DDD05C17CEB592554D60754FDD8E30ED2C1FA826AA2D1CFF", "examinerReport": "307B389CF62997A9F90415DC804DC4FE564A42B9F2B18C8C979CA5FA7CD61E6C" 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"tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set the derivative equal to 2, solve the resulting equation, and report both x-values.", "sourceFiles": { "questionPaper": "WMA11-01-QP.pdf", "markScheme": "WMA11-01-MS.pdf", "examinerReport": "WMA11-01-ER.pdf" }, "sourceSha256": { "questionPaper": "72C32C76D8A62B99818C69BD360A93D68BBE510AA0120B2DE53AD6DF7B6E2207", "markScheme": "2717CBF3427A5EA0DDD05C17CEB592554D60754FDD8E30ED2C1FA826AA2D1CFF", "examinerReport": "307B389CF62997A9F90415DC804DC4FE564A42B9F2B18C8C979CA5FA7CD61E6C" } }, { "id": "ial-2021-10-WMA11-01-q3-i", "paperId": "ial-2021-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2021, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Solve the rational inequality in x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20211008.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220113.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply through by x² to avoid reversing an inequality for an unknown-sign denominator, factor the resulting expression, and identify the valid interval 00.", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "WMA12-01-MS.pdf", "examinerReport": "WMA12-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E10D2F11CB2B89161DF6CD21E4771C1CD0F7AC9E583F789A11285EAB90A397F5", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220113.pdf#page=15", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the midpoint of QR as the centre and its half-length as the radius, then intersect the circle with the x-axis and choose the positive root.", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "WMA12-01-MS.pdf", "examinerReport": "WMA12-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E10D2F11CB2B89161DF6CD21E4771C1CD0F7AC9E583F789A11285EAB90A397F5", "markScheme": "15A2A1D7725DF228B5AA38B24C3DBB2D7B99EE11EB9BCA40C221B11D0A8B84FE", "examinerReport": "094D4EB3BEC1DEFCCFD5A706F563C5E26E94329D53B659D19133B0E89D045175" } }, { "id": "ial-2021-10-WMA12-01-q7-i", "paperId": "ial-2021-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 3, "topicIds": [ "sequences", "indices" ], "primaryTopicId": "sequences", "summary": "Find the least n for which the geometric sequence term exceeds the stated large bound.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20211013.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220113.pdf#page=17", "qpPages": [ 20, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the nth-term inequality, take logarithms, solve for n, and round up to the least integer satisfying it.", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "WMA12-01-MS.pdf", "examinerReport": "WMA12-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E10D2F11CB2B89161DF6CD21E4771C1CD0F7AC9E583F789A11285EAB90A397F5", "markScheme": "15A2A1D7725DF228B5AA38B24C3DBB2D7B99EE11EB9BCA40C221B11D0A8B84FE", "examinerReport": "094D4EB3BEC1DEFCCFD5A706F563C5E26E94329D53B659D19133B0E89D045175" } }, { "id": "ial-2021-10-WMA12-01-q7-ii-a", "paperId": "ial-2021-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Derive the quadratic equation for the common ratio r.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_que_20211013.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220113.pdf#page=17", "qpPages": [ 20, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use ar=−6 and the infinite-sum equation a/(1−r)=25 to eliminate a and rearrange.", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "WMA12-01-MS.pdf", "examinerReport": "WMA12-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E10D2F11CB2B89161DF6CD21E4771C1CD0F7AC9E583F789A11285EAB90A397F5", "markScheme": "15A2A1D7725DF228B5AA38B24C3DBB2D7B99EE11EB9BCA40C221B11D0A8B84FE", "examinerReport": "094D4EB3BEC1DEFCCFD5A706F563C5E26E94329D53B659D19133B0E89D045175" } }, { "id": 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"reviewed", "answerGuide": "Solve the quadratic exactly or to the required form and list both roots.", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "WMA12-01-MS.pdf", "examinerReport": "WMA12-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E10D2F11CB2B89161DF6CD21E4771C1CD0F7AC9E583F789A11285EAB90A397F5", "markScheme": "15A2A1D7725DF228B5AA38B24C3DBB2D7B99EE11EB9BCA40C221B11D0A8B84FE", "examinerReport": "094D4EB3BEC1DEFCCFD5A706F563C5E26E94329D53B659D19133B0E89D045175" } }, { "id": "ial-2021-10-WMA12-01-q7-ii-c", "paperId": "ial-2021-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2021, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)(c)", "marks": 1, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "sequences", "summary": "Select the valid common ratio and 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20220113.pdf#page=18", "qpPages": [ 20, 21 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the base radius into the log-linear model, solve for log₁₀M, then exponentiate.", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "WMA13-01-MS.pdf", "examinerReport": "WMA13-01-ER.pdf" }, "sourceSha256": { "questionPaper": "0A5266CADFF9EB3BC87E353C94EC66E6CB30DB5EAE8DDF746BBDA82326342331", "markScheme": "A32C66719B8AE36DB4F48007AE70CE7D8F911D3316C035F8EA349ADA34D884CA", "examinerReport": "E927EC5581D0B49E689D94DEF78C070EEAA0066CCE1A9413FA7B4BE2C4DB3168" } }, { "id": "ial-2021-10-WMA13-01-q7-b", "paperId": "ial-2021-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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2021, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent equation at the point with the stated y-coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_que_20211022.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20220113.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find x and dy/dx at that point, then use point-gradient form and express the line as y=mx+c.", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_que_20211022.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA13_01_rms_20220113.pdf#page=23", "qpPages": [ 26, 29 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the product rule to the polynomial and exponential factors, then simplify.", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "WMA13-01-MS.pdf", "examinerReport": "WMA13-01-ER.pdf" }, "sourceSha256": { "questionPaper": "0A5266CADFF9EB3BC87E353C94EC66E6CB30DB5EAE8DDF746BBDA82326342331", "markScheme": "A32C66719B8AE36DB4F48007AE70CE7D8F911D3316C035F8EA349ADA34D884CA", "examinerReport": "E927EC5581D0B49E689D94DEF78C070EEAA0066CCE1A9413FA7B4BE2C4DB3168" } }, { "id": 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and give y explicitly.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate the y- and x-terms, integrate both sides, apply the initial point to determine the constant, then rearrange for y.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": 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"difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up the identity, use convenient x-values to find constants, then compare coefficients for the remaining values.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } }, { "id": "ial-2021-10-WMA14-01-q3-b", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate g(x) using the decomposition from part (a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each polynomial or reciprocal term in the partial-fraction form and simplify.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this explanatory part was frequently omitted and demanding for many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the derivative from part (b) and show the remaining fraction is positive for x>0.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": 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"questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "binomial", "indices" ], "primaryTopicId": "binomial", "summary": "Use the expansion at x=1/4 to estimate the requested root to four decimal places.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=11", "qpPages": [ 12, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x=1/4 into the series, evaluate the resulting approximation, and round to four decimal places.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form dy/dx=(dy/dt)/(dx/dt), determine the parameter value corresponding to x=3, then evaluate the gradient.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } }, { "id": "ial-2021-10-WMA14-01-q5-b", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the vertex and the sign of the leading coefficient to state the minimum or maximum bound.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } 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"Find the point obtained by reflecting A across the line.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=15", "qpPages": [ 22, 25 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the closest point as the midpoint between A and its reflection, then calculate the reflected position vector.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": 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"msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply integration by parts with the logarithm differentiated and the polynomial integrated; include the constant.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } }, { "id": "ial-2021-10-WMA14-01-q8-b", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "integration", "geometry" ], "primaryTopicId": "integration", "summary": "Find the exact volume formed by rotating the region under y=x ln x about the x-axis.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=17", "qpPages": [ 26, 29 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls the volume part demanding and a strong discriminator, with integration limits and repeated integration common errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the disc formula π∫(x ln x)²dx from the positive x-intercept to e, applying the integral from part (a).", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" }, "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2021-10-WMA14-01-q9-a", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Derive the differential equation for water depth in the cylindrical tank.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=19", "qpPages": [ 30, 33 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Relate volume to depth using the tank's cross-sectional area, equate the net inflow rate to dV/dt, and rearrange for dh/dt.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } }, { "id": "ial-2021-10-WMA14-01-q9-b", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the time for the water depth to rise from 0.5 m to 3.5 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=19", "qpPages": [ 30, 33 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables in the depth equation, integrate, apply the initial depth at t=0, and substitute h=3.5 to solve for t.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } }, { "id": "ial-2021-10-WMA14-01-q10-a", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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cube.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "WMA14-01-MS.pdf", "examinerReport": "WMA14-01-ER.pdf" }, "sourceSha256": { "questionPaper": "E9E6302B8600A0EE6AEB28F2DBE0679D11DE460AB8D4A3578CFB35BFB1B25DE9", "markScheme": "A82203B9EF2AD47591D1D583C3DFF4E8E5BEEDD7C377425259147E193A92B7A8", "examinerReport": "B515CB30C9106CCBFC035BFE3D03D4C95F2710EA9568C1A5DE68FF547780347C" } }, { "id": "ial-2021-10-WMA14-01-q10-b", "paperId": "ial-2021-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2021, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "proof", "indices" ], "primaryTopicId": "proof", "summary": "Prove by contradiction that the cube root of 2 is irrational.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_que_20211029.pdf#page=34", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_rms_20220113.pdf#page=21", "qpPages": [ 34, 36 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson reports mixed performance on the contradiction proof, with many failing to use a valid lowest-terms contradiction.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA14_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Assume the root is a reduced fraction, cube both sides, use the even-cube result to show both numerator and denominator are even, and contradict lowest terms.", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical force equilibrium with the stated reaction ratio, then take moments about one support to solve for the distance d." }, { "id": "ial-2021-10-WME01-01-q2a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find particle P’s speed after a direct collision with particle Q.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of linear momentum along the line of motion, using the given masses and Q’s final speed." }, { "id": "ial-2021-10-WME01-01-q2b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "State the direction in which P moves after the collision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the signed velocity obtained from momentum to identify whether P continues forward or reverses direction." }, { "id": "ial-2021-10-WME01-01-q2c", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of friction as Q slides to rest after the collision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use constant-deceleration motion over the given distance to find the deceleration, then equate it to the frictional acceleration μg." }, { "id": "ial-2021-10-WME01-01-q3a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the car’s speed at a road sign under constant deceleration.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use v²=u²+2as with the initial speed, deceleration and 48 m distance." }, { "id": "ial-2021-10-WME01-01-q3b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the arrival time when the car decelerates to the limit speed, then continues at that speed.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the distance and time needed to slow from 25 to 13 m/s, then cover the remaining distance at 13 m/s." }, { "id": "ial-2021-10-WME01-01-q3c", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the arrival time when the car coasts for 0.2 seconds before braking.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the coasting distance, use suvat to reach 13 m/s, then cover the remaining distance at the speed limit and total elapsed time." }, { "id": "ial-2021-10-WME01-01-q4a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find the angle between the particle’s position vector and the upward direction at t=2.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute t=2 into r, then use a suitable trigonometric ratio or scalar product to find the angle to the nearest degree." }, { "id": "ial-2021-10-WME01-01-q4b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find the times when the particle is 2.5 m from the origin.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the squared magnitude of r equal to 2.5², solve the resulting quadratic in t, and retain both valid times." }, { "id": "ial-2021-10-WME01-01-q5a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the size and direction of friction on a bead held against a rough vertical wire.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces vertically and horizontally using tan α=4/3 and T=2.5 N, then report the friction magnitude and direction." }, { "id": "ial-2021-10-WME01-01-q5b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction when the rod thrust reaches its limiting value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At limiting equilibrium set friction=μR; resolve the force balance on the bead using T=6.125 N, then solve for μ." }, { "id": "ial-2021-10-WME01-01-q6a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the launch speed from the velocity-time graph.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=11", "qpPages": [ 22, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the gradient of the straight-line velocity graph to determine the initial velocity." }, { "id": "ial-2021-10-WME01-01-q6b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the launch height from the ball's velocity-time graph and its time to hit the ground.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=11", "qpPages": [ 22, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says part (b) was less accessible than finding the launch speed; candidates often lost the height calculation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use signed area under the velocity-time graph to find net displacement; the final position is ground level, so reverse the displacement to get the initial height.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2021-10-WME01-01-q7a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Write the equations of motion for the two masses in the pulley system.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=12", "qpPages": [ 24, 27 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the plane for A and vertically for B, including friction on A and a common acceleration and tension." }, { "id": "ial-2021-10-WME01-01-q7b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration after the particles are released.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=12", "qpPages": [ 24, 27 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the two equations from part (a) with tan α=3/4 and μ=1/2, eliminate tension, and solve for a." }, { "id": "ial-2021-10-WME01-01-q7c", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the resultant force exerted on the pulley by the two string segments.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=12", "qpPages": [ 24, 27 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use equal string tensions from the smooth-pulley model and resolve the two tension vectors at the pulley to find their resultant." }, { "id": "ial-2021-10-WME01-01-q7d", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "State how the smooth-pulley assumption is used.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=12", "qpPages": [ 24, 27 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the tension is the same on both sides of a light smooth pulley." }, { "id": "ial-2021-10-WME01-01-q8a", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Show that the ship’s velocity is 10i−5j km/h.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=13", "qpPages": [ 28, 32 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the port position from the 7:36 position and divide the displacement by 0.6 hours." }, { "id": "ial-2021-10-WME01-01-q8b", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Give the ship’s position vector t hours after departure.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=13", "qpPages": [ 28, 32 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add t times the constant velocity vector to the port’s initial position vector." }, { "id": "ial-2021-10-WME01-01-q8c", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Determine the lighthouse position from its bearings relative to the moving ship at two observation times.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=13", "qpPages": [ 28, 32 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says locating the lighthouse was a stumbling block and many candidates stopped at this stage.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the ship positions from its constant velocity, then impose the due-east and northeast alignment conditions to solve for the lighthouse coordinates.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2021-10-WME01-01-q8d", "paperId": "ial-2021-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2021, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find the ship’s position vector when it is due south of the lighthouse.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_que_20211014.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME01_01_rms_20220113.pdf#page=13", "qpPages": [ 28, 32 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the ship’s east coordinate equal to the lighthouse’s and use the ship’s position equation to calculate its north coordinate." }, { "id": "ial-2021-10-WME02-01-q1a", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show the work done against friction as the particle moves up the inclined plane.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the normal reaction, calculate friction μR, and multiply the friction force by distance d." }, { "id": "ial-2021-10-WME02-01-q1b", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use work-energy to find the speed as the particle returns from B to A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the gain in kinetic energy to the loss in gravitational potential energy less the work against friction, then solve for speed." }, { "id": "ial-2021-10-WME02-01-q2", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 8, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the engine power on an inclined road from two journeys in opposite directions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write Newton’s second-law equations for uphill and downhill travel, express power as force times speed, eliminate the constant resistance, and solve for P." }, { "id": "ial-2021-10-WME02-01-q3a", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the non-zero times at which the particle is instantaneously at rest.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this motion question was very difficult for some candidates, especially finding the non-zero rest times.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the position function to obtain velocity and solve v(t)=0, retaining only positive roots.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2021-10-WME02-01-q3b", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled from t=0 to t=4.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the turning time from v=0, evaluate the position at the turning time and at t=4, and add the absolute changes in position." }, { "id": "ial-2021-10-WME02-01-q3c", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the particle’s acceleration at t=4.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the velocity function and substitute t=4." }, { "id": "ial-2021-10-WME02-01-q4", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find both values of the scalar impulse parameter from the before-and-after velocity information.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report notes that many treated vector impulse equations as scalar equations, which prevented a correct solution.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply vector impulse-momentum, express the final velocity using the scalar parameter and set its magnitude equal to the stated speed.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2021-10-WME02-01-q5a", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the cable tension in the pole system is 56 N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about B using the pole’s weight and cable geometry, with tan α=3/4." }, { "id": "ial-2021-10-WME02-01-q5b", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the wall-friction coefficient when the pole is in limiting equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve horizontal and vertical forces to find the wall reaction and friction, use friction=μR, and solve for μ." }, { "id": "ial-2021-10-WME02-01-q6a", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of restitution for the direct collision of A and B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use impulse-momentum to determine the initial velocities from the given impulse, then compare relative speed of separation with approach." }, { "id": "ial-2021-10-WME02-01-q6b", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the coefficient of restitution between B and the wall from the kinetic-energy losses.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express B’s kinetic-energy loss in each collision in terms of its speeds, relate the two losses through E, and solve for the rebound-speed ratio." }, { "id": "ial-2021-10-WME02-01-q7a", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the relationship between the rectangle dimensions when the composite lamina's centre of mass lies on a specified side.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says candidates found the sector-centre geometry and required angle unfamiliar; this part was a notable source of error.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate the sector and rectangle masses and centroid positions, then equate their weighted moments along EC.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2021-10-WME02-01-q7b", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the angle β when the composite lamina is suspended from B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the composite centre of mass from the masses and centroids, then use its position relative to B to determine the equilibrium angle." }, { "id": "ial-2021-10-WME02-01-q8a", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use energy to find the launch height h from the change in speed.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the change in kinetic energy to the loss of gravitational potential energy over the vertical drop." }, { "id": "ial-2021-10-WME02-01-q8b", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the horizontal distance from O to the impact point B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical suvat over 2.5 s to find the initial vertical velocity, then use the final speed to find the horizontal component and distance." }, { "id": "ial-2021-10-WME02-01-q8c", "paperId": "ial-2021-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2021, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time for which the particle’s speed is at most 8 m/s during its flight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20211019.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220113.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the constant horizontal component and vertical velocity to solve when the speed equals 8 m/s, then take the interval between those times." }, { "id": "ial-2021-10-WME03-01-q1a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show that the particle’s SHM amplitude is 4/π m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use ω=2π/T and the speed formula for SHM at the stated displacement to solve for amplitude." }, { "id": "ial-2021-10-WME03-01-q1b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the maximum speed in the SHM.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the maximum-speed relation vmax=aω with the amplitude and angular frequency." }, { "id": "ial-2021-10-WME03-01-q2a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the resultant force when the particle’s speed is 13 m/s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a=v dv/dx from the given speed-position relation, then apply F=ma." }, { "id": "ial-2021-10-WME03-01-q2b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time for the particle to travel 7.5 m from rest.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange the speed-position relation to express dt in terms of dx, integrate from x=0, and apply the initial condition." }, { "id": "ial-2021-10-WME03-01-q3a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the spring modulus constant from the equilibrium extension.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Balance the particle’s weight with the elastic tension at the stated equilibrium position and solve for k." }, { "id": "ial-2021-10-WME03-01-q3b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle moves with SHM after being displaced vertically.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to the displacement from equilibrium and show the acceleration is proportional to the negative displacement." }, { "id": "ial-2021-10-WME03-01-q3c", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the speed as the particle passes through equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the SHM angular frequency and amplitude in vmax=aω, with the displacement amplitude from release." }, { "id": "ial-2021-10-WME03-01-q3d", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the time from release to the first passage through equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the SHM quarter-period from an extreme position to the equilibrium position." }, { "id": "ial-2021-10-WME03-01-q4a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 7, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use work-energy to find the air-resistance constant k during upward motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Account for work against air resistance, gravitational potential-energy change, kinetic-energy change and elastic-energy change over distance a." }, { "id": "ial-2021-10-WME03-01-q4b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the distance OB where the particle reaches maximum speed on descent.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set acceleration to zero at maximum speed, balance weight and elastic tension, and solve for the extension and distance from O." }, { "id": "ial-2021-10-WME03-01-q5a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the tension in the lower half of the rotating string has the stated value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical equilibrium for the particle attached midway along the string and the geometry of the two equal string segments." }, { "id": "ial-2021-10-WME03-01-q5b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the period of the rotating particle in the two-string circular-motion system.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this circular-motion part was often poorly answered or left blank.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use vertical force balance to find the lower tension, horizontal centripetal force to determine angular speed, then convert to period.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2021-10-WME03-01-q6a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show the speed relation for the particle attached to a rotating rod.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of mechanical energy between the initial and general angular positions, including the change in gravitational potential energy." }, { "id": "ial-2021-10-WME03-01-q6b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find cos β when the force exerted by the rod is zero.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the radial equation at the zero-force point, substitute the speed relation and given initial speed, then solve for cos β." }, { "id": "ial-2021-10-WME03-01-q7a", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the centre of mass of a uniform cone lies 3h/4 from its vertex.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a thin-disc integration or the standard centroid integral, weighting each cross-section by its volume." }, { "id": "ial-2021-10-WME03-01-q7b", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of the truncated solid F from its larger plane face.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the removed smaller cone’s mass moment from the original cone’s moment and divide by the remaining mass." }, { "id": "ial-2021-10-WME03-01-q7c", "paperId": "ial-2021-10-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2021, "session": "October", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Derive the condition on the cone's dimensions for the composite solid to remain in equilibrium on its curved surface.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_que_20211026.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220113.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the least successful part of the question; many candidates did not attempt it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the composite centre of mass from part (b) and the geometry of the contact/support condition to obtain and simplify the required inequality.", "sourceEvidencePages": { "examinerReport": [ 7 ] } }, { "id": "ial-2021-10-WST01-01-q1a", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(B) from the Venn diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the probabilities in all regions belonging to B." }, { "id": "ial-2021-10-WST01-01-q1b", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether events A and B are independent.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate P(A), P(B) and P(A∩B), then compare P(A∩B) with P(A)P(B)." }, { "id": "ial-2021-10-WST01-01-q1c", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find p and q given the conditional-probability relationship.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P(C|B)=P(C) to form an equation, combine it with total probability 1, and solve for both unknown regions." }, { "id": "ial-2021-10-WST01-01-q1d", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show a possible location for event D under the two stated conditions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw D entirely outside A while overlapping a positive-probability part of B." }, { "id": "ial-2021-10-WST01-01-q2a", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the value of Sxp from the summary statistics.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Sxp=Σxp−(ΣxΣp)/n and substitute the given totals." }, { "id": "ial-2021-10-WST01-01-q2b", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation coefficient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find Spp from the supplied totals, then divide Sxp by √(SxxSpp)." }, { "id": "ial-2021-10-WST01-01-q2c", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the regression line of paper spending p on employee count x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the regression gradient Sxp/Sxx and intercept from the sample means, then write p=a+bx." }, { "id": "ial-2021-10-WST01-01-q2d", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the regression gradient in context.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the predicted average increase in annual paper spending, in hundreds of dollars, for one additional employee." }, { "id": "ial-2021-10-WST01-01-q2e", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the company's percentage saving when it applies the revised paper-spending model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this part was more challenging; many used the wrong employee total or could not form the percentage.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply each model to the total of 93 employees, compare the predicted spend, and divide the difference by the original estimate.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2021-10-WST01-01-q3a", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Find the median age from the male passengers’ stem-and-leaf data.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the 18th ordered observation among the 35 ages." }, { "id": "ial-2021-10-WST01-01-q3b", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Show that the male ages have interquartile range 16.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the lower and upper quartiles from the ordered observations and subtract Q1 from Q3." }, { "id": "ial-2021-10-WST01-01-q3c", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Show that there are three outliers among the male passengers.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the 1.5×IQR fences and count the ages outside them." }, { "id": "ial-2021-10-WST01-01-q3d", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Draw a box plot for the male passengers’ ages.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot the quartiles and median, then show whiskers to the most extreme non-outliers and mark each outlier separately." }, { "id": "ial-2021-10-WST01-01-q3e", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 1, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "Compare the age distributions of male and female passengers using a statistic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State a valid comparison such as median, spread or skew, quoting the relevant box-plot values." }, { "id": "ial-2021-10-WST01-01-q3f", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the unchanged female box plot and the family age relationships to state what can be inferred about the granddaughter and Anja.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says almost all candidates scored zero; many left this reasoning part blank or confused outliers, quartiles and age ranges.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate unchanged quartiles and whiskers into order-statistic bounds for the four added ages, then combine those bounds with the 23-year age gap.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2021-10-WST01-01-q4a", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the tree for the sequential draws from bags A, B and C.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add each red/yellow branch probability, updating bag C after the first two draws before writing the final draw probabilities." }, { "id": "ial-2021-10-WST01-01-q4b", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show the probability that exactly two recorded balls are yellow.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sum the probabilities of the three colour sequences with exactly two yellows, accounting for the changed contents of bag C." }, { "id": "ial-2021-10-WST01-01-q4c", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability that the first draw from bag A was red, given that exactly two recorded balls were yellow.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most challenging part of the question; many used only two probability factors instead of all three.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the tree to sum the paths matching the condition, and divide the paths with A red by the total probability of exactly two yellows.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2021-10-WST01-01-q4d", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the distribution of X, the number of yellow balls recorded in one game.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the probabilities for X=0,1,2,3 by summing the matching tree paths." }, { "id": "ial-2021-10-WST01-01-q4e", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find E(X) for the number of recorded yellow balls.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each possible value of X by its probability from part (d) and add." }, { "id": "ial-2021-10-WST01-01-q5a", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the expectation of the symmetric random variable Y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the symmetric probabilities at ±9 and ±5 to show their contributions cancel, leaving E(Y)=0." }, { "id": "ial-2021-10-WST01-01-q5b", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the cumulative distribution condition to find u.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the cumulative probability at zero using q, r and u, then solve together with total probability one." }, { "id": "ial-2021-10-WST01-01-q5c", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use Var(Y)=37 to find q and r.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(Y²), set it equal to 37 because E(Y)=0, and solve alongside the probability total." }, { "id": "ial-2021-10-WST01-01-q5d", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability distribution of the distance from the origin to P=(12,Y).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was another challenging part that many candidates did not attempt; errors often came from the distance calculation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "For each possible Y value, calculate sqrt(12²+Y²), combine equal distances and transfer the corresponding probabilities.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2021-10-WST01-01-q6a", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show that the probability a book is taller than 30.8 cm is 0.15.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise 30.8 using the normal mean and standard deviation, then evaluate the upper tail." }, { "id": "ial-2021-10-WST01-01-q6b", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the range of book heights retained for shelving.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the fifth percentile for the lower cut-off and pair it with the specified upper limit of 30.8 cm." }, { "id": "ial-2021-10-WST01-01-q6ci", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the upper cut-off d for medium books is 25.8 cm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the category proportions and the normal model to locate the percentile boundary d, then round to one decimal place." }, { "id": "ial-2021-10-WST01-01-q6cii", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the lower medium-book boundary m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the cumulative probability at the lower medium boundary from the 2:3:3 category ratios, then use the inverse normal distribution." }, { "id": "ial-2021-10-WST01-01-q6d", "paperId": "ial-2021-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2021, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate the minimum stacked shelf height for small, medium and large books under the specified normal model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20211012.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates completed this part; common errors used the lower rather than upper height limit or omitted repeated shelves and wood.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the category cut-offs and normal percentiles to get the tallest allowed book in each category, multiply by the number of shelves and add the wood thickness.", "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2021-10-WST02-01-q1a", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability at least 12 of 30 customers do not buy free-range eggs.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model F as Binomial(30,0.35) and evaluate the upper tail from 12." }, { "id": "ial-2021-10-WST02-01-q1b", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that between 8 and 14 customers, inclusive, avoid free-range eggs.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial cumulative probabilities P(F≤14)−P(F≤7)." }, { "id": "ial-2021-10-WST02-01-q1c", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that total loyalty points for 30 customers is below 70.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies this as a difficult part of the opening question; candidates struggled to model the points total with the binomial variable.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Let F count non-free-range purchases; total points are 90−2F, so convert the event to a binomial tail and evaluate it.", "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2021-10-WST02-01-q1d", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 7, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test whether more than 35% of customers avoid free-range eggs using a normal approximation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State one-sided hypotheses, model the count under the null, apply a continuity correction, compare the tail probability or test statistic with 5%, and conclude in context." }, { "id": "ial-2021-10-WST02-01-q2ia", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State P(X>14) for the uniform distribution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given interval probability and the distribution’s equal density to obtain the right-hand tail." }, { "id": "ial-2021-10-WST02-01-q2ib", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find P(6X>a+b) using the uniform interval endpoints.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use E(X)=11 to find the midpoint, combine it with the given interval probability to solve for a and b, then evaluate the required interval proportion." }, { "id": "ial-2021-10-WST02-01-q2iia", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the distribution of the shortest piece S when a 45 cm strip is cut at a random point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use symmetry about the midpoint to identify the support from 0 to 22.5 cm and its constant density." }, { "id": "ial-2021-10-WST02-01-q2iib", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability the shorter pasta piece is under 12 cm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the uniform distribution of S on [0,22.5] and divide the favourable interval length by 22.5." }, { "id": "ial-2021-10-WST02-01-q2iic", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability exactly six of 20 strips produce a piece under 12 cm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with n=20 and the probability from part (b)." }, { "id": "ial-2021-10-WST02-01-q3a", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the cumulative distribution function to establish the value of b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says this was the most challenging part of Question 3; candidates often failed to match the CDF branches at the boundary.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Apply the required CDF boundary condition at the upper endpoint of its support and solve for b.", "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2021-10-WST02-01-q3b", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the exact value of a.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the CDF boundary condition or total probability at the upper end of the support and solve for a." }, { "id": "ial-2021-10-WST02-01-q3c", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find P(X>2.25) from the cumulative distribution function.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate 1−F(2.25) using the correct branch of the piecewise CDF." }, { "id": "ial-2021-10-WST02-01-q3d", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the probability density and calculate its mode.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the CDF on its support to obtain the density, sketch the connected piecewise curve, then differentiate the curved branch and locate its maximum." }, { "id": "ial-2021-10-WST02-01-q4a", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability exactly eight cars enter in ten minutes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a Poisson model with mean 6 and evaluate P(X=8)." }, { "id": "ial-2021-10-WST02-01-q4b", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the smallest n for which the upper Poisson tail is below 0.05.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check successive integer cut-offs using the Poisson(6) cumulative probabilities." }, { "id": "ial-2021-10-WST02-01-q4c", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the smallest integer m for which no cars arrive with probability below 0.05.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a Poisson mean of 0.6m and solve e^(−0.6m)<0.05, then take the smallest integer." }, { "id": "ial-2021-10-WST02-01-q4d", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability the wait after one car arrives is less than five minutes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the exponential waiting-time model with rate 0.6 per minute." }, { "id": "ial-2021-10-WST02-01-q4e", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Given 15 arrivals in a half hour, find the probability exactly one arrived during its first five minutes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report identifies this as one of the most discriminating parts; candidates often did not condition on the fixed total of 15.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_pef_20220113.pdf", "tagConfidence": "reviewed", "answerGuide": "Condition on 15 arrivals and use a binomial model with first-five-minute probability 5/30.", "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2021-10-WST02-01-q4f", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(f)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Find the 5% critical region for testing whether the arrival mean exceeds six per ten minutes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the 15-minute count to the null Poisson mean, then find the smallest upper-tail count with probability at most 0.05." }, { "id": "ial-2021-10-WST02-01-q5a", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show the probability Alif scores 8 points in one game.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=10", "qpPages": [ 18, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the ordered counter values that give 8 and multiply their independent probabilities." }, { "id": "ial-2021-10-WST02-01-q5b", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of Alif’s score from two draws.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=10", "qpPages": [ 18, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List all ordered pairs of counter values, calculate each score 4x−2y, and combine probabilities for equal scores." }, { "id": "ial-2021-10-WST02-01-q5c", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Calculate the expected game score.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=10", "qpPages": [ 18, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each score by its probability from part (b) and sum." }, { "id": "ial-2021-10-WST02-01-q6a", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the piecewise probability density function of Y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw each linear density segment over its stated interval, showing the joins and zero density outside the support." }, { "id": "ial-2021-10-WST02-01-q6b", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Var(2Y−3) using E(Y²).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20211016.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_rms_20220113.pdf#page=11", "qpPages": [ 20, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "First find E(Y) from the density, use Var(Y)=E(Y²)−E(Y)², then apply Var(2Y−3)=4Var(Y)." }, { "id": "ial-2021-10-WST02-01-q6c", "paperId": "ial-2021-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2021, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the stated CDF on −1 0, so the stationary value is a minimum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=19", "qpPages": [ 6, 6 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q07d", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Calculate the minimum perimeter to an appropriate accuracy.", "answerGuide": "Substitute x = √(800/(4 + π)) into the perimeter formula to obtain approximately 37.8 m.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q08a", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Use the geometric-sequence condition to form an equation in sin θ.", "answerGuide": "Equate the square of the middle term to the product of the outer terms and simplify to 3sin²θ + 2sinθ − 1 = 0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=21", "qpPages": [ 7, 7 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q08b", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the acute-angle condition to find the required exact trigonometric value.", "answerGuide": "The valid root is sin θ = 1/3; then cos θ = √(1 − sin²θ) = 2√2/3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=21", "qpPages": [ 7, 7 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q08c", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the specified later term of the geometric sequence.", "answerGuide": "Use the common ratio √2 and the given first term to calculate the 11th term as 64/3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=22", "qpPages": [ 7, 7 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q09", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the product and logarithmic-ratio conditions for the two positive constants.", "answerGuide": "Combine the logarithms to get a/b = 64; with ab = 25 and a,b > 0, obtain a = 40 and b = 5/8.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=23", "qpPages": [ 7, 7 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q10a", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the first two non-constant coefficients in the binomial expansion to find n and k.", "answerGuide": "Equate coefficients to get nk = 36 and n(n − 1)k²/2 = 567; solving gives n = 8 and k = 9/2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=25", "qpPages": [ 8, 8 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA12-01A-q10b", "paperId": "ial-2026-1-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the specified coefficient in the binomial expansion.", "answerGuide": "Use C(8, 3)(9/2)³ to obtain the x³ coefficient 5103.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2A_MS.pdf#page=26", "qpPages": [ 8, 8 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2024-w-11-q01", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use the constant term of a binomial expansion to determine a positive parameter, then find the coefficient of x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the term with zero power of x to solve for the parameter, then substitute it into the term with power x²." }, { "id": "cie-2024-w-11-q02", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use a stationary point of a curve y = x² − xᵃ to determine a and its ordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, set the gradient to zero at x = −3 to find a, then substitute the point into the curve to find b." }, { "id": "cie-2024-w-11-q03", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the area of a shaded annular sector to find its boundary length in terms of π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the difference of sector areas to find the inner radius, then add both arc lengths and the two straight radial sides." }, { "id": "cie-2024-w-11-q04", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Show that a quadratic curve intersects a parameterised line for every real value of the parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the curve to form a quadratic and show its discriminant is positive for every parameter value." }, { "id": "cie-2024-w-11-q05a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the x-coordinate where a curve has a stated gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the given derivative equal to 2 and solve the resulting quadratic, retaining the value allowed by the curve." }, { "id": "cie-2024-w-11-q05b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Recover the curve equation from its derivative and a point on the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each power, include a constant of integration, then use the point (4, 11) to determine the constant." }, { "id": "cie-2024-w-11-q06a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the distance between the centres of two circles from their equations.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to read off both centres, then use the distance formula." }, { "id": "cie-2024-w-11-q06b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the greatest and least possible distances between points on two circles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the centre distance with the two radii by adding for the maximum and subtracting for the minimum." }, { "id": "cie-2024-w-11-q07a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the equation of a tangent to a fractional-power curve at a specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, evaluate the gradient at the given x-coordinate, and use point-gradient form." }, { "id": "cie-2024-w-11-q07b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the area under the curve over a specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve and evaluate the antiderivative at the two vertical boundaries." }, { "id": "cie-2024-w-11-q08a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite an expression involving tan β, sin²β and cos²β using a = sin β.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use tan β = sin β/cos β and cos²β = 1 − a², taking the positive cosine because β is in the second quadrant." }, { "id": "cie-2024-w-11-q08b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over a full turn in degrees.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use double-angle identities to obtain a quadratic in sin θ, solve it, and list all angles in the interval." }, { "id": "cie-2024-w-11-q09a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the intervals on which a cubic curve is decreasing.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and solve the resulting quadratic inequality for the x-values where the derivative is negative." }, { "id": "cie-2024-w-11-q09b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the intercept of a tangent whose gradient is specified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative equal to the line gradient, locate the tangency point, then substitute into the line to find k." }, { "id": "cie-2024-w-11-q10a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Determine the common difference of an arithmetic progression whose selected terms form a geometric progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the second, fifth and eleventh AP terms, then use the square of the middle GP term equal to the product of the outer terms." }, { "id": "cie-2024-w-11-q10b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Calculate the difference between a long AP sum and a GP sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 15, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the AP sum formula for 77 terms and the GP sum formula for 10 terms with the common ratio from the preceding part." }, { "id": "cie-2024-w-11-q11a", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Write a quadratic function in completed-square form and state its range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square, identify the maximum value from the vertex, and state the corresponding range." }, { "id": "cie-2024-w-11-q11b", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe a reflection and translation that move a quadratic graph to a minimum at the origin.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect in the x-axis, then translate the vertex to (0, 0), giving the required vector." }, { "id": "cie-2024-w-11-q11c", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch a restricted quadratic and explain why it is one-to-one.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 17, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the graph on its stated domain and use the horizontal-line test or unique y-values to justify one-to-one behavior." }, { "id": "cie-2024-w-11-q11d", "paperId": "cie-2024-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(d)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch the inverse graph and find the inverse function of a restricted quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_11.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_11.pdf", "qpPages": [ 17, 17 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect the graph in y = x, then solve y = g(x) for x and choose the branch consistent with the restricted domain." }, { "id": "cie-2024-w-12-q01a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Read the amplitude, frequency and vertical shift from a plotted sine curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the maximum and minimum to find the amplitude and midline, then use the period to determine b." }, { "id": "cie-2024-w-12-q01bi", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)(i)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Count intersections of the given sine curve and a straight line on the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch or inspect both sides and count their intersection points within the interval." }, { "id": "cie-2024-w-12-q01bii", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)(ii)", "marks": 1, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Count intersections after shifting the sine curve vertically and reflecting the comparison line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the constants found in part (a), then count intersections on the closed interval." }, { "id": "cie-2024-w-12-q02a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum of the first 20 terms of an arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the first term, common difference and n = 20 into the AP sum formula." }, { "id": "cie-2024-w-12-q02b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find k when the sum of the first 2k AP terms is ten times the first k-term sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write both sums using the AP formula, equate them as stated, solve and use the positive integer condition." }, { "id": "cie-2024-w-12-q03a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of a chord joining two nearby points on a quadratic curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute both x-coordinates into the curve and simplify the difference quotient in h." }, { "id": "cie-2024-w-12-q03b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Explain how the tangent gradient follows from a chord-gradient expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take the limit as the second point approaches the first, then state the resulting derivative value." }, { "id": "cie-2024-w-12-q04a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find a constant term in a binomial expansion with a negative power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the general term whose power of x is zero and simplify its coefficient." }, { "id": "cie-2024-w-12-q04b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the constant term in a product of a polynomial and a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the needed terms from the expansion, multiply by each polynomial term that contributes to x⁰, and add." }, { "id": "cie-2024-w-12-q05ai", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate a rational function at a stated input.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the input into the formula and simplify." }, { "id": "cie-2024-w-12-q05aii", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch the inverse graph of a rational function on given axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect the original graph in y = x and label the mirror line." }, { "id": "cie-2024-w-12-q05aiii", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(iii)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse rule and domain of a rational function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set y equal to the function, rearrange for x, swap variable roles, and restrict the inverse domain to the original range." }, { "id": "cie-2024-w-12-q05b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve an equation involving the composition of a linear function and a rational function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the inner function into the outer rule, clear denominators and solve, checking the domain." }, { "id": "cie-2024-w-12-q06", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Determine a radius and angle for a plate made from several circular sectors given its perimeter and area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write expressions for the straight edges and arcs to form the perimeter equation, then combine sector areas and solve the two equations." }, { "id": "cie-2024-w-12-q07a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Complete the square to find the vertex of a downward-opening quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor the leading coefficient before completing the square and read off the vertex coordinates." }, { "id": "cie-2024-w-12-q07b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the shaded area between a quadratic curve and a line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the intersections, integrate the difference between line and curve, and evaluate over the correct bounds." }, { "id": "cie-2024-w-12-q08a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Rewrite a circle equation in centre-radius form using its parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square in x and y and express the centre and radius in terms of p and q." }, { "id": "cie-2024-w-12-q08bi", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the normal to a circle at a specified point using a tangent line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the tangent gradient from the circle geometry, take its negative reciprocal and use the given point." }, { "id": "cie-2024-w-12-q08bii", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the parameters of a circle from a point and tangent line at that point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 13, 13 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the known point on the circle and the normal direction to locate the centre, then find the radius and match the general equation." }, { "id": "cie-2024-w-12-q09a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 7, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Use a known intersection to determine parameters of a quadratic curve and line, then find their second intersection.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 14, 14 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given point into both equations, solve for k and p, then equate the expressions and solve the intersection quadratic." }, { "id": "cie-2024-w-12-q09b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find parameter values for which a line and quadratic curve do not intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 15, 15 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the intersection quadratic and require its discriminant to be negative." }, { "id": "cie-2024-w-12-q10a", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal to a logarithmic curve at a specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 16, 16 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product and chain rules, evaluate the tangent gradient at the point and take the negative reciprocal." }, { "id": "cie-2024-w-12-q10b", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Recover a function from its derivative and a point on its graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 16, 16 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each term with its coefficient, include a constant, and use the stated point to find it." }, { "id": "cie-2024-w-12-q10c", "paperId": "cie-2024-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Determine whether a function is increasing or decreasing from its derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_12.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_12.pdf", "qpPages": [ 17, 17 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the derivative has one sign throughout the domain by analysing the supplied quadratic and state the monotonicity." }, { "id": "cie-2024-w-13-q01", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find a distant term of an arithmetic progression from its fourth and eighth terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the two terms to find the common difference and first term, then calculate the 30th term." }, { "id": "cie-2024-w-13-q02", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve an exact trigonometric equation involving cos x and tan²x on a restricted interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use tan²x = sec²x − 1, solve for cos x and select the value in the stated interval." }, { "id": "cie-2024-w-13-q03a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find two specified coefficients in a binomial expansion in terms of a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial theorem to identify the x³ and x⁴ terms separately." }, { "id": "cie-2024-w-13-q03b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use a coefficient in a product of expansions to find a positive parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Collect the x⁴ contributions from both products, equate to 240 and solve for positive a." }, { "id": "cie-2024-w-13-q04", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a sine equation over 0° to 360°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor or substitute to obtain a quadratic in sin θ, solve and list all angles in the interval." }, { "id": "cie-2024-w-13-q05a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe three transformations that map one displayed function graph to another.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read reflection, translation and vertical stretch from the plotted features, then give them in the correct order." }, { "id": "cie-2024-w-13-q05b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Express the transformed graph function in terms of the original function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the graph transformations into changes to the input and output of f." }, { "id": "cie-2024-w-13-q06", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": null, "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find two possible sums to infinity from a relation between the first four and first eight GP sums.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write each partial sum using first term 10 and ratio r, use the stated ratio to solve for r, then evaluate S∞ for both convergent roots." }, { "id": "cie-2024-w-13-q07a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the area of a composite plate containing a sector, semicircles and triangles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the sector area, semicircle radii and triangle areas separately, then add all five regions." }, { "id": "cie-2024-w-13-q07b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact perimeter of a sector-based plate from the areas of its semicircles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the semicircle areas to find their radius and the central angle, then add exposed arcs and straight sides." }, { "id": "cie-2024-w-13-q08a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Complete the square for a quadratic and identify its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Group terms to form a perfect square and compare with the requested expression." }, { "id": "cie-2024-w-13-q08b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the least domain boundary that makes a quadratic function one-to-one.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the vertex location to restrict the domain to one monotonic branch." }, { "id": "cie-2024-w-13-q08c", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of a restricted quadratic function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set y equal to f(x), solve the quadratic for x and choose the branch matching the domain restriction." }, { "id": "cie-2024-w-13-q08d", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve a composition equation using the inverse function from the previous part.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the inverse to the output value, then solve for the required input and reject values outside the restricted domain." }, { "id": "cie-2024-w-13-q09a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the x-coordinates where two cubic curves intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the curves, factor the resulting expression and list all real roots." }, { "id": "cie-2024-w-13-q09b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the area between two cubic curves over their bounded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 13, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the lower curve from the upper one and integrate between their intersection abscissae." }, { "id": "cie-2024-w-13-q10a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that a circle centre lies on a specified line given two points on the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the equal-distance property of the centre or the perpendicular bisector of the chord AB to derive the line equation." }, { "id": "cie-2024-w-13-q10b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find both circles of a given radius passing through two points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 15, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Parameterise the centre on the perpendicular bisector, impose the radius condition, then write both circle equations." }, { "id": "cie-2024-w-13-q11a", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a curve with an exponential term twice as requested.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the chain and product rules to obtain the first derivative, then differentiate again for the second derivative." }, { "id": "cie-2024-w-13-q11b", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the stationary point and its nature for a member of a curve family.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative to zero at the specified parameter, find x and y exactly, and use the second derivative or sign change to classify it." }, { "id": "cie-2024-w-13-q11c", "paperId": "cie-2024-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the parameter for which two tangents meet at a point with a specified abscissa.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_13.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_13.pdf", "qpPages": [ 17, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find each tangent equation using the parameter, equate their y-values at x = 0.6, and solve for k." }, { "id": "cie-2024-w-21-q01a", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use logarithms to show that an exponential model becomes linear and identify its gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides and rearrange in the form y = mx + c." }, { "id": "cie-2024-w-21-q01b", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use two points on the transformed graph to find the model parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the two simultaneous linear equations in ln a and k, then recover a and k." }, { "id": "cie-2024-w-21-q02", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an inequality involving two absolute-value expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split at the critical values, solve each linear case and combine the intervals satisfying the original inequality." }, { "id": "cie-2024-w-21-q03a", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate a squared tangent function and evaluate its derivative at a specified angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule and use exact trigonometric values at x = π/3." }, { "id": "cie-2024-w-21-q03b", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact definite integral involving the composite function f(sin x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute sin x into f, simplify using trigonometric identities, integrate the resulting terms and apply the limits." }, { "id": "cie-2024-w-21-q04a", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a known linear factor to determine a coefficient in a cubic polynomial.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the root corresponding to the factor into the polynomial and solve for a." }, { "id": "cie-2024-w-21-q04b", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a cubic polynomial completely after finding its parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide by the known factor, then factor the remaining quadratic over the integers." }, { "id": "cie-2024-w-21-q04c", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosecant equation over an interval of negative and positive angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factorisation from part (b), convert the nonzero factor to a sine equation and list the permitted angles." }, { "id": "cie-2024-w-21-q05a", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Derive a stated exponential equation from a differential relation and boundary values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate, apply the given limits and rearrange the logarithmic equation to the required exponential form." }, { "id": "cie-2024-w-21-q05b", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use a prescribed iteration to solve the exponential equation to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate from 2, retain five significant figures for each value, and show enough iterations to justify rounding." }, { "id": "cie-2024-w-21-q06a", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for a curve defined parametrically by exponential functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t and divide dy/dt by dx/dt, simplifying the result." }, { "id": "cie-2024-w-21-q06b", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradient where the parametric curve crosses the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set x = 0 to find the parameter value, substitute it into dy/dx and simplify exactly." }, { "id": "cie-2024-w-21-q07a", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a product-to-sum identity involving cosines of shifted angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand both compound-angle terms, substitute exact values for 30° and 60°, and simplify to the stated expression." }, { "id": "cie-2024-w-21-q07b", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosine equation over the first quadrant using the identity from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity to obtain an equation in sin 4α, solve, and retain the solutions in the interval." }, { "id": "cie-2024-w-21-q07c", "paperId": "cie-2024-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove an exact sum of products of cosines of specified angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_21.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_21.pdf", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply compound-angle identities to each product and combine the resulting sine terms to obtain the exact value." }, { "id": "cie-2024-w-22-q01", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use logarithms to express an exponential relationship as a power law and find its constant.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logs, compare the resulting straight-line form and evaluate the gradient to three significant figures." }, { "id": "cie-2024-w-22-q02a", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate sin⁴(3x) and evaluate at x = π/4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule and substitute the exact angle value." }, { "id": "cie-2024-w-22-q02b", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact integral of a trigonometric function over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a double-angle identity to write the integrand as a constant plus a cosine term, integrate and evaluate the limits." }, { "id": "cie-2024-w-22-q03", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of an implicitly defined exponential curve at a specified point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides using product and chain rules, collect dy/dx terms and substitute the point coordinates." }, { "id": "cie-2024-w-22-q04a", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "functions", "exponentials" ], "primaryTopicId": "functions", "summary": "Sketch an exponential curve and a modulus graph on common axes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the exponential growth, the V-shaped modulus graph and their relative positions." }, { "id": "cie-2024-w-22-q04b", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Derive a logarithmic equation for the intersection abscissa of two graphs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the two functions and rearrange using logarithms to the required form." }, { "id": "cie-2024-w-22-q04c", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use an iteration to find the graph intersection to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate from 0.45, show successive values to five significant figures and verify the rounded answer." }, { "id": "cie-2024-w-22-q05a", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find two coefficients of a cubic from a factor and a remainder condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor theorem at −2 and the remainder theorem at 2, then solve the simultaneous equations." }, { "id": "cie-2024-w-22-q05b", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a polynomial and show that it has exactly one real root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor the polynomial, identify the known root, then show the remaining quadratic has negative discriminant." }, { "id": "cie-2024-w-22-q05c", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosecant equation using the polynomial factorisation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the polynomial argument equal to its roots, convert the nonzero case to a sine equation and retain values in range." }, { "id": "cie-2024-w-22-q06a", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Approximate a bounded area using the trapezium rule with two intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the function values at the three equally spaced ordinates and apply the trapezium rule with h = 1." }, { "id": "cie-2024-w-22-q06b", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find the exact combined area of two regions under exponential and logarithmic curves.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each relevant curve over its interval and combine the exact expressions, including logarithmic terms." }, { "id": "cie-2024-w-22-q06c", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Deduce an approximate area by subtracting the trapezium estimate from an exact total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the estimate for region A from the exact combined area and round to three significant figures." }, { "id": "cie-2024-w-22-q06d", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Decide whether the trapezium-based area estimate is an overestimate or underestimate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the concavity of the curve relative to the trapezia to explain the direction of the error for region B." }, { "id": "cie-2024-w-22-q07a", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express a sum of sine terms as a single shifted sine function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand sin(2θ − α), compare sine and cosine coefficients, and find the amplitude and phase in the required ranges." }, { "id": "cie-2024-w-22-q07b", "paperId": "cie-2024-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the smallest positive solution of an equation using the phase form from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_22.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_22.pdf", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the single-sine form, solve within one cycle and choose the smallest positive angle." }, { "id": "cie-2024-w-23-q01a", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use logarithms to show that an exponential model becomes linear and identify its gradient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms of both sides and rearrange in the form y = mx + c." }, { "id": "cie-2024-w-23-q01b", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use two points on the transformed graph to find the model parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the two simultaneous linear equations in ln a and k, then recover a and k." }, { "id": "cie-2024-w-23-q02", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an inequality involving two absolute-value expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split at the critical values, solve each linear case and combine the intervals satisfying the original inequality." }, { "id": "cie-2024-w-23-q03a", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate a squared tangent function and evaluate its derivative at a specified angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule and use exact trigonometric values at x = π/3." }, { "id": "cie-2024-w-23-q03b", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate an exact definite integral involving the composite function f(sin x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute sin x into f, simplify using trigonometric identities, integrate the resulting terms and apply the limits." }, { "id": "cie-2024-w-23-q04a", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a known linear factor to determine a coefficient in a cubic polynomial.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the root corresponding to the factor into the polynomial and solve for a." }, { "id": "cie-2024-w-23-q04b", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor a cubic polynomial completely after finding its parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide by the known factor, then factor the remaining quadratic over the integers." }, { "id": "cie-2024-w-23-q04c", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosecant equation over an interval of negative and positive angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factorisation from part (b), convert the nonzero factor to a sine equation and list the permitted angles." }, { "id": "cie-2024-w-23-q05a", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Derive a stated exponential equation from a differential relation and boundary values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate, apply the given limits and rearrange the logarithmic equation to the required exponential form." }, { "id": "cie-2024-w-23-q05b", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use a prescribed iteration to solve the exponential equation to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate from 2, retain five significant figures for each value, and show enough iterations to justify rounding." }, { "id": "cie-2024-w-23-q06a", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for a curve defined parametrically by exponential functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t and divide dy/dt by dx/dt, simplifying the result." }, { "id": "cie-2024-w-23-q06b", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradient where the parametric curve crosses the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 11, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set x = 0 to find the parameter value, substitute it into dy/dx and simplify exactly." }, { "id": "cie-2024-w-23-q07a", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a product-to-sum identity involving cosines of shifted angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand both compound-angle terms, substitute exact values for 30° and 60°, and simplify to the stated expression." }, { "id": "cie-2024-w-23-q07b", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosine equation over the first quadrant using the identity from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity to obtain an equation in sin 4α, solve, and retain the solutions in the interval." }, { "id": "cie-2024-w-23-q07c", "paperId": "cie-2024-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove an exact sum of products of cosines of specified angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_23.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_23.pdf", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply compound-angle identities to each product and combine the resulting sine terms to obtain the exact value." }, { "id": "cie-2024-w-31-q01", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find two polynomial coefficients using a known factor and a relation between remainders.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem at −2 and the remainder theorem at 2 and 4 to form two equations, then solve for a and b." }, { "id": "cie-2024-w-31-q02", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate exactly an integral of x² ln(3x) over a finite interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use integration by parts with the logarithm as the differentiated factor, then substitute the limits and express the result in the requested exact form." }, { "id": "cie-2024-w-31-q03", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of an implicitly defined logarithmic curve at (1, 0).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each side with respect to x, collect dy/dx and substitute the given point." }, { "id": "cie-2024-w-31-q04a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove an identity involving sec²(θ/4) and tan²(θ/4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the secant-tangent identity and double-angle relations to simplify the left-hand side." }, { "id": "cie-2024-w-31-q04b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over 0° to 180° using part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the identity to form an equation in tan²(α/2), solve and list every angle in range." }, { "id": "cie-2024-w-31-q05a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use two graphs to establish that a logarithmic-exponential equation has one root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the two functions with their key behaviour and show that their intersection is unique." }, { "id": "cie-2024-w-31-q05b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Verify by calculation that the unique root is between 7 and 9.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate a suitable difference at each endpoint and show a sign change." }, { "id": "cie-2024-w-31-q05c", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use the stated recurrence to approximate the root to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate from the prescribed start, record four decimal places and show enough terms to justify the rounded root." }, { "id": "cie-2024-w-31-q06a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates of a minimum point of a trigonometric curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve the stationary equation on the given interval and substitute the angle into the curve." }, { "id": "cie-2024-w-31-q06b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of the region between a trigonometric curve and the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 11, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the positive portion of the curve and integrate between the relevant roots using exact limits." }, { "id": "cie-2024-w-31-q07a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express a rational function as partial fractions with a repeated linear factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up the repeated-factor form, clear denominators and solve for all constants." }, { "id": "cie-2024-w-31-q07b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use partial fractions to find the coefficient of x³ in an expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 13, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand each partial-fraction term as a binomial or geometric series and collect the x³ coefficient." }, { "id": "cie-2024-w-31-q08a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Write the reciprocal of a complex number in Cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply numerator and denominator by the complex conjugate and separate real and imaginary parts." }, { "id": "cie-2024-w-31-q08b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Verify a relationship between the real and imaginary parts of a complex reciprocal.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the expressions from part (a), simplify both sides and show they agree." }, { "id": "cie-2024-w-31-q08c", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch a vertical line and circle loci on an Argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 15, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the real-part condition to a vertical line and identify the circle centre and radius from its modulus equation." }, { "id": "cie-2024-w-31-q08d", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(d)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Describe geometrically the locus of a reciprocal complex number under a real-part constraint.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 15, 15 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity from part (b) to express the reciprocal locus as a circle and state its centre and radius." }, { "id": "cie-2024-w-31-q09a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Write a vector equation for a line through a point parallel to a given vector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated point vector as the position vector and the given direction vector as the parameter coefficient." }, { "id": "cie-2024-w-31-q09b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find where a parametric point lies on the vector line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 16, 16 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate corresponding components in the two position-vector expressions and solve for the parameter." }, { "id": "cie-2024-w-31-q09c", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find parameter values that give a specified acute angle between two 3D line directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 17, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the scalar product and direction-vector magnitudes to form the cosine equation, then solve the resulting quadratic." }, { "id": "cie-2024-w-31-q10a", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation", "mechanics" ], "primaryTopicId": "differentiation", "summary": "Derive a differential equation for a spherical balloon whose net volume inflow is specified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate V = 4πr³/3, express net volume rate as inflow minus outflow, and use the chain rule to obtain dr/dt." }, { "id": "cie-2024-w-31-q10b", "paperId": "cie-2024-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the balloon-radius differential equation using two measured radii, then find the emptying time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_31.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_31.pdf", "qpPages": [ 19, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables and integrate, use both boundary conditions to find constants, then set r = 0 to obtain the time." }, { "id": "cie-2024-w-32-q01", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand a fractional binomial in ascending powers through x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the generalized binomial series and retain the constant, x and x² terms, simplifying their coefficients." }, { "id": "cie-2024-w-32-q02a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use graph sketches to show that a trigonometric equation has exactly one root on an interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the two functions with their asymptotes and identify a single intersection in the stated interval." }, { "id": "cie-2024-w-32-q02b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Show that a convergent sequence from a given iteration has the required root as its limit.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set successive terms equal at the limit and rearrange to the original equation." }, { "id": "cie-2024-w-32-q03", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find exact Cartesian square roots of a complex number by forming a quartic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Let the root be x + iy, square it, equate real and imaginary parts, eliminate one variable and state both roots." }, { "id": "cie-2024-w-32-q04", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": null, "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation to three decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the laws of indices to isolate the exponential term, then take logarithms and round to three decimal places." }, { "id": "cie-2024-w-32-q05a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the exact argument of a quotient of two complex numbers in trigonometric form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the denominator argument from the numerator argument, adjusting to the principal argument range." }, { "id": "cie-2024-w-32-q05b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Describe the transformation from a complex number to its conjugate and find the conjugate argument.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Reflect in the real axis and reverse the sign of the argument." }, { "id": "cie-2024-w-32-q06", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find a and b from two points on a straight-line plot of ln y against x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the gradient and intercept of the ln y graph to determine ln b and ln a, then round each model constant to one significant figure." }, { "id": "cie-2024-w-32-q07a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Show an equation involving tan x can be rewritten in a specified factorised form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the double-angle formula, clear denominators and simplify, noting the stated nonzero condition." }, { "id": "cie-2024-w-32-q07b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a tangent equation exactly over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factorised equation to find tan x, then list all exact angles in the interval." }, { "id": "cie-2024-w-32-q08a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for a trigonometric curve given parametrically.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate both coordinates with respect to t and simplify their ratio to the required cosine form." }, { "id": "cie-2024-w-32-q08b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation at a specified parameter value on a parametric curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the point coordinates and tangent gradient, take the negative reciprocal and use point-gradient form." }, { "id": "cie-2024-w-32-q09a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a fourth vertex so that a vector quadrilateral is a trapezium with a specified side relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the required parallel-side vector relation to express D from A, B and C." }, { "id": "cie-2024-w-32-q09b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection point of the diagonals of a vector trapezium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 12, 12 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write vector equations for both diagonals, equate components to find their parameters, and substitute into either diagonal." }, { "id": "cie-2024-w-32-q09c", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find an angle in a 3D vector triangle using a scalar product.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the two vectors meeting at B, evaluate their dot product and magnitudes, and take the inverse cosine." }, { "id": "cie-2024-w-32-q10a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation", "mechanics" ], "primaryTopicId": "differentiation", "summary": "Derive a radius differential equation for a spherical balloon with a net rate of air inflow.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Relate volume rate to radius rate using V = 4πr³/3 and substitute the given pumping and leakage rates." }, { "id": "cie-2024-w-32-q10b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder in a polynomial division involving r².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Perform division by r − 50 and identify the linear quotient and constant remainder." }, { "id": "cie-2024-w-32-q10c", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(c)", "marks": 6, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the balloon differential equation for time as a function of radius.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 15, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, use the division from part (b) to integrate, and apply the initial condition r = 0 at t = 0." }, { "id": "cie-2024-w-32-q10d", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(d)", "marks": 1, "topicIds": [ "differentiation", "mechanics" ], "primaryTopicId": "differentiation", "summary": "Evaluate the time when the balloon reaches a specified radius.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 15, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute r = 12 into the expression for t obtained in part (c)." }, { "id": "cie-2024-w-32-q11a", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate an exponential rational function and find its stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the quotient rule, set the derivative to zero, solve for x and substitute to find y." }, { "id": "cie-2024-w-32-q11b", "paperId": "cie-2024-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 9, "topicIds": [ "integration", "exponentials", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate an exact logarithmic integral using u = eˣ and partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_32.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_32.pdf", "qpPages": [ 17, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Change variables to u, decompose the rational integrand, integrate logarithmic terms and apply the transformed limits." }, { "id": "cie-2024-w-33-q01a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the locus of a complex number with fixed modulus and restricted argument.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw the circle arc centred at the origin with radius 2 over the stated argument sector." }, { "id": "cie-2024-w-33-q01b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the locus of the square of a complex number from its modulus and argument range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Square the modulus and double the argument interval, then draw the resulting arc with radius 4." }, { "id": "cie-2024-w-33-q02a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Show that a sequence from the stated recurrence converges to a root of a cubic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the iteration's fixed point equal to x and rearrange to the original cubic equation." }, { "id": "cie-2024-w-33-q02b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "numerical-methods", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration from 1.2 to find a root to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the recurrence repeatedly, record four decimal places and justify the rounded root." }, { "id": "cie-2024-w-33-q03a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the parameters of an exponential population model from a log plot.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the gradient as k and the vertical intercept as ln a, then calculate a to two significant figures." }, { "id": "cie-2024-w-33-q03b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the doubling time of an exponential population.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the model value to twice its initial value and solve for t, rounding to the nearest hour." }, { "id": "cie-2024-w-33-q04", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a linear equation involving a complex number and its conjugate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write z = x + iy, substitute z and its conjugate, equate real and imaginary parts and solve the simultaneous equations." }, { "id": "cie-2024-w-33-q05a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity involving fourth powers and double angles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use double-angle and power-reduction identities to transform both sides to the same expression." }, { "id": "cie-2024-w-33-q05b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over a half-turn using part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the identity to form a quadratic in cos²α, solve and list the angles in range." }, { "id": "cie-2024-w-33-q06a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "State the coordinates of the intersection of two vector lines.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the parameter value at the common point into either line equation." }, { "id": "cie-2024-w-33-q06b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact cosine of the acute angle between two 3D lines.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take the scalar product of direction vectors and divide by the product of their magnitudes." }, { "id": "cie-2024-w-33-q06c", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of a triangle defined by points on two vector lines.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Construct the two vectors from P, use their cross product or the included-angle area formula and simplify exactly." }, { "id": "cie-2024-w-33-q07a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for a curve given parametrically by trigonometric functions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t, divide to form dy/dx and simplify to the required expression." }, { "id": "cie-2024-w-33-q07b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation at a specified parameter value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 11, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the point and tangent gradient, take the negative reciprocal, then form the integer-coefficient line equation." }, { "id": "cie-2024-w-33-q08a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express a rational function as partial fractions with two linear factors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 12, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set up constants over each factor, clear denominators and compare coefficients." }, { "id": "cie-2024-w-33-q08b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the partial-fraction expression in ascending powers through x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 13, 13 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply geometric-series expansions to both terms and collect coefficients through x²." }, { "id": "cie-2024-w-33-q08c", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "State the interval where the power-series expansion converges.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 13, 13 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the convergence condition to each geometric expansion and intersect the resulting bounds." }, { "id": "cie-2024-w-33-q09a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder when a quartic polynomial is divided by a quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Carry out polynomial division and state the quotient and constant remainder." }, { "id": "cie-2024-w-33-q09b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use polynomial division to evaluate an exact definite integral with arctangent terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 15, 15 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split the integrand into a polynomial plus a reciprocal quadratic, integrate each term and evaluate the limits exactly." }, { "id": "cie-2024-w-33-q10a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation", "mechanics" ], "primaryTopicId": "differentiation", "summary": "Derive a differential equation for a water tank whose leakage rate depends on depth.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 16, 16 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Relate volume rate to depth rate using the base area, subtract the leak rate from the inflow and simplify." }, { "id": "cie-2024-w-33-q10b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Solve the tank depth differential equation using an initial depth and find a later time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 17, 17 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate with the initial condition and substitute the target depth to find t." }, { "id": "cie-2024-w-33-q11a", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the minimum of a trigonometric curve to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 18, 18 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve the stationary equation numerically within the interval and select the minimum." }, { "id": "cie-2024-w-33-q11b", "paperId": "cie-2024-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact area of a shaded trigonometric region using a substitution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_33.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_33.pdf", "qpPages": [ 19, 19 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the specified substitution to transform both the integrand and limits, integrate the resulting expression and simplify exactly." }, { "id": "cie-2024-w-41-q01", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration and string tension for two hanging particles connected over a smooth pulley.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write Newton's second-law equations for both masses with a shared acceleration and tension, then solve simultaneously." }, { "id": "cie-2024-w-41-q02a", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed at the foot of a smooth incline after a particle descends through a stated vertical height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the loss of gravitational potential energy to the gain in kinetic energy." }, { "id": "cie-2024-w-41-q02b", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the constant friction force from the speed at the bottom of a rough incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use work-energy: gravitational work minus work against friction equals the kinetic-energy gain." }, { "id": "cie-2024-w-41-q03", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction of a third force that balances two coplanar forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the known forces into perpendicular components; set the resultant components to zero and recover the unknown magnitude and angle." }, { "id": "cie-2024-w-41-q04a", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph for a bus that accelerates, cruises and decelerates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Show the three straight-line stages, with the cruise at the stated speed and the final segment returning to zero velocity." }, { "id": "cie-2024-w-41-q04b", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Express the bus's constant-speed travel time in terms of its acceleration rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the acceleration and deceleration durations from the velocity changes, then subtract both from the total journey time." }, { "id": "cie-2024-w-41-q04c", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the bus's acceleration when its total journey distance and duration are given.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the area under the velocity-time graph for total distance, substituting the constant-speed duration from part (b), then solve." }, { "id": "cie-2024-w-41-q05a", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Show when two vertically projected particles of equal mass collide after the second is launched one second later.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write each particle's displacement from the launch point using constant-acceleration equations and equate their heights." }, { "id": "cie-2024-w-41-q05b", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the height at which the two projected particles collide.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the collision time into either particle's displacement equation." }, { "id": "cie-2024-w-41-q05c", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find when the coalesced particle returns to the launch point after the collision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conservation of momentum at impact to find the joined particle's velocity, then use vertical-motion equations for its return to the origin." }, { "id": "cie-2024-w-41-q06", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the smallest horizontal force that can keep a particle at rest on a rough incline.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and normal to the plane, include the horizontal force's components, and apply limiting friction in the direction opposing impending motion." }, { "id": "cie-2024-w-41-q07ai", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 1, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the stated maximum power and terminal speed on a level road to show the resistance coefficient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At the greatest constant speed, engine power balances the resistive force times speed." }, { "id": "cie-2024-w-41-q07aii", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest acceleration of a car at a given speed when its engine power is at its maximum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert power to driving force using P = Fv, subtract resistance, then divide the resultant force by the car's mass." }, { "id": "cie-2024-w-41-q07b", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the maximum steady speed of the car while climbing a gradient with resistive force proportional to speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At steady speed set engine power equal to speed times the sum of resistance and the component of weight down the slope, then solve." }, { "id": "cie-2024-w-41-q08", "paperId": "cie-2024-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 8, "questionPart": null, "marks": 7, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled between two instants of rest when acceleration is given as a function of time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_41.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_41.pdf", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration to obtain velocity, use the initial velocity to find its zeros, then integrate the speed over the intervals between them." }, { "id": "cie-2024-w-42-q01a", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the peak speed from a velocity-time graph with a stated uniform deceleration interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the gradient of the decelerating segment to relate its speed change to elapsed time." }, { "id": "cie-2024-w-42-q01b", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the final time on a piecewise velocity-time graph when total distance is known.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the areas representing distance on each forward and return-motion segment, then solve for the remaining interval." }, { "id": "cie-2024-w-42-q02", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed of a block descending a plane when work is done against resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set initial kinetic energy plus gravitational work minus resistance work equal to final kinetic energy." }, { "id": "cie-2024-w-42-q03a", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the constant resistance acting on a cyclist from power, speed and acceleration on level ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use power equals driving force times speed, then apply Newton's second law along the road." }, { "id": "cie-2024-w-42-q03b", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cyclist's steady uphill speed when power and resistance are unchanged.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At steady speed balance the driving force against resistance and the component of weight down the slope, using P = Fv." }, { "id": "cie-2024-w-42-q04a", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal connecting-string tension for two suspended particles in equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces on particle A horizontally and vertically to find the tension in the horizontal string." }, { "id": "cie-2024-w-42-q04b", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a string angle and its tension for two suspended particles joined by a horizontal string.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces on particle B and use the known connecting tension with vertical equilibrium to obtain the angle and sloping-string tension." }, { "id": "cie-2024-w-42-q05a", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the speeds of two vertically moving particles immediately before they collide.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "First find the collision time by equating heights, then use constant-acceleration velocity equations for each particle." }, { "id": "cie-2024-w-42-q05b", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find one particle's speed when it reaches the ground after an impact with a stated rebound velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 9, 9 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply momentum conservation at the collision to find the other particle's post-impact velocity, then use vertical motion to the ground." }, { "id": "cie-2024-w-42-q06a", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find velocity at a specified time when acceleration is piecewise-defined.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the first acceleration rule and apply the initial velocity." }, { "id": "cie-2024-w-42-q06b", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find a velocity expression after the acceleration rule changes, given continuity of velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the second acceleration expression and choose its constant to match the velocity at the changeover time." }, { "id": "cie-2024-w-42-q06c", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled over a time interval when velocity stays positive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 11, 11 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity function over each time interval and add the displacements." }, { "id": "cie-2024-w-42-q07a", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension and acceleration for connected particles where one is on a rough incline and the other hangs freely.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 12, 12 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the weight and limiting friction along the plane, write Newton's second-law equations for both masses, and solve together." }, { "id": "cie-2024-w-42-q07b", "paperId": "cie-2024-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the further distance the inclined-plane particle travels after the hanging particle reaches the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_42.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_42.pdf", "qpPages": [ 13, 13 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the speed at impact and the post-impact acceleration from the remaining forces to find the distance before rest." }, { "id": "cie-2024-w-43-q01", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find an athlete's mass from work done against resistance while increasing speed over a known distance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the athlete's work to the kinetic-energy increase plus work against the constant resistance." }, { "id": "cie-2024-w-43-q02", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the single force that balances a set of four coplanar forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force horizontally and vertically, then take the magnitude and direction opposite to their resultant." }, { "id": "cie-2024-w-43-q03a", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find engine power for a car climbing a slope at constant speed against resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "At steady speed the engine force balances resistance and the downslope weight component; multiply by speed." }, { "id": "cie-2024-w-43-q03b", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a car's acceleration when it travels uphill at a reduced engine power and a stated speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert available power to driving force, subtract resistance and the component of weight down the slope, then divide by mass." }, { "id": "cie-2024-w-43-q04", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the greater possible kinetic-energy loss in a one-dimensional collision of two particles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use momentum conservation with the two possible directions of A's post-impact velocity, find B's velocity in each case and compare energy losses." }, { "id": "cie-2024-w-43-q05a", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension pulling a particle that is on the point of moving across a rough horizontal plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve vertically to find the normal reaction, set friction to its limiting value, then resolve horizontally." }, { "id": "cie-2024-w-43-q05b", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the pulling-string tension when the particle accelerates along the rough horizontal plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the vertical balance for the normal reaction, calculate friction and apply Newton's second law horizontally." }, { "id": "cie-2024-w-43-q06a", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show the speed reached after a stated constant-acceleration interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the given acceleration over the first stage from rest." }, { "id": "cie-2024-w-43-q06b", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch a velocity-time graph for acceleration, constant-speed travel and deceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw a rising straight segment, a horizontal segment and a falling straight segment with the stated times and speed." }, { "id": "cie-2024-w-43-q06c", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find velocity as a function of time during the final constant-deceleration stage.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the change from the cruising speed to rest over the final interval to find the gradient and line equation." }, { "id": "cie-2024-w-43-q06d", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find total distance travelled during the three-stage motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the areas under the velocity-time graph for acceleration, constant speed and deceleration." }, { "id": "cie-2024-w-43-q07a", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find string tension and the time for two connected particles to move until the heavier one reaches the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton's second law to each mass to find acceleration and tension, then use constant-acceleration displacement for the descent." }, { "id": "cie-2024-w-43-q07b", "paperId": "cie-2024-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find how long the lighter particle is at least a specified height above the plane after the other mass reaches the ground.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_43.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_43.pdf", "qpPages": [ 11, 11 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find its speed and height at the transition, then model its subsequent vertical motion under gravity and solve the height inequality." }, { "id": "cie-2024-w-51-q01a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the first success in repeated fair-die throws occurs on the eighth throw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric model: seven failures followed by a success." }, { "id": "cie-2024-w-51-q01b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the second six occurs on the eighth die throw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose which of the first seven throws is the first six, require the other six throws to be non-sixes, then include a six on throw eight." }, { "id": "cie-2024-w-51-q02a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Complete a probability table for a discrete random variable whose probabilities depend linearly on its possible values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute each value into the given rule and use the total-probability condition to determine the constant." }, { "id": "cie-2024-w-51-q02b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of the discrete random variable from its probability table.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(X) and E(X²), then use Var(X) = E(X²) − E(X)²." }, { "id": "cie-2024-w-51-q03a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a cumulative-frequency graph for grouped walking-time data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals against the upper class boundaries and join with a smooth increasing curve." }, { "id": "cie-2024-w-51-q03b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median and interquartile range from a cumulative-frequency graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the 50th and 25th/75th percentile times at cumulative frequencies 100 and 50/150, then subtract the quartiles." }, { "id": "cie-2024-w-51-q03c", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean walking time from grouped frequency data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use class midpoints to find the estimated total time, then divide by the number of pupils." }, { "id": "cie-2024-w-51-q04a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that two marbles drawn under a coin-dependent bag rule have the same colour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Condition on the coin outcome, calculate the same-colour probability for each bag with its replacement rule, and combine by total probability." }, { "id": "cie-2024-w-51-q04b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability both marbles came from one bag given that both are blue.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conditional probability by dividing the joint probability of choosing that bag and two blues by the total probability of two blues." }, { "id": "cie-2024-w-51-q05a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a probability interval for the weight of a normally distributed green apple.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise both bounds using the given mean and standard deviation, then subtract the cumulative probabilities." }, { "id": "cie-2024-w-51-q05b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the probability that fewer than a stated number of red apples exceed a weight threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the success probability from the stated percentage, model the count binomially, apply a continuity correction and standardise the normal approximation." }, { "id": "cie-2024-w-51-q06a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the mean and standard deviation of heights from two normal-distribution percentiles.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert each percentile to a standard normal z-score, form two equations and solve for the parameters." }, { "id": "cie-2024-w-51-q06b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that fewer than eight of ten students exceed a height threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the percentile from the normal model to obtain the individual success probability, then calculate the relevant binomial tail." }, { "id": "cie-2024-w-51-q07a", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of a repeated-letter word with the two copies of one letter adjacent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat the adjacent repeated letters as one block, then divide by factorials for the remaining repeated letters." }, { "id": "cie-2024-w-51-q07b", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count word arrangements with a specified letter at both ends and two other repeated letters separated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Fix the end letters, count the remaining multiset arrangements, and subtract cases where the two target letters are adjacent." }, { "id": "cie-2024-w-51-q07c", "paperId": "cie-2024-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the percentage of four-letter selections containing at least one E and exactly one T.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_51.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_51.pdf", "qpPages": [ 13, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count the favourable selections by the number of Es chosen, respecting repeated copies, then divide by all four-letter selections." }, { "id": "cie-2024-w-52-q01a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of a joint event from a two-way table of sport and year group.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the number in the relevant row-and-column cell by the total population." }, { "id": "cie-2024-w-52-q01b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability from the two-way table.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Restrict the denominator to the stated conditioning group and take the corresponding favourable fraction." }, { "id": "cie-2024-w-52-q01c", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Decide whether two events from the table are independent.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the joint probability with the product of the marginal probabilities." }, { "id": "cie-2024-w-52-q01d", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a binomial probability for fewer than three successes in eight independent attempts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated success probability in B(8,p) and sum the probabilities for 0, 1 and 2 successes." }, { "id": "cie-2024-w-52-q02a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count distinct arrangements of the letters in a word with one repeated letter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide 9! by 2! for the repeated As." }, { "id": "cie-2024-w-52-q02b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of the word with no more than two letters between the two As.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Treat the two As as a pair separated by zero, one or two interior letters, count each placement and arrange the remaining distinct letters." }, { "id": "cie-2024-w-52-q03a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Construct the distribution of a random score determined by a fair coin and a fair die.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate the head and tail cases, enumerate resulting values and combine probabilities for equal scores." }, { "id": "cie-2024-w-52-q03b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of the coin-and-die score variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate its expectation and second moment from the distribution, then subtract the square of the mean." }, { "id": "cie-2024-w-52-q04a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Estimate how many white pine trees fall below a height threshold under a normal model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the height, find its lower-tail probability and multiply by the number of trees." }, { "id": "cie-2024-w-52-q04b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation of red pine heights from a stated upper-tail percentage and mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the tail probability to a standard normal quantile, then solve the standardisation equation for the spread." }, { "id": "cie-2024-w-52-q05a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count groups with fixed numbers of violinists, guitarists and pianists.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the three combinations for choosing the specified category counts." }, { "id": "cie-2024-w-52-q05b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count groups meeting minimum numbers of two student types and a maximum number of pianists.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the feasible category compositions and sum the products of combinations for each case." }, { "id": "cie-2024-w-52-q06a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Display two teams' race times in a back-to-back stem-and-leaf diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Order each team's unit digits about shared tens stems, put the Falcons on the left and add a key." }, { "id": "cie-2024-w-52-q06b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the median and interquartile range of one team's 15 race times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the middle observation and the two quartile positions from the ordered data, then subtract the quartiles." }, { "id": "cie-2024-w-52-q06c", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the overall mean and standard deviation for both teams from summary totals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine both teams' sums and sums of squares and calculate the moments for all 30 times." }, { "id": "cie-2024-w-52-q07a", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the first goal occurs on the fifth attempt.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a geometric model with four failures followed by one success." }, { "id": "cie-2024-w-52-q07b", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the first goal occurs within a stated range of attempts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric probability for the allowed attempt values in the stated range and sum them." }, { "id": "cie-2024-w-52-q07c", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that the second goal occurs by the fifth attempt.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial count for the first five attempts and calculate the probability of at least two goals." }, { "id": "cie-2024-w-52-q07d", "paperId": "cie-2024-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(d)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Approximate the probability that a player scores between two goal-count limits in 75 attempts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_52.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_52.pdf", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a normal approximation to B(75,0.4), apply continuity corrections and evaluate the probability between the adjusted bounds." }, { "id": "cie-2024-w-53-q01a", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that three residents either all own an electric car or none does.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the probabilities of three successes and three failures under the binomial model." }, { "id": "cie-2024-w-53-q01b", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Use a normal approximation to find the probability that more than a threshold number in a sample own electric cars.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Approximate the binomial count by a normal distribution, apply a continuity correction and standardise the upper tail." }, { "id": "cie-2024-w-53-q02a", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Construct the probability distribution for the product of scores from two nonstandard fair dice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the six equally likely face-pair outcomes for each die, tally each distinct product and divide by 36." }, { "id": "cie-2024-w-53-q02b", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected product of the two dice scores.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each product value by its probability and add the results." }, { "id": "cie-2024-w-53-q03a", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that a normally distributed adult male exceeds a stated height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the threshold and evaluate the upper tail of the standard normal distribution." }, { "id": "cie-2024-w-53-q03b", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a height bound when a specified proportion of adults lie between that bound and a given height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given interval probability with the normal model to identify the matching z-score and solve for the height." }, { "id": "cie-2024-w-53-q04a", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a cumulative-frequency graph from grouped plant-height measurements.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply class boundaries for measurements rounded to the nearest centimetre, calculate cumulative totals and plot them." }, { "id": "cie-2024-w-53-q04b", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the 30th percentile from a cumulative-frequency graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the height corresponding to cumulative frequency 45 from the curve." }, { "id": "cie-2024-w-53-q04c", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean and standard deviation from grouped plant-height data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use class midpoints with frequencies to calculate the estimated first and second moments, then obtain the standard deviation." }, { "id": "cie-2024-w-53-q05a", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that a sample of chocolates contains no more than two red-wrapped pieces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with the stated red-wrapper probability and sum the probabilities for 0, 1 and 2." }, { "id": "cie-2024-w-53-q05b", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability the first red-wrapped chocolate appears before the seventh choice.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the complement of getting no red-wrapped chocolates in the first six choices." }, { "id": "cie-2024-w-53-q05c", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability about the first gold-wrapped chocolate given the first unwrapped chocolate occurs fifth.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Condition on the fifth choice being the first unwrapped one, then compare the relevant ordered wrapper sequences using the remaining category probabilities." }, { "id": "cie-2024-w-53-q06a", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count distinct arrangements of a word with repeated letters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the factorial of the total number of letters by the factorials of each repeated-letter count." }, { "id": "cie-2024-w-53-q06b", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count word arrangements whose first and last letters differ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count all multiset arrangements, then subtract arrangements with equal end letters by considering each possible repeated letter." }, { "id": "cie-2024-w-53-q06c", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements with one repeated pair adjacent and another pair separated by exactly two letters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 11, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Place the constrained repeated letters first, then arrange the remaining multiset while accounting for overlapping cases." }, { "id": "cie-2024-w-53-q06d", "paperId": "cie-2024-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that two repeated-letter pairs are separated into different groups when the word is split into groups of five and four.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_53.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_53.pdf", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Count the valid choices of positions for the two pairs across the groups and divide by the total ways to choose the group of five." }, { "id": "cie-2024-w-61-q01a", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate a confidence interval for a population mean from a sample with known standard deviation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal critical value for 96% confidence and the standard error σ/√n around the sample mean." }, { "id": "cie-2024-w-61-q01b", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Assess whether a confidence interval for the population mean contains 96% of individual deer heights.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that the interval estimates the population mean; its confidence level does not describe the proportion of individual observations inside it." }, { "id": "cie-2024-w-61-q02", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that three independent small-bag masses together exceed one large-bag mass when both sizes are normally distributed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the difference of the independent normal variables, add means and variances appropriately, and standardise the upper-tail event." }, { "id": "cie-2024-w-61-q03a", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates of a population mean and variance from a sample's sum and sum of squares.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use x-bar = Σx/n and s² = (Σx² − (Σx)²/n)/(n−1)." }, { "id": "cie-2024-w-61-q03b", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a cutoff for a second sample mean given a specified lower-tail probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the estimates from part (a) as the population parameters, form the standard error for the new sample mean, and invert the normal distribution." }, { "id": "cie-2024-w-61-q04a", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine a parameter in a probability density function by using its total integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density over the stated support and set the integral equal to one." }, { "id": "cie-2024-w-61-q04b", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Evaluate the expectation of the logarithm of a continuous random variable with the given density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate ln(x)f(x) over the support, using integration by parts and the parameter from part (a)." }, { "id": "cie-2024-w-61-q05a", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Choose between a one-tailed and two-tailed test for a claim that a population mean differs from a stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match the direction of the alternative hypothesis to the wording 'different from'." }, { "id": "cie-2024-w-61-q05b", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test a stated normal-population mean using a sample of worm lengths at the 1% significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses, calculate the standardised sample-mean statistic, compare with the two-tailed critical value and give a contextual conclusion." }, { "id": "cie-2024-w-61-q06a", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of more than two arrivals at a service desk during a 15-minute period.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Scale the Poisson mean from ten to fifteen minutes and sum the probabilities for zero, one and two arrivals before taking the complement." }, { "id": "cie-2024-w-61-q06b", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that fewer than four customers arrive at either of two independent service desks during five minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Scale both Poisson means to five minutes, combine independent counts by adding their means, then sum probabilities for 0 to 3." }, { "id": "cie-2024-w-61-q06c", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the minimum wait for a 90% chance of at least one arrival at a service desk.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson zero-arrival probability over t minutes and solve 1 − e^(−rate·t) ≥ 0.9, rounding up to a whole minute." }, { "id": "cie-2024-w-61-q07a", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the Type I error probability for a one-sided Poisson test of whether an accident rate has decreased.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine the critical region at the 5% level under the null rate for the two-year total, then calculate its null probability." }, { "id": "cie-2024-w-61-q07b", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the Poisson rate test when the observed two-year accident count is given.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the observed count with the critical region and state the test decision in context." }, { "id": "cie-2024-w-61-q07c", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the Type II error probability for a repeat Poisson test when the true accident rate is lower.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the same acceptance region as the original test, but calculate its probability under the specified alternative mean for two years." }, { "id": "cie-2024-w-61-q07d", "paperId": "cie-2024-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the probability of more than ten accidents over 30 years at the specified reduced rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_61.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_61.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a suitable normal approximation to the Poisson total, apply a continuity correction and standardise the upper tail." }, { "id": "cie-2024-w-62-q01", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a Poisson approximation to estimate a binomial upper-tail probability for a rare event.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take λ = np and sum the Poisson probabilities for the requested count range." }, { "id": "cie-2024-w-62-q02a", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Recover a sample mean from the midpoint of a confidence interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take the average of the interval's lower and upper limits." }, { "id": "cie-2024-w-62-q02b", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the population standard deviation from a confidence interval for a mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the interval half-width, the 95% normal critical value and standard error σ/√n to solve for σ." }, { "id": "cie-2024-w-62-q02c", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain whether normality of the population changes the estimate derived from the confidence interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State whether the interval method uses the normal critical value under the given sample size and explain the effect of population normality on that calculation." }, { "id": "cie-2024-w-62-q03a", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether a binomial success probability increased after a menu change, using the observed count at a 10% level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State null and alternative hypotheses, find the upper-tail critical region under the null, calculate its probability and compare the observed count." }, { "id": "cie-2024-w-62-q03b", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Comment on whether a binomial model is suitable when the number of employees is known.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the model's maximum possible count with the total number of employees available." }, { "id": "cie-2024-w-62-q04a", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a probability for the mean of a large sample from a normal population.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sampling distribution with the same mean and standard deviation divided by √n, then standardise the event." }, { "id": "cie-2024-w-62-q04b", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a sample-mean threshold corresponding to a specified upper-tail probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the probability to a standard normal quantile and transform back using the sampling distribution of the mean." }, { "id": "cie-2024-w-62-q05a", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a Poisson probability for the number of lemon sweets in a bag.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the Poisson probabilities for the allowed counts and add them." }, { "id": "cie-2024-w-62-q05b", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the combined number of two independent types of sweets is below a threshold.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the independent Poisson means and evaluate the lower-tail probability for the total." }, { "id": "cie-2024-w-62-q05c", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use approximating distributions to compare total lemon and orange sweet counts across ten bags.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Aggregate each type's independent Poisson counts over ten bags, approximate the difference by a normal distribution, and apply a continuity correction." }, { "id": "cie-2024-w-62-q06a", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Interpret the lower and upper support limits of a continuous waiting-time density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the constants as the minimum and maximum possible completion times in context." }, { "id": "cie-2024-w-62-q06b", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the relation between the two endpoints of a density's support from its total probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the given density over its support and set the result to one, then rearrange for the stated relation." }, { "id": "cie-2024-w-62-q06c", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a stated expectation to find the support limits of a continuous density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate E(ln X) as an integral, combine it with the relation from part (b), and solve for both constants." }, { "id": "cie-2024-w-62-q06d", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the median of a continuous distribution with determined support limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the cumulative probability up to the median equal to one half and solve within the support." }, { "id": "cie-2024-w-62-q07a", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State the probability of a Type I error for a test at the 2% significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the significance level as the probability of rejecting the null hypothesis when it is true." }, { "id": "cie-2024-w-62-q07b", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test whether a population mean tree height is greater than a known value using sample summaries.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses, estimate the standard error from the supplied sample totals, calculate the test statistic and compare with the one-tailed 2% critical value." }, { "id": "cie-2024-w-62-q07c", "paperId": "cie-2024-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Comment on whether a Type II error can be identified after the test decision is known.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_62.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_62.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that a Type II error means failing to reject a false null and cannot be determined solely from a correct test procedure or its observed decision." }, { "id": "cie-2024-w-63-q01a", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate a confidence interval for a population mean from a sample with known standard deviation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal critical value for 96% confidence and the standard error σ/√n around the sample mean." }, { "id": "cie-2024-w-63-q01b", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Assess whether a confidence interval for the population mean contains 96% of individual deer heights.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that the interval estimates the population mean; its confidence level does not describe the proportion of individual observations inside it." }, { "id": "cie-2024-w-63-q02", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that three independent small-bag masses together exceed one large-bag mass when both sizes are normally distributed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form the difference of the independent normal variables, add means and variances appropriately, and standardise the upper-tail event." }, { "id": "cie-2024-w-63-q03a", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates of a population mean and variance from a sample's sum and sum of squares.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use x-bar = Σx/n and s² = (Σx² − (Σx)²/n)/(n−1)." }, { "id": "cie-2024-w-63-q03b", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a cutoff for a second sample mean given a specified lower-tail probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the estimates from part (a) as the population parameters, form the standard error for the new sample mean, and invert the normal distribution." }, { "id": "cie-2024-w-63-q04a", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine a parameter in a probability density function by using its total integral.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density over the stated support and set the integral equal to one." }, { "id": "cie-2024-w-63-q04b", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Evaluate the expectation of the logarithm of a continuous random variable with the given density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate ln(x)f(x) over the support, using integration by parts and the parameter from part (a)." }, { "id": "cie-2024-w-63-q05a", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Choose between a one-tailed and two-tailed test for a claim that a population mean differs from a stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match the direction of the alternative hypothesis to the wording 'different from'." }, { "id": "cie-2024-w-63-q05b", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test a stated normal-population mean using a sample of worm lengths at the 1% significance level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses, calculate the standardised sample-mean statistic, compare with the two-tailed critical value and give a contextual conclusion." }, { "id": "cie-2024-w-63-q06a", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of more than two arrivals at a service desk during a 15-minute period.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Scale the Poisson mean from ten to fifteen minutes and sum the probabilities for zero, one and two arrivals before taking the complement." }, { "id": "cie-2024-w-63-q06b", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that fewer than four customers arrive at either of two independent service desks during five minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Scale both Poisson means to five minutes, combine independent counts by adding their means, then sum probabilities for 0 to 3." }, { "id": "cie-2024-w-63-q06c", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the minimum wait for a 90% chance of at least one arrival at a service desk.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson zero-arrival probability over t minutes and solve 1 − e^(−rate·t) ≥ 0.9, rounding up to a whole minute." }, { "id": "cie-2024-w-63-q07a", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the Type I error probability for a one-sided Poisson test of whether an accident rate has decreased.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine the critical region at the 5% level under the null rate for the two-year total, then calculate its null probability." }, { "id": "cie-2024-w-63-q07b", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the Poisson rate test when the observed two-year accident count is given.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the observed count with the critical region and state the test decision in context." }, { "id": "cie-2024-w-63-q07c", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the Type II error probability for a repeat Poisson test when the true accident rate is lower.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the same acceptance region as the original test, but calculate its probability under the specified alternative mean for two years." }, { "id": "cie-2024-w-63-q07d", "paperId": "cie-2024-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the probability of more than ten accidents over 30 years at the specified reduced rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_qp_63.pdf", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w24_ms_63.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a suitable normal approximation to the Poisson total, apply a continuity correction and standardise the upper tail." }, { "id": "cie-2024-s-12-q01", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2024-s-12-q02", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2: Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2024-s-12-q03a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-12-q03b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-12-q04a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-12-q04b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "4(b): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the output of the inner function into the outer function, respect the given domains, and simplify." }, { "id": "cie-2024-s-12-q04c", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-12-q05a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-12-q05bi", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "5(b)(i): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2024-s-12-q05bii", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b)(ii): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-12-q06a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-12-q06b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "6(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2024-s-12-q07a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 12, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-12-q07b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a diameter perpendicular to a given tangent, using a larger parameter value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 12, 13 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this part challenging and notes that around 30% omitted it; a common detour was finding the tangent-circle intersection unnecessarily (9709/12, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 5 ], "tagConfidence": "reviewed", "answerGuide": "Use the perpendicular gradient relationship to find the diameter direction, then substitute the relevant point to determine its equation." }, { "id": "cie-2024-s-12-q08ai", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)(i)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use right-triangle trigonometry to determine a chord length in a plate-geometry setting.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 14, 14 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge reports that about 40% earned no marks; many used an arc-length approach instead of the required right-triangle trigonometry (9709/12, Question 8).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Set up the right triangle and use the appropriate sine, cosine or tangent ratio; do not substitute an arc-length calculation for the chord." }, { "id": "cie-2024-s-12-q08aii", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find a composite plate area by combining a trapezium, triangles and circular sectors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says responses were mixed and identifies the trapezium height as a particular obstacle (9709/12, Question 8).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Split the shape into the central trapezium, adjoining triangles and sectors; find the perpendicular height before adding the component areas." }, { "id": "cie-2024-s-12-q08b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Recalculate a perimeter after the radius changes, including the resulting change to the chord.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 14, 15 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Nearly one fifth omitted the part, and many kept the original chord length after the radius changed (9709/12, Question 8).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Recompute the chord from the new radius and fixed angle, then add the updated straight and curved boundary lengths." }, { "id": "cie-2024-s-12-q09a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 16, 16 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-12-q09b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 16, 17 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-12-q10a", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 18, 18 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-12-q10b", "paperId": "cie-2024-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Use a gradient condition to locate a point on a curve, then form a perpendicular bisector.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_12.pdf", "qpPages": [ 18, 19 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "About one fifth omitted the part; many who formed the equation lost the final accuracy/form mark (9709/12, Question 10).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate to apply the gradient condition, solve for the point, then use the midpoint and negative reciprocal gradient for the bisector." }, { "id": "cie-2024-s-13-q01", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2024-s-13-q02a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Identify a translated cosine graph from its intercept and minimum features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a minority answered well and the part was often omitted; candidates who recognised a simple translation were more successful (9709/13, Question 2).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Compare the key intercept and minimum with the basic cosine curve, then describe the translation that maps one graph to the other." }, { "id": "cie-2024-s-13-q02b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-13-q03a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "3(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2024-s-13-q03b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "3(b): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2024-s-13-q04a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-13-q04b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-13-q05a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2024-s-13-q05b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2024-s-13-q05c", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 7, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-13-q06a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-13-q06b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Apply a horizontal stretch and translation to a curve obtained by integrating a given derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this transformation sequence caused difficulty; candidates often reversed the order or direction of the transformations (9709/13, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 8 ], "tagConfidence": "reviewed", "answerGuide": "Integrate first to identify the base curve, then apply the stated horizontal stretch and translation in the correct order." }, { "id": "cie-2024-s-13-q07a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "7(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2024-s-13-q07b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Count arithmetic-sequence terms lying inside a specified interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority used the nth-term method successfully; errors included using partial sums and failing to convert a decimal index into an integer count (9709/13, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 9 ], "tagConfidence": "reviewed", "answerGuide": "Write the interval inequalities for the nth term, solve for the integer range of n, and count the admissible indices." }, { "id": "cie-2024-s-13-q08", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8", "marks": 8, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2024-s-13-q09a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a curve with fractional powers and use the result to determine a tangent gradient and line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes differentiation here as very challenging for some candidates (9709/13, Question 9).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 9 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite fractional powers consistently, differentiate term by term, evaluate the derivative at the stated x-value, then use point-gradient form." }, { "id": "cie-2024-s-13-q09b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(b): Set up a rotation integral to calculate the stated volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 12, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the radius in terms of the integration variable, set up the disc or washer integral with the correct limits, and evaluate it." }, { "id": "cie-2024-s-13-q10a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "10(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2024-s-13-q10b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "10(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2024-s-13-q10c", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "10(c): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2024-s-13-q11a", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "11(a): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2024-s-13-q11b", "paperId": "cie-2024-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2024, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_13.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_13.pdf", "qpPages": [ 16, 17 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-22-q01", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2024-s-22-q02", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation by taking logarithms and simplifying the resulting expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes very few full-mark responses, often because the logarithm on the right-hand side was combined incorrectly (9709/22, Question 2).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms on both sides, apply log laws carefully, and solve for the unknown." }, { "id": "cie-2024-s-22-q03a", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2024-s-22-q03b", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2024-s-22-q04", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-22-q05a", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2024-s-22-q05b", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS 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"cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-22-q06b", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2024-s-22-q06c", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(c): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-22-q06d", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Continue an iterative process to obtain the required x-coordinate to the specified precision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], 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stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-22-q07b", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-22-q07c", "paperId": "cie-2024-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate a definite trigonometric integral using an identity established earlier in the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_22.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few responses were correct; many left it blank or did not use the identity established earlier (9709/22, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 16 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the earlier identity into the integrand, simplify to a standard integral, and evaluate using the given limits." }, { "id": "cie-2024-s-23-q01", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2024-s-23-q02", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation by taking logarithms and simplifying the resulting expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes very few full-mark responses, often because the logarithm on the 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"https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2024-s-23-q03b", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2024-s-23-q04", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-23-q05a", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2024-s-23-q05b", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "5(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2024-s-23-q06a", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-23-q06b", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "6(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2024-s-23-q06c", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(c): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-23-q06d", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Continue an iterative process to obtain the required x-coordinate to the specified precision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many responses were blank or did not give sufficient accuracy, as noted in the report's comments on Question 6(d) (9709/23).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Substitute each iterate into the given recurrence until the stopping requirement is met, retaining enough unrounded digits before rounding the final value." }, { "id": "cie-2024-s-23-q07a", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-23-q07b", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-23-q07c", "paperId": "cie-2024-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2024, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate a definite trigonometric integral using an identity established earlier in the question.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_23.pdf", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few responses were correct; many left it blank or did not use the identity established earlier (9709/23, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Substitute the earlier identity into the integrand, simplify to a standard integral, and evaluate using the given limits." }, { "id": "cie-2024-s-32-q01a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2024-s-32-q01b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2024-s-32-q02", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-32-q03a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2024-s-32-q03b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "3(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-32-q04", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate an exponential curve and find its gradient at a point with a stated y-coordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Finding the required x-value was a problem, and the report says fully correct responses were unusual (9709/32, Question 4).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 25 ], "tagConfidence": "reviewed", "answerGuide": "Take logarithms or otherwise solve the curve equation for x, differentiate implicitly, then substitute the point coordinates into dy/dx." }, { "id": "cie-2024-s-32-q05a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-32-q05b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2024-s-32-q05c", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "5(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2024-s-32-q06a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-32-q06b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate an exact integral involving a product of an exponential and an algebraic factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes sign and coefficient errors in integration by parts; many candidates gained only method marks (9709/32, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 26 ], "tagConfidence": "reviewed", "answerGuide": "Choose the exponential as the factor to integrate, apply integration by parts with signed coefficients carefully, and evaluate the exact endpoints." }, { "id": "cie-2024-s-32-q07a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-32-q07b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Evaluate an exact trigonometric integral using a suitable identity and double-angle form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report found this part substantially more challenging; direct integration attempts commonly led to errors (9709/32, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 26 ], "tagConfidence": "reviewed", "answerGuide": "Use the relevant trigonometric identity to rewrite the integrand, apply a double-angle identity where needed, then integrate exactly." }, { "id": "cie-2024-s-32-q08a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "8(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2024-s-32-q08b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-32-q08c", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a point on a line whose distance from another fixed point has a specified value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 12, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority solved the resulting distance equation; many misread the notation or what the condition required (9709/32, Question 8).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 27 ], "tagConfidence": "reviewed", "answerGuide": "Parameterise a point on the line, form the squared-distance equation, solve for the parameter, and check each solution against the condition." }, { "id": "cie-2024-s-32-q09a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2024-s-32-q09b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2024-s-32-q09c", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "9(c): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2024-s-32-q09d", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Prove an exact tangent relation using the arguments of complex numbers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates produced a correct proof; the report says the product-of-arguments property was generally not used (9709/32, Question 9).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 27 ], "tagConfidence": "reviewed", "answerGuide": "Express each complex argument, use the argument rule for a product, and connect the resulting angle relation to the tangent identity." }, { "id": "cie-2024-s-32-q10a", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2024-s-32-q10b", "paperId": "cie-2024-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 8, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-33-q01", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=2", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 2, 2 ], "msPages": [ 102, 102 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-33-q02", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=3", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 3, 3 ], "msPages": [ 104, 104 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2024-s-33-q03", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=4", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 4, 4 ], "msPages": [ 105, 105 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-33-q04", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4: Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=5", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 5, 5 ], "msPages": [ 106, 106 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2024-s-33-q05", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Decompose a rational expression into partial fractions when the numerator degree first requires division.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=6", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 6, 6 ], "msPages": [ 107, 107 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes this as highly challenging for many; an unsuitable partial-fraction setup was a common obstacle (9709/33, Question 5).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 30 ], "tagConfidence": "reviewed", "answerGuide": "Perform polynomial division first, then decompose the proper rational remainder using coefficients matched across the identity." }, { "id": "cie-2024-s-33-q06a", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=7", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 7, 7 ], "msPages": [ 108, 108 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2024-s-33-q06b", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Determine the greatest argument of a complex number whose locus lies in a specified Argand region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=7", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 7, 7 ], "msPages": [ 108, 108 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few candidates earned full marks; the report points to difficulty interpreting the region and using tangent geometry (9709/33, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 30 ], "tagConfidence": "reviewed", "answerGuide": "Sketch the Argand region and the origin-based angle, then identify the limiting ray or boundary point that gives the greatest argument." }, { "id": "cie-2024-s-33-q07a", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=8", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 8, 8 ], "msPages": [ 109, 109 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2024-s-33-q07b", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=8", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 8, 8 ], "msPages": [ 109, 109 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-33-q07c", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(c): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=8", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 8, 9 ], "msPages": [ 110, 110 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-33-q08a", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=10", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 10, 10 ], "msPages": [ 110, 110 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-33-q08b", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=10", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 10, 11 ], "msPages": [ 111, 111 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-33-q09a", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Model a changing cuboid quantity with inverse proportionality and use the chain rule to find a rate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=12", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 12, 12 ], "msPages": [ 112, 112 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many candidates did not know how to start and sometimes omitted the proportionality constant needed for a valid model (9709/33, Question 9).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 31 ], "tagConfidence": "reviewed", "answerGuide": "Write the proportional relation with a constant, relate the changing variables, then differentiate with respect to time using the chain rule." }, { "id": "cie-2024-s-33-q09b", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=12", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 12, 13 ], "msPages": [ 113, 113 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2024-s-33-q10a", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Use the stated coordinates to determine the required angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=14", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 14, 14 ], "msPages": [ 114, 114 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2024-s-33-q10b", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=14", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 14, 15 ], "msPages": [ 115, 115 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2024-s-33-q11", "paperId": "cie-2024-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2024, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11", "marks": 9, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "11: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_33.pdf#page=16", "msUrl": "https://satprep.co.in/wp-content/uploads/2020/10/A_Pure_Math_Paper-3-Answer-1-2.pdf", "qpPages": [ 16, 16 ], "msPages": [ 116, 116 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2024-s-42-q01", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q02a", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2024-s-42-q02b", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q03", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2024-s-42-q04ai", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a)(i): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q04aii", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a)(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2024-s-42-q04b", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q05", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5: Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2024-s-42-q06a", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q06b", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show a stated speed using the total kinetic-energy loss over two collisions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Only a minority completed the demonstration correctly, according to the report's discussion of Question 6(b) (9709/42).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Apply the collision energy-loss relationship at both impacts, combine the kinetic-energy changes, and solve for the requested speed." }, { "id": "cie-2024-s-42-q06ci", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time interval between two successive collisions in a vertical-motion scenario.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was not well answered; very few progressed beyond an initial travel-time calculation because they missed the motion before the first collision (9709/42, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Track each particle's motion from the initial projection through the first impact, then calculate the next collision time from the correct post-impact velocities and positions." }, { "id": "cie-2024-s-42-q06cii", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)(ii)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(c)(ii): Use the motion equations to calculate the requested elapsed time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q07a", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-42-q07b", "paperId": "cie-2024-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use conservation of energy to find the speed of connected particles when they reach the same vertical height.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this the most challenging part of the paper; many did not establish the required displacement relationship (9709/42, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 38 ], "tagConfidence": "reviewed", "answerGuide": "Relate the particles' displacements using the string constraint, compare initial and final gravitational potential energy, then solve the energy equation for speed." }, { "id": "cie-2024-s-43-q01", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q02", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2024-s-43-q03a", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q03b", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q04a", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q04b", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q05a", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q05bi", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b)(i): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 6, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q05bii", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b)(ii): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 6, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2024-s-43-q06a", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2024-s-43-q06b", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2024-s-43-q06c", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy balance for a cyclist descending with variable resistance and constant power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many responses gained only partial credit; common omissions included work, potential-energy changes, and correct signs (9709/43, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 40 ], "tagConfidence": "reviewed", "answerGuide": "Account for the work done by the cyclist, the change in gravitational potential energy, and the distance-dependent resistance with consistent signs." }, { "id": "cie-2024-s-43-q07a", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the distance travelled along a path containing a rough incline from an energy relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Both parts of Question 7 were described as very challenging; the report notes errors in propagating the speed through the stages (9709/43, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Write the energy balance from the initial point to the required position, include gravitational change and friction work, and solve for the distance." }, { "id": "cie-2024-s-43-q07b", "paperId": "cie-2024-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2024, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find a friction coefficient and the later speed after motion over rough and inclined sections.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_43.pdf", "qpPages": [ 10, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says this was even more challenging than part (a); many gained at most one mark, omitted resistance terms, or used the wrong initial speed (9709/43, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Apply energy changes section by section, include work against friction on each rough section, and carry the resulting speed into the next stage." }, { "id": "cie-2024-s-52-q01a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-52-q01b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-52-q01c", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-52-q02a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of drawing at least one red marble when the number available depends on a coin result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says candidates found the context challenging; successful solutions separated the coin cases before calculating the draw probability (9709/52, Question 2).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 47 ], "tagConfidence": "reviewed", "answerGuide": "Condition on each possible coin outcome, calculate the chance of at least one red in that case, and combine using the case probabilities." }, { "id": "cie-2024-s-52-q02b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2024-s-52-q03a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2024-s-52-q03b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(b): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2024-s-52-q04a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2024-s-52-q04b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "4(b): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2024-s-52-q04c", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2024-s-52-q05a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2024-s-52-q05b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the distribution of the total from two coin outcomes sampled without replacement.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes replacement errors, impossible totals, and probabilities that did not sum correctly (9709/52, Question 5).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 48 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate the first-draw and second-draw outcomes without replacing the first result, combine outcomes by total, and check probabilities sum to one." }, { "id": "cie-2024-s-52-q05c", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2024-s-52-q06a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-52-q06b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-52-q06c", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that three selected residents include one person from each of three rating groups.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes that many found this challenging or omitted it; treating the selection as a binomial event was a common mismatch (9709/52, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 49 ], "tagConfidence": "reviewed", "answerGuide": "Count selections with one from each group and divide by the total number of equally likely selections." }, { "id": "cie-2024-s-52-q07a", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2024-s-52-q07b", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2024-s-52-q07c", "paperId": "cie-2024-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of repeated letters in which vowels and consonants alternate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_52.pdf", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Many candidates were challenged and the part was surprisingly often omitted; the report indicates that a systematic count was the key approach (9709/52, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 49 ], "tagConfidence": "reviewed", "answerGuide": "Arrange the vowel positions and consonant positions separately, account for repeated letters in each group, then combine the valid arrangements without double-counting." }, { "id": "cie-2024-s-53-q01a", "paperId": "cie-2024-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2024-s-53-q01b", "paperId": "cie-2024-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_53.pdf", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2024-s-53-q01c", "paperId": "cie-2024-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_53.pdf", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2024-s-53-q02a", "paperId": "cie-2024-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2024, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-53-q02b", "paperId": "cie-2024-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level 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"topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a conditional probability for a split of goals with a specified total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_62.pdf", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes confusion between an exact total and an at-least condition, and between adding and multiplying the relevant probabilities (9709/62, Question 5).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Use the specified condition to define the sample space, identify the joint event, then form the conditional probability as a ratio." }, { "id": "cie-2024-s-62-q06a", "paperId": "cie-2024-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "6(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2024-s-62-q06b", "paperId": "cie-2024-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Calculate the probability of a Type II error for a second one-sided hypothesis test when the true mean is specified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report describes this as not well attempted; many did not find the critical value or follow the Type II error procedure (9709/62, Question 6).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 58 ], "tagConfidence": "reviewed", "answerGuide": "Find the critical boundary under the null hypothesis, then evaluate the probability of failing to reject the null under the stated true mean." }, { "id": "cie-2024-s-62-q07a", "paperId": "cie-2024-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_62.pdf", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2024-s-62-q07b", "paperId": "cie-2024-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_62.pdf", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2024-s-62-q07c", "paperId": "cie-2024-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "integration", "distributions" ], "primaryTopicId": "integration", "summary": "Evaluate an exact expected value by integrating x times a given density function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many candidates struggled with the integration by parts required; the report notes errors in the product with a cosine term and in signs (9709/62, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 58 ], "tagConfidence": "reviewed", "answerGuide": "Set up E(X) as the integral of x times the density over its support, apply integration by parts carefully, and evaluate the exact limits." }, { "id": "cie-2024-s-63-q01a", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2024-s-63-q01b", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2024-s-63-q02a", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the appropriate z or t critical value with the sample standard error, then calculate and round both interval endpoints as requested." }, { "id": "cie-2024-s-63-q02b", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Explain why the Central Limit Theorem is needed when the population distribution is not specified.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this was not generally well answered; a valid explanation needed both the decision and the reason tied to the unknown population distribution (9709/63, Question 2).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 59 ], "tagConfidence": "reviewed", "answerGuide": "State that the Central Limit Theorem is used because the population distribution is unspecified, allowing the sample-mean distribution to be approximated for a sufficiently large sample." }, { "id": "cie-2024-s-63-q03", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2024-s-63-q04", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "4: Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2024-s-63-q05a", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required 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"sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2024-s-63-q06a", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-63-q06b", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2024-s-63-q06c", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(c): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2024-s-63-q06d", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "6(d): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2024-s-63-q07a", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2024-s-63-q07b", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate a conditional probability for a Poisson variable by identifying compatible paired values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only around one fifth achieved full marks; the report highlights the need to identify compatible pairs and their joint probabilities (9709/63, Question 7).", "difficultySourceUrl": "https://www.cambridgeinternational.org/Images/567737-june-2024-examiner-report.pdf", "sourceEvidencePages": [ 60 ], "tagConfidence": "reviewed", "answerGuide": "List the value pairs that meet the condition, calculate each joint Poisson probability, sum them, then divide by the probability of the conditioning event." }, { "id": "cie-2024-s-63-q07c", "paperId": "cie-2024-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2024, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_s24_ms_63.pdf", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "ial-2023-1-WDM11-01-q1a", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Show both nearest-neighbour routes from A and give their lengths.", "answerGuide": "Follow the nearest available edge at each step, return to A, and compare the two possible routes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q1b", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound for the classroom tour by deleting A.", "answerGuide": "Construct a minimum spanning tree on the remaining vertices and add the two shortest links needed to connect A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q1c", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the smallest interval containing the optimal tour length.", "answerGuide": "Use the nearest-neighbour route lengths as upper bounds and the deleted-vertex result as a lower bound.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only the strongest candidates gave a correct interval for the final bound, with the required inequalities.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20230302.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" }, "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2023-1-WDM11-01-q2a", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra’s algorithm to find the shortest route from A to J.", "answerGuide": "Label vertices in increasing permanent value, record each working value in order, then trace back the shortest path and state its length.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q2b", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the shortest route from J to H via A and its length.", "answerGuide": "Combine the shortest J-to-A and A-to-H paths and add their distances.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q2c", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the closed route-inspection tour and list the roads to repeat.", "answerGuide": "Identify the odd vertices, compare all pairings, repeat the minimum-weight pairing, and add its cost to the network total.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q2d", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Choose the best finishing vertex for a route starting at G and give its length.", "answerGuide": "Compare the relevant odd-vertex pairings for each allowable endpoint, choose the least extra distance, and add it to the network weight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this part discriminated well and that few justified the required shortest path explicitly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20230302.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" }, "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2023-1-WDM11-01-q3a", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Pack the listed decimal lengths using first fit.", "answerGuide": "Scan the items in order and place each in the first bin with enough remaining capacity.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q3b", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Perform one bubble-sort pass and report its comparisons and swaps.", "answerGuide": "Compare adjacent values from left to right, show the resulting list, and count each comparison and interchange.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q3c", "paperId": "ial-2023-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2023, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use quick sort to complete the descending order of the updated list.", "answerGuide": "Choose an allowed middle pivot consistently, partition each sublist around it, and show the successive passes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-que-20230126.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wdm11-01-rms-20230302.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-01-que-20230126.pdf", "markScheme": "wdm11-01-rms-20230302.pdf", "examinerReport": "wdm11-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "e78329ded0a4425fbd61a14d9d5c2b8e606ad8d3ba10390c0a4dd9829d85f4f6", "markScheme": "a10b15b77c33b4ba9dc9d0e1015df82c48c9bf0c1f71d923393b25edbd0d3205", "examinerReport": "54e94ff071423d0393822102fab1f41e46ee4ca27a404c3812bd0148aadab70b" } }, { "id": "ial-2023-1-WDM11-01-q3d", "paperId": "ial-2023-1-WDM11-01", "board": 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"wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": "7773a7f1ad10f55662ac8e5bf5d547548191fecfacef52ef360720d370e5bc5e", "examinerReport": "11bd164a2867bdac786e5af57b727b5121032a19ffa15df3affc3ff5af258630" } }, { "id": "ial-2023-1-WMA12-01-q2b", "paperId": "ial-2023-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the surface-area function.", "answerGuide": "Apply the power rule to each term of S(x).", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": "7773a7f1ad10f55662ac8e5bf5d547548191fecfacef52ef360720d370e5bc5e", "examinerReport": "11bd164a2867bdac786e5af57b727b5121032a19ffa15df3affc3ff5af258630" } }, { "id": "ial-2023-1-WMA12-01-q2c", "paperId": "ial-2023-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the stationary value of x.", "answerGuide": "Set dS/dx to zero and solve, keeping the positive length.", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": 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"https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": "7773a7f1ad10f55662ac8e5bf5d547548191fecfacef52ef360720d370e5bc5e", "examinerReport": "11bd164a2867bdac786e5af57b727b5121032a19ffa15df3affc3ff5af258630" } }, { "id": "ial-2023-1-WMA12-01-q2e", "paperId": "ial-2023-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the minimum surface area.", "answerGuide": "Substitute the stationary x-value into S(x).", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": "7773a7f1ad10f55662ac8e5bf5d547548191fecfacef52ef360720d370e5bc5e", "examinerReport": "11bd164a2867bdac786e5af57b727b5121032a19ffa15df3affc3ff5af258630" } }, { "id": "ial-2023-1-WMA12-01-q3a", "paperId": "ial-2023-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the first four terms of the binomial expansion.", "answerGuide": "Use the generalized binomial theorem and collect the constant through cubic terms.", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, 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coefficients form an arithmetic progression.", "answerGuide": "Extract the coefficients, equate consecutive differences and solve the resulting quadratic.", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": "7773a7f1ad10f55662ac8e5bf5d547548191fecfacef52ef360720d370e5bc5e", "examinerReport": "11bd164a2867bdac786e5af57b727b5121032a19ffa15df3affc3ff5af258630" } }, { "id": "ial-2023-1-WMA12-01-q4i", "paperId": "ial-2023-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation using logarithm laws.", "answerGuide": "Combine the logs, equate positive arguments and reject values outside the original domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": "7773a7f1ad10f55662ac8e5bf5d547548191fecfacef52ef360720d370e5bc5e", "examinerReport": "11bd164a2867bdac786e5af57b727b5121032a19ffa15df3affc3ff5af258630" } }, { "id": "ial-2023-1-WMA12-01-q4ii", "paperId": "ial-2023-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Express y in terms of a from the logarithmic identity.", "answerGuide": "Apply change of base and logarithm laws, then use the stated restrictions.", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson identifies the second part of Question 4 among the most challenging parts of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-pef-20230302.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": 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"Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Use the remainder theorem to show the second relation.", "answerGuide": "Evaluate f(−p) and set it equal to the given remainder 9.", "qpUrl": "https://www.alevelcopilot.com/whfile/54/53/2d/8d/ff/88/8a/a0/25/6d/76/cd/0c/7e/cb/20.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/1c/1c/0f/2a/c2/78/7d/31/6a/0b/9f/f0/60/f4/1a/84.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20230117.pdf", "markScheme": "wma12-01-rms-20230302.pdf", "examinerReport": "wma12-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "cd9168109964df91cc6691f25b383b38689be8c1d295510395e1a670b3a460c6", "markScheme": 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constant-speed section, and a straight-line deceleration to rest.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q1b", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration time T from the total journey distance.", "answerGuide": "Use the area under the speed-time graph for 4.8 km and the given acceleration-to-deceleration ratio.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q1c", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the train’s acceleration during the first T seconds.", "answerGuide": "Use the change in speed divided by T.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q2a", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find both particles’ speeds immediately after their direct collision.", "answerGuide": "Choose a consistent positive direction, conserve momentum, use the given speed difference, and solve the simultaneous equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q2b", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the impulse on particle B.", "answerGuide": "Calculate B’s change in momentum from its signed initial and final velocities.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q3a", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed two seconds after starting.", "answerGuide": "Add acceleration multiplied by time to the initial velocity vector, then take the magnitude.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q3b", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the angle between the i direction and the particle’s motion at t = 2.", "answerGuide": "Use the velocity components at t = 2 and an appropriate inverse trigonometric ratio.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q3c", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find when the velocity is parallel to the stated vector.", "answerGuide": "Write the velocity as a function of time and equate the component ratio to the direction vector’s ratio.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q4", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the rod’s weight and centre-of-mass position from the two tipping conditions.", "answerGuide": "Take moments about C for upward tipping and D for downward tipping, then solve the equations for W and x.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q5a", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Show that the distance PQ satisfies x = 3u.", "answerGuide": "Apply constant-acceleration motion over the first 12 seconds using the initial and final speeds.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q5b", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find u using the second acceleration stage and the distance QR.", "answerGuide": "Use the speed change and 1.5 m/s² acceleration over QR, then substitute x = 3u.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q5c", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled during the first 14 seconds.", "answerGuide": "Add the 12-second distance from P to Q to the distance covered in the next two seconds with the new acceleration.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q6", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the two rope directions and tension in the boat model.", "answerGuide": "Resolve forces parallel and perpendicular to the motion; use equilibrium with the water resistance to solve for P and α.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q7a", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the combined mass of the two boxes in the accelerating lift.", "answerGuide": "Apply Newton’s second law to the lift and boxes together, using the cable tension and upward acceleration.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=17", "qpPages": [ 22, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q7b", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the mass of the upper box from its contact force.", "answerGuide": "Apply Newton’s second law to box P alone and use the stated normal contact force.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=17", "qpPages": [ 22, 23 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q8a", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the parcel’s acceleration while it is pulled up the rough incline.", "answerGuide": "Resolve the pulling force, weight and normal reaction; calculate friction and apply Newton’s second law parallel to the plane.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=18", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q8b", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the parcel’s speed after moving 5 m up the plane.", "answerGuide": "Use constant-acceleration motion from A to B with the acceleration from part (a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=18", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME01-01-q8c", "paperId": "ial-2023-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Decide whether the parcel stays at rest after reaching C.", "answerGuide": "Compare the downslope weight component with the maximum available friction at C and justify the conclusion.", "qpUrl": "https://www.alevelcopilot.com/whfile/fe/99/bc/83/03/f5/7d/03/00/f1/46/b3/4f/8a/eb/fb.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/31/37/a0/8e/a7/f5/b4/e9/d7/6b/4c/e2/ba/fd/a9/57.pdf#page=18", "qpPages": [ 24, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20230118.pdf", "markScheme": "wme01-01-rms-20230302.pdf", "examinerReport": "wme01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "954406780c1e33d892e44fa026c9c074bcfcc8973cf360b478f1719047f21010", "markScheme": "8cf01dbe90a42e218da137c6427ee83e9b0f228b1f890098fc2e8e36aa96494d", "examinerReport": "84195b989a2b3a0a015c6653c7fad73a8bfd59e5f2c013c59c20eae9a8e475ed" } }, { "id": "ial-2023-1-WME02-01-q1a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the truck’s resistance force on the horizontal road.", "answerGuide": "Use power = driving force × speed and apply Newton’s second law to the horizontal motion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q1b", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the truck’s speed while climbing the inclined road.", "answerGuide": "Resolve weight down the slope, include resistance and deceleration in Newton’s second law, then use the engine power relation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report explicitly says the inclined-road equation of motion was more challenging and identifies omitted terms and sign errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20230302.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" }, "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2023-1-WME02-01-q2", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the possible values of the vertical impulse parameter from the kinetic-energy change.", "answerGuide": "Use impulse-momentum to express the final velocity components, then equate the change in kinetic energy to 22 J and solve.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q3a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Find the template’s centre of mass measured from AE.", "answerGuide": "Subtract the triangular lamina’s area moment from the rectangle’s area moment about AE, then divide by the remaining area.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, 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"answerGuide": "Use the line from the suspension point to the centre of mass and resolve its horizontal and vertical offsets.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q4a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Show that the constant k equals 4.", "answerGuide": "Use the direction condition at t = 2 to set the velocity components equal.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q4b", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the force magnitude acting on Q when t = 2.", "answerGuide": "Differentiate the velocity to obtain acceleration and multiply its magnitude by the particle’s mass.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q4c", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "mechanics", "integration", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find Q’s position vector at t = 2.", "answerGuide": "Integrate each velocity component and use the initial position vector to determine constants.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q5a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the work done against friction on the ramp.", "answerGuide": "Find the normal reaction, calculate friction as μR, and multiply by the distance AB.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q5b", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use work-energy to find the package’s projection speed U.", "answerGuide": "Equate initial kinetic energy to gravitational gain plus work against friction as it reaches rest at B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q5c", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the package’s return speed at A.", "answerGuide": "Use work-energy for the downward motion, accounting for gravitational energy gained back and frictional work.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q6a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Show that the attached particle has weight 15 N.", "answerGuide": "Resolve the light-rod thrust and take moments about the hinge, including the pole’s weight and the particle’s weight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q6b", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the resultant hinge force on the pole.", "answerGuide": "Resolve all forces horizontally and vertically, find the hinge reaction components, then take their resultant.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q7a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the relation between Q’s speed and the mass ratio k.", "answerGuide": "Conserve momentum with signed velocities before and after impact, using the stated unchanged direction of P.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q7b", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the impulse received by Q.", "answerGuide": "Use Q’s change in momentum and substitute the velocity expression from part (a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q7c", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the possible values of k from the restitution coefficient.", "answerGuide": "Apply the coefficient-of-restitution relation, use v ≠ u and the allowed direction of motion, then solve the resulting inequalities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q8a", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Derive the trajectory equation for the freely projected particle.", "answerGuide": "Resolve the initial velocity into horizontal and vertical components, eliminate time, and use 1 + tan²α = sec²α.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q8b", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the initial speed from the projectile’s change in height and final speed.", "answerGuide": "Apply conservation of mechanical energy between launch and impact, including the 20 m vertical drop.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", "examinerReport": "wme02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "66f6c226897f450152bce86aff0ce22daed82411a503a6dafc2a391bc5ab805b", "markScheme": "cc1a2bd3796229701332b7f747c758fd06786a12be1552ee3e6a3cc64ce07ae2", "examinerReport": "6c01c1c6262ce5afe1c9e4e7ad5b7fdc4efbdd2edf4f01e7b5c37908ae0245b8" } }, { "id": "ial-2023-1-WME02-01-q8c", "paperId": "ial-2023-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the launch angle from the horizontal range and flight time.", "answerGuide": "Use the vertical motion to obtain the flight time, then use horizontal motion over BC = 30 m.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20230120.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230302.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20230120.pdf", "markScheme": "wme02-01-rms-20230302.pdf", 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"markScheme": "wme03-01-rms-20230302.pdf", "examinerReport": "wme03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "8df414c750bc11e13233d989b9b9d089e57464f3c13b51dc049e4557f28bd36e", "markScheme": "ec3428767fec4cd09fc20e8d88ceef72f3958ade8d0e9e26b3bec411434e7a9b", "examinerReport": "508ff7b2cc83114bbbe3cfd08c290c59ef4886d67ec4c9e0a670c353d2365b74" } }, { "id": "ial-2023-1-WME03-01-q7b", "paperId": "ial-2023-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2023, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed as it passes through O.", "answerGuide": "Use the SHM amplitude and angular frequency in v = ω√(a² − x²), at x = 0.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20230111.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20230302.pdf#page=13", "qpPages": [ 20, 24 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme03-01-que-20230111.pdf", "markScheme": "wme03-01-rms-20230302.pdf", "examinerReport": "wme03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "8df414c750bc11e13233d989b9b9d089e57464f3c13b51dc049e4557f28bd36e", "markScheme": "ec3428767fec4cd09fc20e8d88ceef72f3958ade8d0e9e26b3bec411434e7a9b", "examinerReport": "508ff7b2cc83114bbbe3cfd08c290c59ef4886d67ec4c9e0a670c353d2365b74" } }, { "id": "ial-2023-1-WME03-01-q7c", "paperId": "ial-2023-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2023, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the maximum acceleration in the SHM.", "answerGuide": "Use the maximum displacement and amax = ω²A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20230111.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20230302.pdf#page=13", "qpPages": [ 20, 24 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { 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condition, solve for the first relevant phase angle, and convert phase to time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20230111.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20230302.pdf#page=13", "qpPages": [ 20, 24 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme03-01-que-20230111.pdf", "markScheme": "wme03-01-rms-20230302.pdf", "examinerReport": "wme03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "8df414c750bc11e13233d989b9b9d089e57464f3c13b51dc049e4557f28bd36e", "markScheme": "ec3428767fec4cd09fc20e8d88ceef72f3958ade8d0e9e26b3bec411434e7a9b", "examinerReport": 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"answerGuide": "Use each histogram bar's area, frequency density times class width, to obtain the missing frequencies; check that all class frequencies total 100.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q1b", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate how many swimmers took less than 16 minutes.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the frequencies below 14 minutes, then take the appropriate fraction of the 14–18 class using the width 2 interval from 14 to 16.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": 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midpoints and frequencies to estimate the total of time times frequency, then divide by 100.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q1d", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the standard deviation using the supplied sum of squared time products.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the grouped-data estimate of the mean together with the supplied sum of ft² in the variance formula; take the square root and keep the divisor convention consistent.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q1e", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the interquartile range by linear interpolation, given Q3 = 23.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Interpolate Q1 within its containing class using cumulative frequencies, then subtract the estimated Q1 from 23.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q2a", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the transfer-and-draw tree for two marbles moved from bag X to bag Y and a final draw from Y.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Update the bag counts after each transfer, then write the conditional green and blue probabilities for the second and third branches.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q2b", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the two marbles drawn from bag X have different colours.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the probabilities of the green-then-blue and blue-then-green paths, using the changing contents of bag X.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q2c", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that all three selected marbles have the same colour.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the all-green path and the all-blue path, including the final draw from bag Y after the transfers.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q2d", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Given that all three marbles match in colour, find the probability that they are all green.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use conditional probability: divide the all-green path probability by the probability that all three selected marbles match.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q3a", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find E(X) for the three-valued discrete distribution in terms of a.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each x-value by its probability and simplify the sum.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q3b", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the range of possible values of E(X).", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls this part very challenging; many left it blank or used an invalid parameter range.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230302.pdf", "tagConfidence": "reviewed", "answerGuide": "First impose nonnegative probabilities to bound a, then use the linear expression for E(X) to find its endpoint values.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" }, "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2023-1-WST01-01-full-q3c", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 6, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Use Var(X) = 0.56 to find all possible values of a.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate E(X²), subtract [E(X)]², equate to 0.56, solve the resulting quadratic, and retain the admissible parameter values.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q4ia", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(a)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Use the Venn diagram probabilities to show the stated relation among p, q and r.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the given event probabilities in terms of the Venn regions, substitute their stated values, and simplify to the required identity.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q4ib", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(b)", "marks": 5, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Find p, q, r and s from the event probabilities and the complete Venn diagram.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the equations from the event probabilities, then use p+q+r+s=1 to obtain the four region probabilities.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q4ii", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show that events C and D cannot be mutually exclusive for the given positive x.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report stresses that this show-that part was poorly answered when candidates used independence properties instead of the required probability bound.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230302.pdf", "tagConfidence": "reviewed", "answerGuide": "Add the two event probabilities, simplify, and show their sum exceeds 1; mutually exclusive events would have a sum no greater than 1.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" }, "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-2023-1-WST01-01-full-q5a", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a normally distributed housefly wing length is below 3.86 mm.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise 3.86 using the given mean and standard deviation, then use the standard normal lower-tail probability.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q5b", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the upper and lower quartiles of the normal wing-length distribution.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the 0.75 and 0.25 standard normal quantiles and transform each z-value back to a wing length; retain both results under the single printed four-mark bracket.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q5c", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the lower and upper outlier limits defined by the quartiles.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate 1.5 times the interquartile range, subtract it from Q1 and add it to Q3.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q5d", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Show that a randomly selected wing length is not an outlier with probability 0.993.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the two outlier limits, standardise both, and calculate the probability between them, showing sufficient working for the given result.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q5e", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Given the wing length is not an outlier, find the probability it exceeds 5 mm.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes difficulty applying the conditional probability after finding the non-outlier event and records recurring tail-probability errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230302.pdf", "tagConfidence": "reviewed", "answerGuide": "Calculate P(5 < L < upper limit), then divide by the probability that L lies between both outlier limits.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": 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"https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Explain that for each 1°C rise in water temperature, the model predicts an increase of 2.72 grams in the maximum sugar dissolved per 100 grams of water.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q6b", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the regression line to estimate sugar solubility at 90°C.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute 90 for x in the regression equation and evaluate the predicted y-value.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q6c", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Comment on the reliability of the 90°C estimate.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare 90°C with the observed 10–80°C range and explain that this is extrapolation, so the estimate may be unreliable.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST01-01-full-q6d", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 7, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Show from the regression line and summary statistics that the product-moment correlation is 0.988.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes that this calculation required a complete method and that calculator-only solutions and incomplete substitutions lost credit.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230302.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the regression equation to recover the required x-summary, calculate Sxx and Syy, use the regression gradient to find Sxy, then evaluate r = Sxy/√(SxxSyy) to three decimal places.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" }, "sourceEvidencePages": { "examinerReport": [ 6 ] } }, { "id": "ial-2023-1-WST01-01-full-q6e", "paperId": "ial-2023-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Assess whether a linear regression model is suitable using the scatter diagram and correlation coefficient.", "qpUrl": "https://www.alevelcopilot.com/whfile/86/6c/bb/32/7f/14/3d/ac/bb/b4/de/33/c2/08/b5/14.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8b/94/94/1c/db/e9/81/3d/ec/fd/43/e0/c3/b2/7e/23.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Refer to how closely the plotted points follow a straight-line pattern and to r being close to 1, then give a contextual conclusion.", "sourceFiles": { "questionPaper": "wst01-01-que-20230119.pdf", "markScheme": "wst01-01-rms-20230302.pdf", "examinerReport": "wst01-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "D0FF4FF64EABAAC955EF9F7EFA8F60BA8FCF0E471178C1D7F0544EC5C2AA917D", "markScheme": "EE08420C569133FCAD2FC5DEAF02CCE1F5C08CF45FDB482EEB1A03D96BCD73EB", "examinerReport": "AC5E90436B44493A107D4767D7B775BE5D59F8876EA81EA372CFC1F19C936FCB" } }, { "id": "ial-2023-1-WST02-01-full-q1a", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Suggest a distribution for hourly shoe sales at a mean rate of four pairs.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the count in a fixed one-hour interval with a Poisson distribution whose parameter is 4.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q1b", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State one assumption needed for the Poisson shoe-sales model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State a suitable condition in context, such as sales occurring independently and at a constant average rate.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q1c", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probabilities of more than four sales in one hour and in each of three consecutive hours.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For X~Po(4), calculate P(X>4) for one hour, then cube that probability for three independent one-hour periods; both subparts share the printed four-mark bracket.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q1d", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% whether the advert increased the hourly shoe-sales rate after observing seven sales.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0: λ=4 and H1: λ>4, calculate the one-sided tail probability (or critical region) for a Poisson count, compare with 0.05, and conclude in context.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q2a", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "List all possible unordered combinations of three 20p and 50p coins.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the four count combinations from zero to three 50p coins, with the remaining coins worth 20p.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q2b", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the two missing mean values in the sampling distribution of the sample mean.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the mean for each mixed three-coin combination and match them to the missing table entries.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q2c", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probabilities p and q of selecting a 20p or 50p coin.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probabilities of the all-20p and all-50p sample-mean outcomes as p³ and q³, then take cube roots.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q2d", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the two missing probabilities in the sample-mean distribution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the three permutations for each mixed-coin composition to calculate the probabilities for means of 30p and 40p.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q2e", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the sampling distribution of the mode of the three sampled coins.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the most challenging part; some candidates found the median instead of the mode or failed to combine probabilities correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20230302.pdf", "tagConfidence": "reviewed", "answerGuide": "The mode is 20p unless at least two 50p coins are drawn; combine the relevant binomial probabilities for each possible modal value.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" }, "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2023-1-WST02-01-full-q3a", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find two binomial probabilities for the number of failed tennis balls in a sample of ten.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use X~B(10,0.1); calculate the upper tail from four failures and the probability for two, three or four failures. Both subparts share the printed bracket.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q3b", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the changed process reduces the tennis-ball failure probability after two failures in 50.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set H0:p=0.1 against H1:p<0.1, use the exact lower-tail binomial probability (or critical region), and state a contextual conclusion at 5%.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q3c", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the smallest sample size for which zero failures supports the claim at 1%.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set P(X=0)=0.9^n below 0.01, solve the logarithmic inequality and choose the least whole-number n.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q4a", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(X<10k) from the given triangular probability density function.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density from the lower endpoint to 10k, or use the corresponding triangle-area ratio.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q4b", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Show that the density constant is k=1/π.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the area under the triangular density over its support equal to 1 and solve for k.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q4c", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find E(X) and Var(X) in terms of π.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x f(x) and x² f(x) over the support, then use Var(X)=E(X²)-[E(X)]²; the two results share one printed three-mark bracket.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q4d", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the expected area of a circle whose radius depends on X.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite A in terms of X² and X, then substitute E(X²) and E(X), or integrate A(x)f(x) across the support.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q5a", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of at most five defects in an 80-metre cable.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the defect count by Po(5), since the rate is one per 16 metres, then evaluate P(X≤5).", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q5b", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that fewer than two of four cables each contain at most five defects.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a binomial model with n=4 and success probability from part (a), then add the probabilities of zero and one qualifying cable.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q5c", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 8, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to determine the cable length x from the given probability for fewer than 26 defects.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Let the defect count be Po(x/16), approximate by N(x/16,x/16), apply continuity correction at 25.5, standardise to the given normal quantile, solve for x and check the positive length.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q6a", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Show the stated relationship among a, b and k for the cumulative distribution function.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the CDF's value at the upper endpoint and the requirement that the total cumulative probability is 1, then rearrange.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q6b", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "distributions", "integration", "algebra" ], "primaryTopicId": "distributions", "summary": "Use E(X)=6/5 and the result from part (a) to show the stated relation for bk³.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the CDF to obtain the density, integrate x f(x), apply the support limits, substitute ak=1-bk², and simplify.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q6c", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "distributions", "integration", "algebra" ], "primaryTopicId": "distributions", "summary": "Use the variance and mean to show the stated relation for bk⁴.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find E(X²) from Var(X)+[E(X)]², integrate x²f(x), substitute the part (a) relation and simplify.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q6d", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "algebra", "distributions" ], "primaryTopicId": "algebra", "summary": "Solve for k using the two moment relations and the condition k<3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Eliminate b from the two equations to obtain a quadratic in k, factor or solve it, and select the root satisfying k<3.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST02-01-full-q6e", "paperId": "ial-2023-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 2, "topicIds": [ "algebra", "distributions" ], "primaryTopicId": "algebra", "summary": "Find the corresponding positive constants a and b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20230121.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230302.pdf#page=12", "qpPages": [ 20, 24 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute k into either moment relation to find b, then use ak=1-bk² to find a.", "sourceFiles": { "questionPaper": "wst02-01-que-20230121.pdf", "markScheme": "wst02-01-rms-20230302.pdf", "examinerReport": "wst02-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "C3BE93DDE299AD1CF5E3BE00CFC6535D4F1C6B1D846427E257F4487F55BA8854", "markScheme": "A43FA8293886322A9AFBE34504715FB234337BFF88240BBF1E49BA45E9BF63B5", "examinerReport": "D4508D7D88ACC254E9C67F9754DD57B914C095B32051C4C98F83C032DF5D11C4" } }, { "id": "ial-2023-1-WST03-01-full-q1a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Use both coded-data summaries to find the unbiased estimates of the mean and variance of bottle volume.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Undo the coding for the sample mean and variance, use the unbiased divisor n−1 for the variance, and keep the two roman-numeral tasks together under their shared mark bracket.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q1b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State null and alternative hypotheses for a two-sided test of whether the mean bottle volume changed.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write H0: μ=1010 and H1: μ≠1010, with μ defined as the population mean bottle volume.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q1c", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region for the sample mean in the 5% two-sided test.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the standard error based on the sample standard deviation and n=100, apply the two-sided 5% normal critical value, and state both sample-mean limits.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q1d", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Decide whether the filling machine should be stopped using the sample mean and critical region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the estimated sample mean 1008.47 with both critical limits and state the decision in context.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q1e", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Explain why using the sample variance as an estimate of the population variance is reasonable.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Refer to the reasonably large sample of 100 bottles.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q2a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Calculate Spearman’s rank correlation coefficient for season-best times and finishing positions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rank season-best times in the stated direction, calculate rank differences and their squares, then apply the Spearman rank formula.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q2b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at 1% for positive rank correlation between season-best time and finishing position.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses for positive association, compare the Spearman statistic with the one-tailed critical value, and conclude in context.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q2c", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation between season-best and finishing times.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the supplied Sxx, Syy and Sxy into r=Sxy/√(SxxSyy), then round to three decimal places.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q2d", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at 1% for positive product-moment correlation between the two times.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the correlation statistic from part (c) with the one-tailed critical value and state the conclusion in context.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q3a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Calculate the two expected frequencies for customers aged 51 or over.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each claim status, multiply the relevant row total by the column total and divide by 1200.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q3b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 7, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at 1% whether making a claim is independent of age using the supplied partial chi-squared sum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State independence hypotheses, add the two supplied contributions to 7.123, use 2 degrees of freedom and the 1% critical value, then conclude with claim and age in context.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q4a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 8, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% whether Binomial(4, 0.5) is a suitable model for the observed numbers of girls.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State model hypotheses, calculate all expected frequencies, evaluate the chi-squared statistic, use 4 degrees of freedom and the critical value, then conclude.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q4b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Show from the sample data that the estimated probability of a girl is 0.45.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the total number of girls and divide by 200 families of four children, or equate the sample mean to 4p.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q4c", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% whether a binomial model with the estimated probability is suitable, given χ²=2.47.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=12", "qpPages": [ 14, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use 3 degrees of freedom because p was estimated, compare 2.47 with the 5% critical value, and state whether the model is rejected.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q5a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 1% whether fertiliser A gives a higher mean yield than fertiliser B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State a one-sided test for the means, calculate the standard error and z statistic from the sample summaries, compare with the 1% critical value and conclude in context.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q5b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Advise which fertiliser Claire should use to maximise expected profit per plant.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report ranks Question 5 among the most discriminating and discusses the challenge of interpreting the comparison and using its test result.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-pef-20230302.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert mean yield to revenue per plant, subtract each fertiliser's cost per plant, compare the two profits and state the recommendation.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" }, "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2023-1-WST03-01-full-q6a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find a 98% confidence interval for the mean stone weight from the sample total.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the sample mean from the total, then use the known standard deviation and z=2.3263 in the confidence-interval formula.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q6b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain why the Central Limit Theorem is not needed for the stone-weight interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The individual bag weights are stated to follow a normal distribution, so the sample mean is normal directly.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q6c", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use the confidence interval to comment on the manufacturer's 22.5 kg mean claim.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check whether 22.5 lies inside the confidence interval and state the corresponding conclusion about the claim.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q6d", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the mean weight of ten gravel bags is below 848 kg.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal distribution of the sample mean with mean 850 and standard error 5/√10, then standardise 848.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q7a", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that three standard-checkout service times total less than 11 minutes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=15", "qpPages": [ 26, 28 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assuming independent normal service times, add the three means and variances to form the total-time distribution, convert 11 minutes to seconds and standardise.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q7b", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the assumption used for the three standard-checkout service times.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=15", "qpPages": [ 26, 28 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that service times for different customers are independent (or an equivalent suitable assumption in context).", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2023-1-WST03-01-full-q7c", "paperId": "ial-2023-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the three-customer standard queue takes longer than the seven-customer express queue.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-que-20230112.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230302.pdf#page=15", "qpPages": [ 26, 28 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assuming independent service times, form both normal total-time distributions, subtract express total from standard total, and calculate the probability that the difference is positive.", "sourceFiles": { "questionPaper": "wst03-01-que-20230112.pdf", "markScheme": "wst03-01-rms-20230302.pdf", "examinerReport": "wst03-01-pef-20230302.pdf" }, "sourceSha256": { "questionPaper": "46A4E391BEDADD8E6642311F4BFAE639E05F3BD9656B9F35EDCF726514D38E80", "markScheme": "5A8B2A7CCC876644FEFE6E418664F1FDE2099072E9EE6EC1B0244AA1C926FF45", "examinerReport": "0E57F3441A8116F891CBF915F7E67634CBB2DB3936504BEAD0F8444D94736EBD" } }, { "id": "ial-2022-1-WDM11-01-q1a", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "1(a): Plan the minimum container count, allocate the weights by first fit, sort them with quick sort, then search the ordered list for a target weight.", "answerGuide": "For the lower bound, divide total weight by capacity and round up. Process boxes in original order for first fit. For quick sort, show each pivot and partition; then binary-search the sorted list, recording rejected ranges.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=5", "qpPages": [ 22, 22 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q1b", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "1(b): Plan the minimum container count, allocate the weights by first fit, sort them with quick sort, then search the ordered list for a target weight.", "answerGuide": "For the lower bound, divide total weight by capacity and round up. Process boxes in original order for first fit. For quick sort, show each pivot and partition; then binary-search the sorted list, recording rejected ranges.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=5", "qpPages": [ 22, 22 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q1c", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "1(c): Plan the minimum container count, allocate the weights by first fit, sort them with quick sort, then search the ordered list for a target weight.", "answerGuide": "For the lower bound, divide total weight by capacity and round up. Process boxes in original order for first fit. For quick sort, show each pivot and partition; then binary-search the sorted list, recording rejected ranges.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=5", "qpPages": [ 22, 22 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q1d", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "1(d): Plan the minimum container count, allocate the weights by first fit, sort them with quick sort, then search the ordered list for a target weight.", "answerGuide": "For the lower bound, divide total weight by capacity and round up. Process boxes in original order for first fit. For quick sort, show each pivot and partition; then binary-search the sorted list, recording rejected ranges.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=5", "qpPages": [ 22, 22 ], "msPages": [ 5, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q2a", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "2(a): Read a graph path and tour, then apply Kruskal’s algorithm to a weighted network and report its minimum spanning tree and weight.", "answerGuide": "List adjacent vertices for a valid path. For a tour, check every consecutive edge and whether the route returns to its start. Sort network edges by weight, add an edge only if it does not make a cycle, draw the selected tree, and total its weights.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=23", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=6", "qpPages": [ 23, 23 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q2b", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Decide whether a listed route is a tour and justify the classification.", "answerGuide": "Check the route against the graph, then distinguish a path that visits the vertices from a tour that also returns to its starting vertex.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=23", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=6", "qpPages": [ 23, 23 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that many candidates treated visiting all vertices as sufficient and missed the return-to-start condition.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" }, "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2022-1-WDM11-01-q2c", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "2(c): Read a graph path and tour, then apply Kruskal’s algorithm to a weighted network and report its minimum spanning tree and weight.", "answerGuide": "List adjacent vertices for a valid path. For a tour, check every consecutive edge and whether the route returns to its start. Sort network edges by weight, add an edge only if it does not make a cycle, draw the selected tree, and total its weights.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=23", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=6", "qpPages": [ 23, 23 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q2d", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "2(d): Read a graph path and tour, then apply Kruskal’s algorithm to a weighted network and report its minimum spanning tree and weight.", "answerGuide": "List adjacent vertices for a valid path. For a tour, check every consecutive edge and whether the route returns to its start. Sort network edges by weight, add an edge only if it does not make a cycle, draw the selected tree, and total its weights.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=23", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=6", "qpPages": [ 23, 23 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q2e", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "2(e): Read a graph path and tour, then apply Kruskal’s algorithm to a weighted network and report its minimum spanning tree and weight.", "answerGuide": "List adjacent vertices for a valid path. For a tour, check every consecutive edge and whether the route returns to its start. Sort network edges by weight, add an edge only if it does not make a cycle, draw the selected tree, and total its weights.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=23", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=6", "qpPages": [ 23, 23 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q3", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Construct an activity-on-arc network from a precedence table using exactly four dummy activities.", "answerGuide": "Represent each activity by an arrow, merge shared predecessor events correctly, and insert the minimum required dummy arrows so every listed precedence relation is enforced. Check that the dummy count is exactly four.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=7", "qpPages": [ 24, 24 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q4a", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "4(a): Use a project network’s float and critical-path information to find two durations, draw a cascade chart, and determine the minimum workforce.", "answerGuide": "Calculate total floats from early and late event times, use the given float ratio and path-length difference to solve for the unknown durations, schedule activities in a cascade chart, then read the maximum simultaneous worker demand and cite its time interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=25", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=8", "qpPages": [ 25, 25 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q4b", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "4(b): Use a project network’s float and critical-path information to find two durations, draw a cascade chart, and determine the minimum workforce.", "answerGuide": "Calculate total floats from early and late event times, use the given float ratio and path-length difference to solve for the unknown durations, schedule activities in a cascade chart, then read the maximum simultaneous worker demand and cite its time interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=25", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=8", "qpPages": [ 25, 25 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q4c", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "4(c): Use a project network’s float and critical-path information to find two durations, draw a cascade chart, and determine the minimum workforce.", "answerGuide": "Calculate total floats from early and late event times, use the given float ratio and path-length difference to solve for the unknown durations, schedule activities in a cascade chart, then read the maximum simultaneous worker demand and cite its time interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=25", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=8", "qpPages": [ 25, 25 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q5a", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "5(a): Minimize the time of a route inspection that covers every road, first as a closed route from G and then as an open route from B to G.", "answerGuide": "For a closed route, identify odd vertices and pair them at minimum added distance. For the open route, adjust the odd-vertex pairing to account for the required endpoints B and G. Add duplicated-edge costs to the network total and compare both routes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=9", "qpPages": [ 26, 26 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q5b", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "5(b): Minimize the time of a route inspection that covers every road, first as a closed route from G and then as an open route from B to G.", "answerGuide": "For a closed route, identify odd vertices and pair them at minimum added distance. For the open route, adjust the odd-vertex pairing to account for the required endpoints B and G. Add duplicated-edge costs to the network total and compare both routes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=9", "qpPages": [ 26, 26 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q5c", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "5(c): Minimize the time of a route inspection that covers every road, first as a closed route from G and then as an open route from B to G.", "answerGuide": "For a closed route, identify odd vertices and pair them at minimum added distance. For the open route, adjust the odd-vertex pairing to account for the required endpoints B and G. Add duplicated-edge costs to the network total and compare both routes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=9", "qpPages": [ 26, 26 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q6a", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "6(a): Find a least-distance route, an upper bound for a travelling-salesperson tour, a lower bound from a reduced-network spanning tree, and a guaranteed interval for the optimum.", "answerGuide": "Run Dijkstra from A and record each permanent label. Apply nearest neighbour for the upper bound. On the reduced network, build Prim’s tree from C and use its total with the relevant incident edges for the lower bound; combine the bounds into the narrowest justified interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=27", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=10", "qpPages": [ 27, 27 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q6b", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "6(b): Find a least-distance route, an upper bound for a travelling-salesperson tour, a lower bound from a reduced-network spanning tree, and a guaranteed interval for the optimum.", "answerGuide": "Run Dijkstra from A and record each permanent label. Apply nearest neighbour for the upper bound. On the reduced network, build Prim’s tree from C and use its total with the relevant incident edges for the lower bound; combine the bounds into the narrowest justified interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=27", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=10", "qpPages": [ 27, 27 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q6ci", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)(i)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "6(c)(i): Find a least-distance route, an upper bound for a travelling-salesperson tour, a lower bound from a reduced-network spanning tree, and a guaranteed interval for the optimum.", "answerGuide": "Run Dijkstra from A and record each permanent label. Apply nearest neighbour for the upper bound. On the reduced network, build Prim’s tree from C and use its total with the relevant incident edges for the lower bound; combine the bounds into the narrowest justified interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=27", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=10", "qpPages": [ 27, 27 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q6cii", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)(ii)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "6(c)(ii): Find a least-distance route, an upper bound for a travelling-salesperson tour, a lower bound from a reduced-network spanning tree, and a guaranteed interval for the optimum.", "answerGuide": "Run Dijkstra from A and record each permanent label. Apply nearest neighbour for the upper bound. On the reduced network, build Prim’s tree from C and use its total with the relevant incident edges for the lower bound; combine the bounds into the narrowest justified interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=27", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=10", "qpPages": [ 27, 27 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q6d", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "6(d): Find a least-distance route, an upper bound for a travelling-salesperson tour, a lower bound from a reduced-network spanning tree, and a guaranteed interval for the optimum.", "answerGuide": "Run Dijkstra from A and record each permanent label. Apply nearest neighbour for the upper bound. On the reduced network, build Prim’s tree from C and use its total with the relevant incident edges for the lower bound; combine the bounds into the narrowest justified interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=27", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=10", "qpPages": [ 27, 27 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q7a", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "decision-mathematics", "algebra" ], "primaryTopicId": "decision-mathematics", "summary": "7(a): Formulate a production-cost linear program, simplify the objective under a fixed total output, graph the constraints, locate the optimum, and state the production plan and cost.", "answerGuide": "Define the decision variables, translate each production, labour, and proportion condition into integer-coefficient inequalities, and state the objective. Substitute the fixed total to simplify cost, graph boundary lines with the correct half-planes, test feasible vertices with the objective line, then interpret the optimum in container counts and pounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=11", "qpPages": [ 28, 28 ], "msPages": [ 11, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q7b", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "decision-mathematics", "algebra" ], "primaryTopicId": "decision-mathematics", "summary": "7(b): Formulate a production-cost linear program, simplify the objective under a fixed total output, graph the constraints, locate the optimum, and state the production plan and cost.", "answerGuide": "Define the decision variables, translate each production, labour, and proportion condition into integer-coefficient inequalities, and state the objective. Substitute the fixed total to simplify cost, graph boundary lines with the correct half-planes, test feasible vertices with the objective line, then interpret the optimum in container counts and pounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=11", "qpPages": [ 28, 28 ], "msPages": [ 11, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q7c", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "decision-mathematics", "algebra" ], "primaryTopicId": "decision-mathematics", "summary": "7(c): Formulate a production-cost linear program, simplify the objective under a fixed total output, graph the constraints, locate the optimum, and state the production plan and cost.", "answerGuide": "Define the decision variables, translate each production, labour, and proportion condition into integer-coefficient inequalities, and state the objective. Substitute the fixed total to simplify cost, graph boundary lines with the correct half-planes, test feasible vertices with the objective line, then interpret the optimum in container counts and pounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=11", "qpPages": [ 28, 28 ], "msPages": [ 11, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q7d", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "decision-mathematics", "algebra" ], "primaryTopicId": "decision-mathematics", "summary": "7(d): Formulate a production-cost linear program, simplify the objective under a fixed total output, graph the constraints, locate the optimum, and state the production plan and cost.", "answerGuide": "Define the decision variables, translate each production, labour, and proportion condition into integer-coefficient inequalities, and state the objective. Substitute the fixed total to simplify cost, graph boundary lines with the correct half-planes, test feasible vertices with the objective line, then interpret the optimum in container counts and pounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=11", "qpPages": [ 28, 28 ], "msPages": [ 11, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WDM11-01-q7e", "paperId": "ial-2022-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2022, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 2, "topicIds": [ "decision-mathematics", "algebra" ], "primaryTopicId": "decision-mathematics", "summary": "7(e): Formulate a production-cost linear program, simplify the objective under a fixed total output, graph the constraints, locate the optimum, and state the production plan and cost.", "answerGuide": "Define the decision variables, translate each production, labour, and proportion condition into integer-coefficient inequalities, and state the objective. Substitute the fixed total to simplify cost, graph boundary lines with the correct half-planes, test feasible vertices with the objective line, then interpret the optimum in container counts and pounds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_que_20220118.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WDM11_01_rms_20220303.pdf#page=11", "qpPages": [ 28, 28 ], "msPages": [ 11, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wdm11-qp.pdf", "markScheme": "wdm11-ms.pdf", "examinerReport": "wdm11-er.pdf" }, "sourceSha256": { "questionPaper": "27b8027ee2d64839f42056e942f11453f8b2990892e165a0e00d60433c78d95a", "markScheme": "2b99f98a0cd9b76c7453b8999f8ba965b405779305f070edbe2a29d44d5fee78", "examinerReport": "196ac0733f74dd100e79add08e704292f17bdf7318a27df26cd15a5f2ff974bc" } }, { "id": "ial-2022-1-WMA11-01-q1", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate an algebraic expression containing positive and negative powers of x, simplifying every term.", "answerGuide": "Rewrite the integrand as a sum of powers, apply the power rule term by term, and express all coefficients and powers in simplest form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q2a", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "functions", "graphs", "transformations" ], "primaryTopicId": "functions", "summary": "2(a): Rewrite a reciprocal function into a transformed standard form, sketch its graph with the y-intercept, and state its symmetry line.", "answerGuide": "Complete the denominator algebraically to identify the horizontal and vertical shifts and scale. Substitute x=0 for the intercept, draw both asymptotic branches, and read the vertical symmetry line from the transformed form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q2b", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions", "graphs", "transformations" ], "primaryTopicId": "functions", "summary": "2(b): Rewrite a reciprocal function into a transformed standard form, sketch its graph with the y-intercept, and state its symmetry line.", "answerGuide": "Complete the denominator algebraically to identify the horizontal and vertical shifts and scale. 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For the equation, isolate the surd, square only after checking signs, solve for the variable, then substitute back to reject extraneous roots.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q3ii", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 4, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "3(ii): Expand a surd expression into a quadratic and solve a related surd equation in exact simplified form.", "answerGuide": "Expand using conjugate products and collect constant, x, and x-squared terms. For the equation, isolate the surd, square only after checking signs, solve for the variable, then substitute back to reject extraneous roots.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q4a", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "quadratics", "graphs" ], "primaryTopicId": "coordinate-geometry", "summary": "4(a): Find the intersection coordinates of a line and quadratic curve, then describe their bounded region using inequalities.", "answerGuide": "Substitute the line equation into the curve to form and solve a quadratic. Use the labelled points to identify the correct x-interval and state which curve is above the x-axis and below the line across the shaded region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q4b", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "quadratics", "graphs" ], "primaryTopicId": "coordinate-geometry", "summary": "4(b): Find the intersection coordinates of a line and quadratic curve, then describe their bounded region using inequalities.", "answerGuide": "Substitute the line equation into the curve to form and solve a quadratic. 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Find the chord with trigonometry, subtract the triangle area from the sector area, and add the chord to the arc length for the perimeter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q5b", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "5(b): Use a semicircular garden diagram to find a central angle, the area of a circular segment-like flowerbed, and its perimeter.", "answerGuide": "Use symmetry and the given radian angle to establish the required central angle. 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Differentiate the expanded cubic, then solve f′(x)<−18 using the resulting quadratic and present the valid interval(s).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q7b", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "functions", "differentiation", "quadratics" ], "primaryTopicId": "functions", "summary": "7(b): Apply vertical and horizontal function shifts, differentiate a cubic, and solve an inequality for its gradient.", "answerGuide": "Use the given point to determine the vertical shift and solve the root condition for the horizontal shifts. Differentiate the expanded cubic, then solve f′(x)<−18 using the resulting quadratic and present the valid interval(s).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q7c", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "functions", "differentiation", "quadratics" ], "primaryTopicId": "functions", "summary": "7(c): Apply vertical and horizontal function shifts, differentiate a cubic, and solve an inequality for its gradient.", "answerGuide": "Use the given point to determine the vertical shift and solve the root condition for the horizontal shifts. Differentiate the expanded cubic, then solve f′(x)<−18 using the resulting quadratic and present the valid interval(s).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q7d", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "functions", "differentiation", "quadratics" ], "primaryTopicId": "functions", "summary": "7(d): Apply vertical and horizontal function shifts, differentiate a cubic, and solve an inequality for its gradient.", "answerGuide": "Use the given point to determine the vertical shift and solve the root condition for the horizontal shifts. Differentiate the expanded cubic, then solve f′(x)<−18 using the resulting quadratic and present the valid interval(s).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q8a", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Find line gradients, construct a perpendicular line, solve the intersection, and recover the opposite square vertex from the diagonals’ midpoint.", "answerGuide": "Read the first gradient from standard form and take its negative reciprocal. Use point-slope form through A, solve the two line equations for M, then apply the midpoint relation to the known diagonal endpoint and M.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q8b", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "8(b): Find line gradients, construct a perpendicular line, solve the intersection, and recover the opposite square vertex from the diagonals’ midpoint.", "answerGuide": "Read the first gradient from standard form and take its negative reciprocal. 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Use point-slope form through A, solve the two line equations for M, then apply the midpoint relation to the known diagonal endpoint and M.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q8d", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 2, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the opposite vertex of a square from its diagonal midpoint and a known vertex.", "answerGuide": "Solve for the intersection M of the perpendicular lines, then use M as the midpoint of diagonal AC and recover the opposite vertex coordinate by coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The question-specific report calls the final part a good discriminator and notes errors with midpoint and vector methods. 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Use the period and phase shift to locate the indicated maximum and calculate its ordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=23", "qpPages": [ 28, 29 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q10a", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "functions", "algebra", "graphs" ], "primaryTopicId": "functions", "summary": "10(a): Sketch a reciprocal-square curve with its intercepts and asymptotes, then find parameter values giving four intersections with a line family.", "answerGuide": "Mark the intercepts and asymptotes from the formula. Equate the two curves, substitute u=x² to form a quadratic, and require two distinct positive u-roots for four distinct x-intersections; solve the resulting parameter inequalities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA11-01-q10b", "paperId": "ial-2022-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "functions", "algebra", "graphs" ], "primaryTopicId": "functions", "summary": "10(b): Sketch a reciprocal-square curve with its intercepts and asymptotes, then find parameter values giving four intersections with a line family.", "answerGuide": "Mark the intercepts and asymptotes from the formula. Equate the two curves, substitute u=x² to form a quadratic, and require two distinct positive u-roots for four distinct x-intersections; solve the resulting parameter inequalities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_que_20220112.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA11_01_rms_20220303.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WMA11-qp2.pdf", "markScheme": "verified-WMA11-ms.pdf", "examinerReport": "wma11-er.pdf" }, "sourceSha256": { "questionPaper": "7a5e99df0f1940553cdedc84f7330841466e32ea070864b1f918e9e8d5a76742", "markScheme": "a60e5ac7f16ec3d668c34b8d4356e6823e087981b9ee271015246c48180561ed", "examinerReport": "f37d23e66bb72f6ce3c4da871ac2f9a418ba9393157da6f491c991e2aa927c57" } }, { "id": "ial-2022-1-WMA12-01-q1a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "1(a): Use tabulated values of a reciprocal expression in the trapezium rule and estimate two related definite integrals.", "answerGuide": "Set the interval width from the table, substitute all ordinates into the trapezium formula, and round as required. Double the first estimate for the scaled integral; for the shifted integral, add the known constant-area contribution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q1bi", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)(i)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "1(b)(i): Use tabulated values of a reciprocal expression in the trapezium rule and estimate two related definite integrals.", "answerGuide": "Set the interval width from the table, substitute all ordinates into the trapezium formula, and round as required. Double the first estimate for the scaled integral; for the shifted integral, add the known constant-area contribution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q1bii", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)(ii)", "marks": 2, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "1(b)(ii): Use tabulated values of a reciprocal expression in the trapezium rule and estimate two related definite integrals.", "answerGuide": "Set the interval width from the table, substitute all ordinates into the trapezium formula, and round as required. Double the first estimate for the scaled integral; for the shifted integral, add the known constant-area contribution.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q2a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "2(a): Differentiate a power-function curve, find its stationary coordinates, then classify the stationary point with a second derivative.", "answerGuide": "Apply the power rule to each term. Set the derivative to zero and solve under x>0, substitute for y, then evaluate the second derivative at the stationary x and state its sign and the resulting nature.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q2b", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "2(b): Differentiate a power-function curve, find its stationary coordinates, then classify the stationary point with a second derivative.", "answerGuide": "Apply the power rule to each term. Set the derivative to zero and solve under x>0, substitute for y, then evaluate the second derivative at the stationary x and state its sign and the resulting nature.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q2c", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "2(c): Differentiate a power-function curve, find its stationary coordinates, then classify the stationary point with a second derivative.", "answerGuide": "Apply the power rule to each term. Set the derivative to zero and solve under x>0, substitute for y, then evaluate the second derivative at the stationary x and state its sign and the resulting nature.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q3a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "3(a): Expand a generalized binomial expression through the first four terms, then determine its parameter from a coefficient condition on a product.", "answerGuide": "Use the binomial coefficients through x³ and simplify each coefficient. Multiply only the terms needed to obtain the product’s constant and x coefficient, impose the stated three-to-one relation, and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q3b", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "3(b): Expand a generalized binomial expression through the first four terms, then determine its parameter from a coefficient condition on a product.", "answerGuide": "Use the binomial coefficients through x³ and simplify each coefficient. Multiply only the terms needed to obtain the product’s constant and x coefficient, impose the stated three-to-one relation, and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q4", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "indices", "algebra", "quadratics" ], "primaryTopicId": "indices", "summary": "Solve a logarithmic equation with a linear argument and a squared logarithm, retaining only values in the logarithms’ domains.", "answerGuide": "Use logarithm laws to combine the right side, convert to an equation of positive arguments, solve the resulting quadratic, and test every root against both original argument restrictions.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q5a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "5(a): Use remainder conditions and a known linear factor to determine two polynomial coefficients, then find the quotient quadratic.", "answerGuide": "Apply the remainder theorem at x=1 and x=−1 to obtain the required linear relation and a second equation. Use the given factor root to solve for A and B, then divide the cubic by the factor and verify the product.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q5b", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "5(b): Use remainder conditions and a known linear factor to determine two polynomial coefficients, then find the quotient quadratic.", "answerGuide": "Apply the remainder theorem at x=1 and x=−1 to obtain the required linear relation and a second equation. Use the given factor root to solve for A and B, then divide the cubic by the factor and verify the product.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q5c", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "5(c): Use remainder conditions and a known linear factor to determine two polynomial coefficients, then find the quotient quadratic.", "answerGuide": "Apply the remainder theorem at x=1 and x=−1 to obtain the required linear relation and a second equation. Use the given factor root to solve for A and B, then divide the cubic by the factor and verify the product.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q6a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(a): Show three points form a right angle, obtain the circle’s centre and radius, and find the tangent at the point diametrically opposite Q.", "answerGuide": "Use gradients, Pythagoras, or the cosine rule for the right angle. Take the midpoint of the hypotenuse as the centre, calculate the radius from a point, locate the opposite endpoint of the diameter, then use the negative reciprocal radius gradient in a line equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q6bi", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(b)(i): Show three points form a right angle, obtain the circle’s centre and radius, and find the tangent at the point diametrically opposite Q.", "answerGuide": "Use gradients, Pythagoras, or the cosine rule for the right angle. Take the midpoint of the hypotenuse as the centre, calculate the radius from a point, locate the opposite endpoint of the diameter, then use the negative reciprocal radius gradient in a line equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q6bii", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 2, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(b)(ii): Show three points form a right angle, obtain the circle’s centre and radius, and find the tangent at the point diametrically opposite Q.", "answerGuide": "Use gradients, Pythagoras, or the cosine rule for the right angle. Take the midpoint of the hypotenuse as the centre, calculate the radius from a point, locate the opposite endpoint of the diameter, then use the negative reciprocal radius gradient in a line equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q6c", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(c): Show three points form a right angle, obtain the circle’s centre and radius, and find the tangent at the point diametrically opposite Q.", "answerGuide": "Use gradients, Pythagoras, or the cosine rule for the right angle. Take the midpoint of the hypotenuse as the centre, calculate the radius from a point, locate the opposite endpoint of the diameter, then use the negative reciprocal radius gradient in a line equation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q7i", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "7(i): Solve two trigonometric equations on specified intervals, one using a shifted double angle and the other a sine-cosine combination.", "answerGuide": "For the first, set the shifted angle as a single variable, solve the tangent ratio and enumerate only roots in the given open interval. For the second, combine sine and cosine as one shifted sinusoid, then list all admissible angles to the requested precision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=17", "qpPages": [ 22, 25 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q7ii", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "7(ii): Solve two trigonometric equations on specified intervals, one using a shifted double angle and the other a sine-cosine combination.", "answerGuide": "For the first, set the shifted angle as a single variable, solve the tangent ratio and enumerate only roots in the given open interval. For the second, combine sine and cosine as one shifted sinusoid, then list all admissible angles to the requested precision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=17", "qpPages": [ 22, 25 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q8a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "sequences", "algebra", "exponentials" ], "primaryTopicId": "sequences", "summary": "8(a): Model repeated driving distances first by an arithmetic sequence and then by a geometric sequence to find a threshold hit number.", "answerGuide": "Use the common difference to get term 20 and the arithmetic-series total. Treat hit 21 as the first geometric term and use hit 22 to solve for r. Add the arithmetic and geometric totals without double-counting, then solve the strict inequality and choose the least valid integer N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q8b", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "sequences", "algebra", "exponentials" ], "primaryTopicId": "sequences", "summary": "8(b): Model repeated driving distances first by an arithmetic sequence and then by a geometric sequence to find a threshold hit number.", "answerGuide": "Use the common difference to get term 20 and the arithmetic-series total. Treat hit 21 as the first geometric term and use hit 22 to solve for r. Add the arithmetic and geometric totals without double-counting, then solve the strict inequality and choose the least valid integer N.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" } }, { "id": "ial-2022-1-WMA12-01-q8c", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "sequences", "algebra", "exponentials" ], "primaryTopicId": "sequences", "summary": "Determine the common ratio for the geometric distances after the twentieth hit.", "answerGuide": "Use the twentieth arithmetic-sequence distance as the first reference amount; the twenty-second hit is two geometric steps later, so form an equation involving r squared and solve for the required ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The examiner report says the context and index in this transition from an arithmetic to a geometric sequence caused difficulty, especially for candidates who restarted the sequence or used the wrong term number.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" }, "sourceEvidencePages": { "examinerReport": [ 3, 9 ] } }, { "id": "ial-2022-1-WMA12-01-q8d", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 4, "topicIds": [ "sequences", "algebra", "exponentials" ], "primaryTopicId": "sequences", "summary": "Find the first hit count for which the accumulated driving distance exceeds three metres.", "answerGuide": "Combine the arithmetic total for the first twenty hits with a geometric sum that does not double-count the twentieth distance; use logarithms to solve the strict inequality and select the least integer hit count.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20220119.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_rms_20220303.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that very few candidates earned full marks, often double-counting the shared twentieth term or omitting the arithmetic part before solving with logarithms.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WMA12_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-qp.pdf", "markScheme": "wma12-ms.pdf", "examinerReport": "wma12-er.pdf" }, "sourceSha256": { "questionPaper": "95421c7683be0086f7c4811981b6930b710561167a28f2717d1b6a2fc9704aa8", "markScheme": "4a946f4254e7f27d5bd4ba66179f2fea3e71ed850422dfeb611493be6982e1b2", "examinerReport": "3425d2e812525482ef24984d57bce4f8abd6afb2e227c59ed66872bcd01d8bd9" }, "sourceEvidencePages": { "examinerReport": [ 3, 9, 10 ] } }, { "id": "ial-2022-1-WMA12-01-q9a", "paperId": "ial-2022-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "9(a): Compare two shaded areas formed by a parabola, a line through the origin, and the x-axis to find an exact slope parameter.", "answerGuide": "Find the nonzero intersection in terms of m. Integrate curve minus line over the correct interval for the first area, express the second region with the correct bounds, set areas equal, and solve using the restriction 02 at x=10 to solve the angle interval. For θ=40°, find crossing time and velocity components at the pole for speed; use the full flight time for OC.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20220122.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220303.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-qp.pdf", "markScheme": "wme02-ms.pdf", "examinerReport": "wme02-er.pdf" }, "sourceSha256": { "questionPaper": "4a0ee6669aaed08f1403f91026c47a0404ff6d4862f80637cda14545962c6cc1", "markScheme": "268071ed2e8933f69b733bb238ba14d35e844c7b681133dd40e72e1599f2f962", "examinerReport": "bec4b30e719fcde85c83b13b368384828b2d2f101d6f7057fafcecdc46ba887e" } }, { "id": "ial-2022-1-WME02-01-q7b", "paperId": "ial-2022-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics", "energy-momentum", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "7(b): Derive a projectile trajectory, find the launch-angle range that clears a pole, then calculate speed over the pole and the full range.", "answerGuide": "Resolve initial velocity into horizontal and vertical components and eliminate time to obtain y(x). Apply y>2 at x=10 to solve the angle interval. For θ=40°, find crossing time and velocity components at the pole for speed; use the full flight time for OC.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20220122.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220303.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-qp.pdf", "markScheme": "wme02-ms.pdf", "examinerReport": "wme02-er.pdf" }, "sourceSha256": { "questionPaper": "4a0ee6669aaed08f1403f91026c47a0404ff6d4862f80637cda14545962c6cc1", "markScheme": "268071ed2e8933f69b733bb238ba14d35e844c7b681133dd40e72e1599f2f962", "examinerReport": "bec4b30e719fcde85c83b13b368384828b2d2f101d6f7057fafcecdc46ba887e" } }, { "id": "ial-2022-1-WME02-01-q7c", "paperId": "ial-2022-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "kinematics", "energy-momentum", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find a projectile's speed as it passes over a pole after determining the valid launch-angle interval.", "answerGuide": "Use the projectile trajectory to identify the crossing time and velocity components at the pole, then combine the components for speed and retain only angles satisfying the height condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_que_20220122.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_rms_20220303.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies the final projectile part as difficult; most candidates used an incorrect height or did not complete the speed calculation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME02_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-qp.pdf", "markScheme": "wme02-ms.pdf", "examinerReport": "wme02-er.pdf" }, "sourceSha256": { "questionPaper": "4a0ee6669aaed08f1403f91026c47a0404ff6d4862f80637cda14545962c6cc1", "markScheme": "268071ed2e8933f69b733bb238ba14d35e844c7b681133dd40e72e1599f2f962", "examinerReport": "bec4b30e719fcde85c83b13b368384828b2d2f101d6f7057fafcecdc46ba887e" }, "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-1-WME02-01-q7d", "paperId": "ial-2022-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "kinematics", "energy-momentum", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "7(d): Derive a projectile trajectory, find the launch-angle range that clears a pole, then calculate speed over the pole and the full range.", "answerGuide": "Resolve initial velocity into horizontal and vertical components and eliminate time to obtain y(x). Apply y>2 at x=10 to solve the angle interval. 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Divide the moment by the area and simplify the y-coordinate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q2", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 8, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the upper bound on angular speed for a particle moving in a horizontal circle on a floor while attached to an elevated point by a taut string.", "answerGuide": "Use the string geometry to resolve tension vertically and horizontally. Equate vertical forces to determine the normal reaction and impose the limiting condition that contact with the floor is just maintained; relate horizontal tension to centripetal acceleration.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q3a", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "3(a): Use work-energy for a particle sliding against a position-dependent force to find its speed and stopping distance.", "answerGuide": "Equate the loss in gravitational potential energy to kinetic energy plus the work done against the variable opposing force. Integrate the force over displacement, substitute x=2 for speed, then set speed to zero to find OA.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q3b", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "3(b): Use work-energy for a particle sliding against a position-dependent force to find its speed and stopping distance.", "answerGuide": "Equate the loss in gravitational potential energy to kinetic energy plus the work done against the variable opposing force. Integrate the force over displacement, substitute x=2 for speed, then set speed to zero to find OA.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q4a", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "4(a): Find the centre of mass of a composite open container and its toppling condition on an inclined plane.", "answerGuide": "Calculate component masses from the common surface density, use the cylinder and hemispherical shell centres to form the weighted centroid, then set the centre-of-mass vertical through the downhill rim at impending toppling and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q4b", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "4(b): Find the centre of mass of a composite open container and its toppling condition on an inclined plane.", "answerGuide": "Calculate component masses from the common surface density, use the cylinder and hemispherical shell centres to form the weighted centroid, then set the centre-of-mass vertical through the downhill rim at impending toppling and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q5a", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "5(a): Use a cosine displacement model to verify simple harmonic motion and find its period, amplitude, maximum speed, and travel time between two positions.", "answerGuide": "Differentiate displacement twice and show acceleration is proportional to negative displacement. Read period and amplitude from the model, differentiate once for the maximum speed, then determine the phase angles at A and B and divide their phase difference by angular frequency.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q5b", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "5(b): Use a cosine displacement model to verify simple harmonic motion and find its period, amplitude, maximum speed, and travel time between two positions.", "answerGuide": "Differentiate displacement twice and show acceleration is proportional to negative displacement. Read period and amplitude from the model, differentiate once for the maximum speed, then determine the phase angles at A and B and divide their phase difference by angular frequency.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q5c", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "5(c): Use a cosine displacement model to verify simple harmonic motion and find its period, amplitude, maximum speed, and travel time between two positions.", "answerGuide": "Differentiate displacement twice and show acceleration is proportional to negative displacement. Read period and amplitude from the model, differentiate once for the maximum speed, then determine the phase angles at A and B and divide their phase difference by angular frequency.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q5d", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "5(d): Use a cosine displacement model to verify simple harmonic motion and find its period, amplitude, maximum speed, and travel time between two positions.", "answerGuide": "Differentiate displacement twice and show acceleration is proportional to negative displacement. Read period and amplitude from the model, differentiate once for the maximum speed, then determine the phase angles at A and B and divide their phase difference by angular frequency.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q5e", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 4, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "5(e): Use a cosine displacement model to verify simple harmonic motion and find its period, amplitude, maximum speed, and travel time between two positions.", "answerGuide": "Differentiate displacement twice and show acceleration is proportional to negative displacement. Read period and amplitude from the model, differentiate once for the maximum speed, then determine the phase angles at A and B and divide their phase difference by angular frequency.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q6a", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "6(a): Use elastic-string extension to find its modulus, the initial acceleration of an attached particle, and its speed before reaching the ceiling.", "answerGuide": "Apply Hooke’s law in the initial horizontal position to obtain λ. Resolve the two tensions and weight at C for the initial acceleration. For the later speed, use conservation of energy with the exact extension of each half-string and its elastic potential-energy change.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q6b", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "6(b): Use elastic-string extension to find its modulus, the initial acceleration of an attached particle, and its speed before reaching the ceiling.", "answerGuide": "Apply Hooke’s law in the initial horizontal position to obtain λ. Resolve the two tensions and weight at C for the initial acceleration. For the later speed, use conservation of energy with the exact extension of each half-string and its elastic potential-energy change.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q6c", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "6(c): Use elastic-string extension to find its modulus, the initial acceleration of an attached particle, and its speed before reaching the ceiling.", "answerGuide": "Apply Hooke’s law in the initial horizontal position to obtain λ. Resolve the two tensions and weight at C for the initial acceleration. For the later speed, use conservation of energy with the exact extension of each half-string and its elastic potential-energy change.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q7a", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "7(a): Analyze a particle on a freely rotating vertical rod, establish a complete-circle condition, and find the ratio of maximum to minimum tension.", "answerGuide": "Use energy conservation between the initial point and a general angular position. Check the speed and rod force at the critical top position for a complete circle. Apply the radial equation at the top and bottom, use energy to relate speeds, and form the exact tension ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q7b", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "7(b): Analyze a particle on a freely rotating vertical rod, establish a complete-circle condition, and find the ratio of maximum to minimum tension.", "answerGuide": "Use energy conservation between the initial point and a general angular position. Check the speed and rod force at the critical top position for a complete circle. Apply the radial equation at the top and bottom, use energy to relate speeds, and form the exact tension ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" } }, { "id": "ial-2022-1-WME03-01-q7c", "paperId": "ial-2022-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2022, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 9, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the exact ratio of greatest to least tension during a complete vertical-circle motion.", "answerGuide": "Use energy conservation to relate speeds at the top and bottom, apply the radial equation of motion at both positions, and form the exact tension ratio.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20220120.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WME03_01_rms_20220303.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the final part distinguished candidates; weaker solutions often failed to use earlier results for the bottom speed and maximum tension.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-pef-20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WME03-qp.pdf", "markScheme": "wme03-ms.pdf", "examinerReport": "verified-WME03-er.pdf" }, "sourceSha256": { "questionPaper": "80dba0d7a6172dacfa80c5e2890933225d0ec2b73236c493f649352b2f207d84", "markScheme": "df23f8251f5f968f1ee755b7f1c1e84ead0122fa3dd3dceae811ad6ecf964022", "examinerReport": "080de3e51d45677bb2bb2825e89c9790b73e1d6409eada3b24372d7e4135b5dc" }, "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-1-WST01-01-q1a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability", "data-analysis", "distributions" ], "primaryTopicId": "probability", "summary": "1(a): Use a Venn diagram for three defect events to find probabilities, conditional probability, and the expected number of defects.", "answerGuide": "Read each region count and divide by the sample total. Apply complements and unions carefully, restrict the denominator for the conditional probability to the qualifying shoes, then compute E(X) by summing xP(X=x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=4", "qpPages": [ 2, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q1b", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "probability", "data-analysis", "distributions" ], "primaryTopicId": "probability", "summary": "1(b): Use a Venn diagram for three defect events to find probabilities, conditional probability, and the expected number of defects.", "answerGuide": "Read each region count and divide by the sample total. Apply complements and unions carefully, restrict the denominator for the conditional probability to the qualifying shoes, then compute E(X) by summing xP(X=x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=4", "qpPages": [ 2, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q1c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability", "data-analysis", "distributions" ], "primaryTopicId": "probability", "summary": "1(c): Use a Venn diagram for three defect events to find probabilities, conditional probability, and the expected number of defects.", "answerGuide": "Read each region count and divide by the sample total. Apply complements and unions carefully, restrict the denominator for the conditional probability to the qualifying shoes, then compute E(X) by summing xP(X=x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=4", "qpPages": [ 2, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q1d", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "probability", "data-analysis", "distributions" ], "primaryTopicId": "probability", "summary": "1(d): Use a Venn diagram for three defect events to find probabilities, conditional probability, and the expected number of defects.", "answerGuide": "Read each region count and divide by the sample total. Apply complements and unions carefully, restrict the denominator for the conditional probability to the qualifying shoes, then compute E(X) by summing xP(X=x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=4", "qpPages": [ 2, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q1e", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 2, "topicIds": [ "probability", "data-analysis", "distributions" ], "primaryTopicId": "probability", "summary": "1(e): Use a Venn diagram for three defect events to find probabilities, conditional probability, and the expected number of defects.", "answerGuide": "Read each region count and divide by the sample total. Apply complements and unions carefully, restrict the denominator for the conditional probability to the qualifying shoes, then compute E(X) by summing xP(X=x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=4", "qpPages": [ 2, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q1f", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(f)", "marks": 3, "topicIds": [ "probability", "data-analysis", "distributions" ], "primaryTopicId": "probability", "summary": "1(f): Use a Venn diagram for three defect events to find probabilities, conditional probability, and the expected number of defects.", "answerGuide": "Read each region count and divide by the sample total. Apply complements and unions carefully, restrict the denominator for the conditional probability to the qualifying shoes, then compute E(X) by summing xP(X=x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=4", "qpPages": [ 2, 5 ], "msPages": [ 4, 4 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q2a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Calculate a product-moment correlation, express remaining fuel in terms of amount used, and obtain the two related correlations.", "answerGuide": "Use S_dp divided by the product of the standard-deviation totals for r. Write w=50−p; this reverses correlation sign with p and preserves its magnitude. Apply the same sign reversal to the distance relationship.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=5", "qpPages": [ 6, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q2bi", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(b)(i): Calculate a product-moment correlation, express remaining fuel in terms of amount used, and obtain the two related correlations.", "answerGuide": "Use S_dp divided by the product of the standard-deviation totals for r. Write w=50−p; this reverses correlation sign with p and preserves its magnitude. Apply the same sign reversal to the distance relationship.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=5", "qpPages": [ 6, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q2bii", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(b)(ii): Calculate a product-moment correlation, express remaining fuel in terms of amount used, and obtain the two related correlations.", "answerGuide": "Use S_dp divided by the product of the standard-deviation totals for r. Write w=50−p; this reverses correlation sign with p and preserves its magnitude. Apply the same sign reversal to the distance relationship.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=5", "qpPages": [ 6, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q2c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Determine the correlation between distance travelled and remaining fuel from the given correlation.", "answerGuide": "Use the fact that replacing p with 50 − p reverses the sign of its correlation with distance while preserving its magnitude.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=5", "qpPages": [ 6, 9 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson lists part 2(c) among the more challenging parts of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" }, "sourceEvidencePages": { "examinerReport": [ 3 ] } }, { "id": "ial-2022-1-WST01-01-q3a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use an outlier condition on a stem-and-leaf display, then undo coding to find the mean, standard deviation, and a conditional probability.", "answerGuide": "Find Q1 and Q3 from the ordered 24 values and apply the 1.5-IQR upper fence to determine c. Reverse x=d−125 for the mean and standard deviation. For the conditional probability, count values satisfying both events and divide by the count under the condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=6", "qpPages": [ 10, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q3b", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use an outlier condition on a stem-and-leaf display, then undo coding to find the mean, standard deviation, and a conditional probability.", "answerGuide": "Find Q1 and Q3 from the ordered 24 values and apply the 1.5-IQR upper fence to determine c. Reverse x=d−125 for the mean and standard deviation. For the conditional probability, count values satisfying both events and divide by the count under the condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=6", "qpPages": [ 10, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q3c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(c): Use an outlier condition on a stem-and-leaf display, then undo coding to find the mean, standard deviation, and a conditional probability.", "answerGuide": "Find Q1 and Q3 from the ordered 24 values and apply the 1.5-IQR upper fence to determine c. Reverse x=d−125 for the mean and standard deviation. For the conditional probability, count values satisfying both events and divide by the count under the condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=6", "qpPages": [ 10, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q3d", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(d): Use an outlier condition on a stem-and-leaf display, then undo coding to find the mean, standard deviation, and a conditional probability.", "answerGuide": "Find Q1 and Q3 from the ordered 24 values and apply the 1.5-IQR upper fence to determine c. Reverse x=d−125 for the mean and standard deviation. For the conditional probability, count values satisfying both events and divide by the count under the condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=6", "qpPages": [ 10, 13 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q4a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(a): Work with a discrete uniform variable and affine transformations to find probabilities, expectations, a transformed distribution, and variances.", "answerGuide": "List the five equally likely W values. Substitute into X and Y to obtain their possible values and probabilities, then calculate expectations and variances from sums. Use Var(a+bY)=b²Var(Y) for the final transformation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=7", "qpPages": [ 14, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q4b", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(b): Work with a discrete uniform variable and affine transformations to find probabilities, expectations, a transformed distribution, and variances.", "answerGuide": "List the five equally likely W values. Substitute into X and Y to obtain their possible values and probabilities, then calculate expectations and variances from sums. Use Var(a+bY)=b²Var(Y) for the final transformation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=7", "qpPages": [ 14, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q4c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(c): Work with a discrete uniform variable and affine transformations to find probabilities, expectations, a transformed distribution, and variances.", "answerGuide": "List the five equally likely W values. Substitute into X and Y to obtain their possible values and probabilities, then calculate expectations and variances from sums. Use Var(a+bY)=b²Var(Y) for the final transformation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=7", "qpPages": [ 14, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q4di", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(d)(i): Work with a discrete uniform variable and affine transformations to find probabilities, expectations, a transformed distribution, and variances.", "answerGuide": "List the five equally likely W values. Substitute into X and Y to obtain their possible values and probabilities, then calculate expectations and variances from sums. Use Var(a+bY)=b²Var(Y) for the final transformation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=7", "qpPages": [ 14, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q4dii", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)(ii)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(d)(ii): Work with a discrete uniform variable and affine transformations to find probabilities, expectations, a transformed distribution, and variances.", "answerGuide": "List the five equally likely W values. Substitute into X and Y to obtain their possible values and probabilities, then calculate expectations and variances from sums. Use Var(a+bY)=b²Var(Y) for the final transformation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=7", "qpPages": [ 14, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q4e", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(e): Work with a discrete uniform variable and affine transformations to find probabilities, expectations, a transformed distribution, and variances.", "answerGuide": "List the five equally likely W values. Substitute into X and Y to obtain their possible values and probabilities, then calculate expectations and variances from sums. Use Var(a+bY)=b²Var(Y) for the final transformation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=7", "qpPages": [ 14, 17 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q5a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(a): Use normal models and independence to calculate a tail probability, solve for an unknown mean, and find a probability for a minimum of five values.", "answerGuide": "Standardize the normal variable for (a). For (b), square the one-value probability to represent two independent results and solve the resulting z-equation for m. For (c), use one minus the probability all five values are non-negative.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=8", "qpPages": [ 18, 21 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q5b", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(b): Use normal models and independence to calculate a tail probability, solve for an unknown mean, and find a probability for a minimum of five values.", "answerGuide": "Standardize the normal variable for (a). For (b), square the one-value probability to represent two independent results and solve the resulting z-equation for m. For (c), use one minus the probability all five values are non-negative.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=8", "qpPages": [ 18, 21 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q5c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(c): Use normal models and independence to calculate a tail probability, solve for an unknown mean, and find a probability for a minimum of five values.", "answerGuide": "Standardize the normal variable for (a). For (b), square the one-value probability to represent two independent results and solve the resulting z-equation for m. For (c), use one minus the probability all five values are non-negative.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=8", "qpPages": [ 18, 21 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q6a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "6(a): Interpret a regression line, assess its validity at an extreme input, read a residual, solve when prediction exceeds the input, and update the slope after correcting one score.", "answerGuide": "Use the regression equation at the mean pre-test score, interpret the slope as predicted final-score change per pre-test mark, and avoid extrapolating beyond the observed range. Identify the largest positive vertical residual. Solve 10.8+0.748p>p, then correct Σpf and the centered cross-product and divide by S_pp for the revised gradient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=9", "qpPages": [ 22, 25 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q6b", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "6(b): Interpret a regression line, assess its validity at an extreme input, read a residual, solve when prediction exceeds the input, and update the slope after correcting one score.", "answerGuide": "Use the regression equation at the mean pre-test score, interpret the slope as predicted final-score change per pre-test mark, and avoid extrapolating beyond the observed range. Identify the largest positive vertical residual. Solve 10.8+0.748p>p, then correct Σpf and the centered cross-product and divide by S_pp for the revised gradient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=9", "qpPages": [ 22, 25 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q6c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "6(c): Interpret a regression line, assess its validity at an extreme input, read a residual, solve when prediction exceeds the input, and update the slope after correcting one score.", "answerGuide": "Use the regression equation at the mean pre-test score, interpret the slope as predicted final-score change per pre-test mark, and avoid extrapolating beyond the observed range. Identify the largest positive vertical residual. Solve 10.8+0.748p>p, then correct Σpf and the centered cross-product and divide by S_pp for the revised gradient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=9", "qpPages": [ 22, 25 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q6d", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "6(d): Interpret a regression line, assess its validity at an extreme input, read a residual, solve when prediction exceeds the input, and update the slope after correcting one score.", "answerGuide": "Use the regression equation at the mean pre-test score, interpret the slope as predicted final-score change per pre-test mark, and avoid extrapolating beyond the observed range. Identify the largest positive vertical residual. Solve 10.8+0.748p>p, then correct Σpf and the centered cross-product and divide by S_pp for the revised gradient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=9", "qpPages": [ 22, 25 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q6e", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Find the pre-test score range for which the regression model predicts a higher final score.", "answerGuide": "Solve 10.8 + 0.748p > p and state the valid range of p in context.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=9", "qpPages": [ 22, 25 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report lists this regression part among the more challenging items and notes the additional algebra needed to interpret the inequality.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" }, "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-2022-1-WST01-01-q6f", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(f)", "marks": 5, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Recalculate a regression gradient after correcting one recorded score.", "answerGuide": "Update Σpf and the corresponding centered cross-product using the corrected student value, then divide the revised S_pf by S_pp to obtain the new gradient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=9", "qpPages": [ 22, 25 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson identifies the final regression recalculation as challenging; responses needed to adjust the summary statistics consistently before finding the gradient.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" }, "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-2022-1-WST01-01-q7a", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "distributions", "probability", "algebra" ], "primaryTopicId": "distributions", "summary": "7(a): Use the cumulative distribution of a hypergeometric count to form an equation for the bag size, verify a candidate size, and calculate exact probabilities.", "answerGuide": "Express the cumulative probabilities through combinations from n marbles with seven green and three draws. Cross-multiply to derive the cubic identity, substitute n=21 to verify it, then use hypergeometric probabilities for zero and one green to find a and b exactly.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=10", "qpPages": [ 26, 28 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q7b", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "distributions", "probability", "algebra" ], "primaryTopicId": "distributions", "summary": "7(b): Use the cumulative distribution of a hypergeometric count to form an equation for the bag size, verify a candidate size, and calculate exact probabilities.", "answerGuide": "Express the cumulative probabilities through combinations from n marbles with seven green and three draws. Cross-multiply to derive the cubic identity, substitute n=21 to verify it, then use hypergeometric probabilities for zero and one green to find a and b exactly.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=10", "qpPages": [ 26, 28 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" } }, { "id": "ial-2022-1-WST01-01-q7c", "paperId": "ial-2022-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "distributions", "probability", "algebra" ], "primaryTopicId": "distributions", "summary": "Find exact cumulative probabilities for the number of green marbles in a sample of three.", "answerGuide": "With n = 21, use hypergeometric combinations to find the probabilities for zero and one green marble, then convert them to the requested cumulative distribution values.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_que_20220113.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST01_01_rms_20220303.pdf#page=10", "qpPages": [ 26, 28 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says Question 7 discriminated well and only a small proportion completed it fully, with errors in the distribution and algebraic probability expressions.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-qp.pdf", "markScheme": "wst01-ms.pdf", "examinerReport": "verified-WST01-er.pdf" }, "sourceSha256": { "questionPaper": "bd4ff04b723e530965114fe967a02de6826cad457c95f3818fbe38b97d910c87", "markScheme": "a122414ffdbce70e6af21c3c25d6cdfff295e564f2d2272e75f05670f5172128", "examinerReport": "47ac86f480f42cdcc93c7f951ec1ff5920beeb4a8e5b9dff3b5589e30f7c3dc0" }, "sourceEvidencePages": { "examinerReport": [ 3, 7 ] } }, { "id": "ial-2022-1-WST02-01-q1a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(a): Use a Poisson model to find a threshold, approximate a sum of weekly counts by a normal distribution, and calculate a binomial probability for poor weeks.", "answerGuide": "Use the Poisson upper tail to identify integer x. For six weeks, add means and variances, apply a continuity correction to the normal approximation, and evaluate the tail. For 50 weeks, model poor weeks as binomial and subtract the probabilities of zero and one.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q1b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(b): Use a Poisson model to find a threshold, approximate a sum of weekly counts by a normal distribution, and calculate a binomial probability for poor weeks.", "answerGuide": "Use the Poisson upper tail to identify integer x. For six weeks, add means and variances, apply a continuity correction to the normal approximation, and evaluate the tail. For 50 weeks, model poor weeks as binomial and subtract the probabilities of zero and one.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q1c", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(c): Use a Poisson model to find a threshold, approximate a sum of weekly counts by a normal distribution, and calculate a binomial probability for poor weeks.", "answerGuide": "Use the Poisson upper tail to identify integer x. For six weeks, add means and variances, apply a continuity correction to the normal approximation, and evaluate the tail. For 50 weeks, model poor weeks as binomial and subtract the probabilities of zero and one.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q2a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "2(a): Derive the density from a piecewise CDF, calculate the mean and variance, and find an expectation after cubing the random variable.", "answerGuide": "Differentiate F(x) on its support and include zero elsewhere. Evaluate the mean from the CDF or density, integrate x²f(x) to obtain the second moment and verify the variance, then integrate x³f(x) and multiply by the scale factor for E(3X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q2b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "2(b): Derive the density from a piecewise CDF, calculate the mean and variance, and find an expectation after cubing the random variable.", "answerGuide": "Differentiate F(x) on its support and include zero elsewhere. Evaluate the mean from the CDF or density, integrate x²f(x) to obtain the second moment and verify the variance, then integrate x³f(x) and multiply by the scale factor for E(3X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q2c", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "2(c): Derive the density from a piecewise CDF, calculate the mean and variance, and find an expectation after cubing the random variable.", "answerGuide": "Differentiate F(x) on its support and include zero elsewhere. Evaluate the mean from the CDF or density, integrate x²f(x) to obtain the second moment and verify the variance, then integrate x³f(x) and multiply by the scale factor for E(3X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q2d", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "2(d): Derive the density from a piecewise CDF, calculate the mean and variance, and find an expectation after cubing the random variable.", "answerGuide": "Differentiate F(x) on its support and include zero elsewhere. Evaluate the mean from the CDF or density, integrate x²f(x) to obtain the second moment and verify the variance, then integrate x³f(x) and multiply by the scale factor for E(3X).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q3a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(a): Set up a two-sided Poisson rate test for a refurbished copier, calculate its critical region and actual size, then interpret the observed count.", "answerGuide": "Give a Poisson reason based on independent random events over a fixed interval. State H0 and two-sided H1 for the rate. Find tail probabilities under λ=8, choose the critical tails totaling at most 5%, calculate the attained size, and classify four against the region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q3b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(b): Set up a two-sided Poisson rate test for a refurbished copier, calculate its critical region and actual size, then interpret the observed count.", "answerGuide": "Give a Poisson reason based on independent random events over a fixed interval. State H0 and two-sided H1 for the rate. Find tail probabilities under λ=8, choose the critical tails totaling at most 5%, calculate the attained size, and classify four against the region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q3c", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(c): Set up a two-sided Poisson rate test for a refurbished copier, calculate its critical region and actual size, then interpret the observed count.", "answerGuide": "Give a Poisson reason based on independent random events over a fixed interval. State H0 and two-sided H1 for the rate. Find tail probabilities under λ=8, choose the critical tails totaling at most 5%, calculate the attained size, and classify four against the region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q3d", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(d): Set up a two-sided Poisson rate test for a refurbished copier, calculate its critical region and actual size, then interpret the observed count.", "answerGuide": "Give a Poisson reason based on independent random events over a fixed interval. State H0 and two-sided H1 for the rate. Find tail probabilities under λ=8, choose the critical tails totaling at most 5%, calculate the attained size, and classify four against the region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q3e", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(e): Set up a two-sided Poisson rate test for a refurbished copier, calculate its critical region and actual size, then interpret the observed count.", "answerGuide": "Give a Poisson reason based on independent random events over a fixed interval. State H0 and two-sided H1 for the rate. Find tail probabilities under λ=8, choose the critical tails totaling at most 5%, calculate the attained size, and classify four against the region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q4a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "4(a): Work with a piecewise probability density to sketch its support, normalize it, construct the CDF, calculate a probability, and describe skewness.", "answerGuide": "Sketch the two supported branches and zero elsewhere. Integrate over the full support to obtain k. Integrate each piece from the lower bound to x and join continuously for F(x). Substitute the given mean for the tail probability; compare mean, median/shape or third moment to justify the skew direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q4b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "4(b): Work with a piecewise probability density to sketch its support, normalize it, construct the CDF, calculate a probability, and describe skewness.", "answerGuide": "Sketch the two supported branches and zero elsewhere. Integrate over the full support to obtain k. Integrate each piece from the lower bound to x and join continuously for F(x). Substitute the given mean for the tail probability; compare mean, median/shape or third moment to justify the skew direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q4c", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 7, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "4(c): Work with a piecewise probability density to sketch its support, normalize it, construct the CDF, calculate a probability, and describe skewness.", "answerGuide": "Sketch the two supported branches and zero elsewhere. Integrate over the full support to obtain k. Integrate each piece from the lower bound to x and join continuously for F(x). Substitute the given mean for the tail probability; compare mean, median/shape or third moment to justify the skew direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q4d", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "4(d): Work with a piecewise probability density to sketch its support, normalize it, construct the CDF, calculate a probability, and describe skewness.", "answerGuide": "Sketch the two supported branches and zero elsewhere. Integrate over the full support to obtain k. Integrate each piece from the lower bound to x and join continuously for F(x). Substitute the given mean for the tail probability; compare mean, median/shape or third moment to justify the skew direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q4e", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "4(e): Work with a piecewise probability density to sketch its support, normalize it, construct the CDF, calculate a probability, and describe skewness.", "answerGuide": "Sketch the two supported branches and zero elsewhere. Integrate over the full support to obtain k. Integrate each piece from the lower bound to x and join continuously for F(x). Substitute the given mean for the tail probability; compare mean, median/shape or third moment to justify the skew direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q5a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "5(a): Model colour-blind applicant counts binomially, justify a Poisson approximation, and test a claimed pilot-success proportion with a normal approximation.", "answerGuide": "State X~Bin(n,0.045), then an assumption such as independent applicants with constant probability. Use a Poisson approximation for the rare-event probability and explain np is moderate while p is small. For the claim, state two-sided hypotheses, check normal-approximation conditions, apply a continuity correction, and compare the test statistic or p-value with the 5% critical value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q5b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "5(b): Model colour-blind applicant counts binomially, justify a Poisson approximation, and test a claimed pilot-success proportion with a normal approximation.", "answerGuide": "State X~Bin(n,0.045), then an assumption such as independent applicants with constant probability. Use a Poisson approximation for the rare-event probability and explain np is moderate while p is small. For the claim, state two-sided hypotheses, check normal-approximation conditions, apply a continuity correction, and compare the test statistic or p-value with the 5% critical value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q5c", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "5(c): Model colour-blind applicant counts binomially, justify a Poisson approximation, and test a claimed pilot-success proportion with a normal approximation.", "answerGuide": "State X~Bin(n,0.045), then an assumption such as independent applicants with constant probability. Use a Poisson approximation for the rare-event probability and explain np is moderate while p is small. For the claim, state two-sided hypotheses, check normal-approximation conditions, apply a continuity correction, and compare the test statistic or p-value with the 5% critical value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q5d", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "5(d): Model colour-blind applicant counts binomially, justify a Poisson approximation, and test a claimed pilot-success proportion with a normal approximation.", "answerGuide": "State X~Bin(n,0.045), then an assumption such as independent applicants with constant probability. Use a Poisson approximation for the rare-event probability and explain np is moderate while p is small. For the claim, state two-sided hypotheses, check normal-approximation conditions, apply a continuity correction, and compare the test statistic or p-value with the 5% critical value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q5e", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 7, "topicIds": [ "probability", "distributions", "statistical-testing" ], "primaryTopicId": "probability", "summary": "5(e): Model colour-blind applicant counts binomially, justify a Poisson approximation, and test a claimed pilot-success proportion with a normal approximation.", "answerGuide": "State X~Bin(n,0.045), then an assumption such as independent applicants with constant probability. Use a Poisson approximation for the rare-event probability and explain np is moderate while p is small. For the claim, state two-sided hypotheses, check normal-approximation conditions, apply a continuity correction, and compare the test statistic or p-value with the 5% critical value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q6a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Define the sampling distribution of a statistic.", "answerGuide": "State that it is the probability distribution of the statistic over all possible random samples of the stated size.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=13", "qpPages": [ 22, 25 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report notes that very few candidates gave the required definition of a sampling distribution correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" }, "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-2022-1-WST02-01-q6b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "6(b): Define a sampling distribution, find the distribution of total breakfast items for two sampled customers, and calculate its expectation.", "answerGuide": "Define the distribution of a statistic over repeated samples of the stated size. List the possible item counts and their probabilities from the 6:3:2 sales ratio, convolve for two independent customers, then sum tP(T=t) for E(T).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=13", "qpPages": [ 22, 25 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q6c", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "6(c): Define a sampling distribution, find the distribution of total breakfast items for two sampled customers, and calculate its expectation.", "answerGuide": "Define the distribution of a statistic over repeated samples of the stated size. List the possible item counts and their probabilities from the 6:3:2 sales ratio, convolve for two independent customers, then sum tP(T=t) for E(T).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=13", "qpPages": [ 22, 25 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q7a", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "7(a): Transform a uniform area distribution into probabilities and the variance of the square’s side length.", "answerGuide": "Translate L≥4.5 into a condition on A and take the corresponding interval length over 20. For the variance, use L=√A to integrate E(L) and E(L²) over the uniform density, then calculate E(L²)−E(L)².", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=14", "qpPages": [ 26, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST02-01-q7b", "paperId": "ial-2022-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "7(b): Transform a uniform area distribution into probabilities and the variance of the square’s side length.", "answerGuide": "Translate L≥4.5 into a condition on A and take the corresponding interval length over 20. For the variance, use L=√A to integrate E(L) and E(L²) over the uniform density, then calculate E(L²)−E(L)².", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST02_01_que_20220118.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20220303.pdf#page=14", "qpPages": [ 26, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-qp.pdf", "markScheme": "verified-WST02-ms.pdf", "examinerReport": "wst02-er.pdf" }, "sourceSha256": { "questionPaper": "33c01375756128918f4aac4cca33c928048849a00a514930200ced2907ecc892", "markScheme": "9a79ff9d4f9d65eb1584dc41a64b6af885ef6e2bc9762fa0c6016e49e00f1111", "examinerReport": "94577f2fc145470940283322d7b2f69910de8e03d481e70c645cf6763a5ee6ef" } }, { "id": "ial-2022-1-WST03-01-q1a", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "1(a): Estimate the population mean and variance from a sample, form a 95% confidence interval, and estimate a proportion using the interval’s lower limit.", "answerGuide": "Use the unbiased sample formulas. Apply the known population standard deviation and normal critical value for the confidence interval. Interpret the lower endpoint as a conservative value for μ and standardize 10.5 to estimate the lower-tail proportion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q1b", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "1(b): Estimate the population mean and variance from a sample, form a 95% confidence interval, and estimate a proportion using the interval’s lower limit.", "answerGuide": "Use the unbiased sample formulas. Apply the known population standard deviation and normal critical value for the confidence interval. Interpret the lower endpoint as a conservative value for μ and standardize 10.5 to estimate the lower-tail proportion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q1c", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "1(c): Estimate the population mean and variance from a sample, form a 95% confidence interval, and estimate a proportion using the interval’s lower limit.", "answerGuide": "Use the unbiased sample formulas. Apply the known population standard deviation and normal critical value for the confidence interval. Interpret the lower endpoint as a conservative value for μ and standardize 10.5 to estimate the lower-tail proportion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q2a", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "2(a): Test whether two large school-year samples have equal mean scores and explain the Central Limit Theorem’s role.", "answerGuide": "State H0: μ7=μ8 and the two-sided alternative. Use the difference in sample means divided by the standard error from both variances and sample sizes, compare with the 1% critical value, and conclude in context. Explain that large samples make the sampling distribution of the mean approximately normal.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q2b", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "2(b): Test whether two large school-year samples have equal mean scores and explain the Central Limit Theorem’s role.", "answerGuide": "State H0: μ7=μ8 and the two-sided alternative. Use the difference in sample means divided by the standard error from both variances and sample sizes, compare with the 1% critical value, and conclude in context. Explain that large samples make the sampling distribution of the mean approximately normal.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q3a", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "3(a): Calculate and test Pearson correlation between metabolic rate and BMI, then calculate and test Spearman correlation with activity rank.", "answerGuide": "Use the supplied centered sums for r. State the one-tailed positive-correlation hypotheses, check the normal-population assumption, and compare r to its critical value. Rank both MR and DPA for Spearman’s coefficient, then use the two-tailed critical table to make the final decision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q3b", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "3(b): Calculate and test Pearson correlation between metabolic rate and BMI, then calculate and test Spearman correlation with activity rank.", "answerGuide": "Use the supplied centered sums for r. State the one-tailed positive-correlation hypotheses, check the normal-population assumption, and compare r to its critical value. Rank both MR and DPA for Spearman’s coefficient, then use the two-tailed critical table to make the final decision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q3c", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "3(c): Calculate and test Pearson correlation between metabolic rate and BMI, then calculate and test Spearman correlation with activity rank.", "answerGuide": "Use the supplied centered sums for r. State the one-tailed positive-correlation hypotheses, check the normal-population assumption, and compare r to its critical value. Rank both MR and DPA for Spearman’s coefficient, then use the two-tailed critical table to make the final decision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q3d", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "3(d): Calculate and test Pearson correlation between metabolic rate and BMI, then calculate and test Spearman correlation with activity rank.", "answerGuide": "Use the supplied centered sums for r. State the one-tailed positive-correlation hypotheses, check the normal-population assumption, and compare r to its critical value. Rank both MR and DPA for Spearman’s coefficient, then use the two-tailed critical table to make the final decision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q3e", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "3(e): Calculate and test Pearson correlation between metabolic rate and BMI, then calculate and test Spearman correlation with activity rank.", "answerGuide": "Use the supplied centered sums for r. State the one-tailed positive-correlation hypotheses, check the normal-population assumption, and compare r to its critical value. Rank both MR and DPA for Spearman’s coefficient, then use the two-tailed critical table to make the final decision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q4a", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Describe how to select a quota sample from the student population.", "answerGuide": "Divide the population into the relevant groups, set the required number from each group, and select students non-randomly until every quota is filled.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says the quota-sampling explanation was not answered well by many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" }, "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2022-1-WST03-01-q4b", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 8, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "4(b): Describe quota sampling, perform a chi-squared independence test, then assess how doubling a percentage table to counts affects each test component.", "answerGuide": "Divide the population into groups and fill each target quota by non-random selection. For the test, state hypotheses, calculate expected counts, sum (O−E)²/E and compare to the appropriate degrees-of-freedom critical value. When counts double, keep hypotheses and critical value fixed, examine how the statistic changes, then state the conclusion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q4ci", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(i)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "4(c)(i): Describe quota sampling, perform a chi-squared independence test, then assess how doubling a percentage table to counts affects each test component.", "answerGuide": "Divide the population into groups and fill each target quota by non-random selection. For the test, state hypotheses, calculate expected counts, sum (O−E)²/E and compare to the appropriate degrees-of-freedom critical value. When counts double, keep hypotheses and critical value fixed, examine how the statistic changes, then state the conclusion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q4cii", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(ii)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "4(c)(ii): Describe quota sampling, perform a chi-squared independence test, then assess how doubling a percentage table to counts affects each test component.", "answerGuide": "Divide the population into groups and fill each target quota by non-random selection. For the test, state hypotheses, calculate expected counts, sum (O−E)²/E and compare to the appropriate degrees-of-freedom critical value. When counts double, keep hypotheses and critical value fixed, examine how the statistic changes, then state the conclusion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q4ciii", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(iii)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "4(c)(iii): Describe quota sampling, perform a chi-squared independence test, then assess how doubling a percentage table to counts affects each test component.", "answerGuide": "Divide the population into groups and fill each target quota by non-random selection. For the test, state hypotheses, calculate expected counts, sum (O−E)²/E and compare to the appropriate degrees-of-freedom critical value. When counts double, keep hypotheses and critical value fixed, examine how the statistic changes, then state the conclusion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q4civ", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)(iv)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "4(c)(iv): Describe quota sampling, perform a chi-squared independence test, then assess how doubling a percentage table to counts affects each test component.", "answerGuide": "Divide the population into groups and fill each target quota by non-random selection. For the test, state hypotheses, calculate expected counts, sum (O−E)²/E and compare to the appropriate degrees-of-freedom critical value. When counts double, keep hypotheses and critical value fixed, examine how the statistic changes, then state the conclusion.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=8", "qpPages": [ 14, 17 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q5a", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(a): Combine independent normal event times to find probabilities and a percentile, then check a calculation for repeated triathlons.", "answerGuide": "Add means and variances to model total time. For the swim-run difference, use independence to form its normal distribution. Invert the stated tail probability for t. Use a binomial complement for six trials, and explain whether the single-triathlon probability applies to the same event.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q5b", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(b): Combine independent normal event times to find probabilities and a percentile, then check a calculation for repeated triathlons.", "answerGuide": "Add means and variances to model total time. For the swim-run difference, use independence to form its normal distribution. Invert the stated tail probability for t. Use a binomial complement for six trials, and explain whether the single-triathlon probability applies to the same event.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q5c", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(c): Combine independent normal event times to find probabilities and a percentile, then check a calculation for repeated triathlons.", "answerGuide": "Add means and variances to model total time. For the swim-run difference, use independence to form its normal distribution. Invert the stated tail probability for t. Use a binomial complement for six trials, and explain whether the single-triathlon probability applies to the same event.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q5d", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(d): Combine independent normal event times to find probabilities and a percentile, then check a calculation for repeated triathlons.", "answerGuide": "Add means and variances to model total time. For the swim-run difference, use independence to form its normal distribution. Invert the stated tail probability for t. Use a binomial complement for six trials, and explain whether the single-triathlon probability applies to the same event.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q5e", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(e): Combine independent normal event times to find probabilities and a percentile, then check a calculation for repeated triathlons.", "answerGuide": "Add means and variances to model total time. For the swim-run difference, use independence to form its normal distribution. Invert the stated tail probability for t. Use a binomial complement for six trials, and explain whether the single-triathlon probability applies to the same event.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q6a", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "6(a): Complete tail expected frequencies for a fixed normal model, test goodness of fit, and state the degrees-of-freedom change when parameters are estimated.", "answerGuide": "Use continuity boundaries and the N(310,4²) model to calculate the two tail frequencies. Add their chi-squared contributions to 9.71 and compare with the 5% critical value using the fixed-parameter degrees of freedom. Reduce degrees of freedom by two when mean and standard deviation are estimated.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { "id": "ial-2022-1-WST03-01-q6b", "paperId": "ial-2022-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2022, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "6(b): Complete tail expected frequencies for a fixed normal model, test goodness of fit, and state the degrees-of-freedom change when parameters are estimated.", "answerGuide": "Use continuity boundaries and the N(310,4²) model to calculate the two tail frequencies. Add their chi-squared contributions to 9.71 and compare with the 5% critical value using the fixed-parameter degrees of freedom. Reduce degrees of freedom by two when mean and standard deviation are estimated.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_que_20220125.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" } }, { 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_rms_20220303.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this a challenging finisher and says only the most able candidates recognized the two-degree reduction.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/WST03_01_pef_20220303.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "verified-WST03_01_que_20220125.pdf", "markScheme": "wst03-ms.pdf", "examinerReport": "wst03-er.pdf" }, "sourceSha256": { "questionPaper": "df65b7a0c6323c731e7d0b78ef1e04cd19a32f53ba6bbea564901f3765a8d708", "markScheme": "6d86ea04025d25872f9977e41be91251965d0e0efaa588b95b12453c652a9619", "examinerReport": "383c600745261d004e80addb0a56401e68974ed0e309126868462d78865dd0ee" }, 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"studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q02a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use the two measured tree heights to determine the constants in the rational growth model.", "answerGuide": "Substitute t=2 and t=7 into the model to form two linear equations, then solve for both constants.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q02b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use the completed model to find the tree’s height at planting.", "answerGuide": "Set t=0 in the model and simplify the resulting value.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q03a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find both possible values of the triangle angle from the given two sides and opposite angle.", "answerGuide": "Apply the sine rule and include both inverse-sine solutions in the permitted angle range.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q03b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Use the obtuse-angle condition to find the remaining side length.", "answerGuide": "Select the obtuse angle from part (a), then use the cosine rule to calculate the third side.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q03c", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Calculate the area of the triangle to the requested precision.", "answerGuide": "Use one half times two known sides times the sine of 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solve the resulting linear equation.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q04ii", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find all real roots of the given quartic equation.", "answerGuide": "Substitute u=x² to obtain a quadratic in u, solve it, then retain non-negative u values and take both real square roots.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q05a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the tangent gradient at the stated point on the curve.", "answerGuide": "Differentiate the curve equation and substitute the point’s x-coordinate.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=13", "qpPages": [ 4, 4 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q05b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of the chord 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"differentiation", "summary": "Explain how the chord gradient relates to the tangent gradient as h approaches zero.", "answerGuide": "State that the secant gradient tends to the tangent gradient as the second point approaches P.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=13", "qpPages": [ 4, 4 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q06a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the angle subtended at the circle’s centre by the right-triangle geometry.", "answerGuide": "Use the sine ratio from the right triangle to determine the central angle in radians.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=15", "qpPages": [ 5, 5 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q06b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Calculate the shaded logo area after removing the triangle from the sector.", "answerGuide": "Find the sector area from radius and angle, subtract the right-triangle area, and round as requested.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=15", "qpPages": [ 5, 5 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q06c", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Calculate the perimeter of the remaining shaded logo.", "answerGuide": "Add the exposed sector arc and the remaining straight boundary lengths, excluding the shared cut edge.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=15", "qpPages": [ 5, 5 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q07ai", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)(i)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Use the given normal line at x=4 to determine the curve parameter k.", "answerGuide": "Use the normal gradient to infer the tangent gradient, differentiate the curve form, and substitute x=4.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=17", "qpPages": [ 6, 6 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q07aii", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)(ii)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at the specified point.", "answerGuide": "Substitute the point into the curve, evaluate the derivative there, and use point-slope form.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=17", "qpPages": [ 6, 6 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q07b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Determine the simplified expression for the function f(x).", "answerGuide": "Insert the parameter from part (a), integrate or simplify the derivative relation given, and use the point condition to fix the constant.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=17", "qpPages": [ 6, 6 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q08a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "State the x-ranges where the sketched cubic is at or below the x-axis.", "answerGuide": "Read the roots and use the curve’s crossing and tangency behaviour to determine the sign intervals.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q08b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the cubic function from its intercepts and the repeated root.", "answerGuide": "Write the cubic in factored form using roots 5 and −4 with −4 repeated, then use the y-intercept to find its coefficient.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q08c", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the exact x-coordinates where the cubic and the second curve intersect.", "answerGuide": "Equate the two expressions, factor the resulting polynomial, and solve every admissible root exactly.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q09a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Identify the constant horizontal shift from the x-intercept.", "answerGuide": "Substitute the stated root into the cosine model and solve for the phase constant.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=21", "qpPages": [ 8, 8 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q09b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Give the minimum point of the shifted cosine curve.", "answerGuide": "Use the phase shift and the cosine minimum value to state both coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=21", "qpPages": [ 8, 8 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q09c", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Solve the cosine equation over the stated interval beyond one full turn.", "answerGuide": "Use the reference angles and periodicity in degrees, then keep only solutions within the given interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=21", "qpPages": [ 8, 8 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q09di", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(d)(i)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Map the minimum point under the vertical stretch and shift.", "answerGuide": "Apply the transformation to the original y-value while keeping the x-coordinate fixed.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=21", "qpPages": [ 8, 8 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q09dii", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(d)(ii)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Map the minimum point under the horizontal compression.", "answerGuide": "Apply the horizontal scaling to the original x-coordinate while retaining the y-value.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=21", "qpPages": [ 8, 8 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q10a", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that tangency forces the stated quadratic condition on the line parameter.", "answerGuide": "Substitute the line into the conic, collect a quadratic in one coordinate, and set its discriminant to zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=9", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=22", "qpPages": [ 9, 9 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA11-01A-q10b", "paperId": "ial-2026-6-WMA11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact contact point using the negative parameter root.", "answerGuide": "Solve the quadratic condition, select the negative parameter, substitute it into the line and conic, and solve for the coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/6c/55/b0/28/1c/77/f8/50/9e/ed/49/f2/13/ac/c5/0f.pdf#page=9", "msUrl": "https://www.alevelcopilot.com/whfile/45/98/27/5a/60/0d/2e/05/71/0f/f9/b2/09/35/7a/7c.pdf#page=22", "qpPages": [ 9, 9 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q01a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Map the given point under the horizontal transformation f(2x).", "answerGuide": "Solve 2x for the original point’s x-coordinate; the output y-value remains unchanged.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q01b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Map the point under the vertical scaling of the function.", "answerGuide": "Keep x fixed and multiply the original y-coordinate by the stated scale.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q01c", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Map the point under the combined horizontal and vertical transformation.", "answerGuide": "Undo the horizontal input scaling to find the new x-coordinate, then apply the vertical multiplier to y.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q02a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Find the non-zero x-intercept of the logarithmic curve.", "answerGuide": "Set y=0 and solve the logarithmic factor equation, checking the stated domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q02b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Show that the stationary point coordinate satisfies the given exponential equation.", "answerGuide": "Differentiate using the product rule, set the derivative to zero, and rearrange to the required form.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q02ci", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)(i)", "marks": 1, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Carry out one iteration from the supplied starting value.", "answerGuide": "Substitute x1 into the iteration rule and round x2 to four decimal places.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q02cii", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)(ii)", "marks": 2, "topicIds": [ "differentiation", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "Repeat the iteration to estimate the stationary-point coordinate.", "answerGuide": "Continue the stated recurrence until the value is stable to four decimal places.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q03a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Differentiate the curve’s implicit relation to find dy/dx in terms of y.", "answerGuide": "Differentiate both sides with respect to x, apply the chain rule to y terms, and rearrange.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q03b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent line where the curve has the stated y-coordinate.", "answerGuide": "Substitute the y-value to find x, evaluate dy/dx, then form the tangent equation and clear denominators.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q04a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the straight-line log plot to determine the two model constants.", "answerGuide": "Use its intercept and gradient to recover log10 p and log10 q, then exponentiate and round.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q04b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the time at which the delivery model reaches 700 thousand.", "answerGuide": "Substitute N=700 into the model and solve the resulting exponential equation with logarithms.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q05a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "State the range of the rational function on its restricted domain.", "answerGuide": "Rewrite the function to expose its horizontal asymptote and excluded values, then state the resulting range.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q05b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the exponential function.", "answerGuide": "Set y=g(x), take logarithms, and rearrange to express the input in terms of y.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q05c", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the exact input a for which the composite f(g(a)) equals 8.", "answerGuide": "Use the definition of f, solve for g(a), then apply the inverse of g and simplify exactly.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q06a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the sine-cosine combination as a single shifted sine.", "answerGuide": "Match coefficients with R sin(x+α), use their squared sum for R, and determine α in the required interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q06bi", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "State the maximum value of the transformed function.", "answerGuide": "Use the amplitude and the stated power transformation to evaluate the maximum.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q06bii", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the greatest x-value in the domain where that maximum is attained.", "answerGuide": "Solve for the sine input that reaches the required extreme, then choose the largest solution in the given interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q07a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the local maximum using calculus and a trig identity.", "answerGuide": "Differentiate, set the derivative to zero, use a double-angle identity, and select the maximum in the specified region.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q07b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the exact area bounded by the curve and the x-axis.", "answerGuide": "Integrate the curve algebraically between the stated limits and simplify the exact result.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q08a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Determine the initial-value constant for the electric-car model.", "answerGuide": "Substitute t=0 and the initial count into the model.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=9", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=19", "qpPages": [ 9, 9 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q08b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Evaluate the electric-car count at t=3.", "answerGuide": "Substitute the time and constant into the exponential model and round to three significant figures.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=9", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=19", "qpPages": [ 9, 9 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q08c", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find the instantaneous rate of increase at t=5.", "answerGuide": "Differentiate the exponential model with respect to time and evaluate at 5.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=9", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=19", "qpPages": [ 9, 9 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q08d", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(d)", "marks": 4, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find when the electric- and petrol-car models predict equal counts.", "answerGuide": "Set both models equal, isolate the exponential terms, take logarithms, and solve for the time.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=9", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=19", "qpPages": [ 9, 9 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q09a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the minimum point of the translated absolute-value graph.", "answerGuide": "Use the vertex of the V-shaped graph, where the expression inside the modulus is zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=21", "qpPages": [ 10, 10 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q09b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the two x-coordinates where the line meets the absolute-value graph.", "answerGuide": "Set the line and graph equal, remove the modulus by considering both signs, and solve for both roots.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=21", "qpPages": [ 10, 10 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q09c", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use the given triangle area to determine the exact area of the second triangle.", "answerGuide": "Express each triangle’s base and height from the intercept geometry, use the known area to relate the parameter, then evaluate the other area.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=21", "qpPages": [ 10, 10 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q10a", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove the stated identity involving cotangent and cosecant.", "answerGuide": "Rewrite cotangent and cosecant using sine and cosine, apply the double-angle identity, and simplify to the right-hand side.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=11", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=23", "qpPages": [ 11, 11 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA13-01A-q10b", "paperId": "ial-2026-6-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on the restricted angle interval.", "answerGuide": "Use the proved identity to reduce the equation, solve the resulting quadratic in a trig ratio, and retain angles within the given range.", "qpUrl": "https://www.alevelcopilot.com/whfile/21/52/0a/c7/69/d9/df/46/a4/f9/30/67/ae/d8/a9/dc.pdf#page=11", "msUrl": "https://www.alevelcopilot.com/whfile/5e/f8/0a/f5/06/66/49/f3/29/86/5a/65/bd/62/f9/c5.pdf#page=23", "qpPages": [ 11, 11 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q01", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand the fractional power to four terms in ascending powers.", "answerGuide": "Apply the generalized binomial theorem through the x³ term and simplify each coefficient.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q02a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Find the constants in the partial-fraction decomposition.", "answerGuide": "Multiply through by the common denominator, compare coefficients or substitute convenient x-values, and solve for A, B and C.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=10", "qpPages": [ 2, 2 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q02b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "Evaluate the resulting rational integral.", "answerGuide": "Integrate each partial-fraction term using logarithmic and power antiderivatives, then include the constant.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=10", "qpPages": [ 2, 2 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q03", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": null, "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use the cube’s volume growth rate to find the instantaneous side-length rate.", "answerGuide": "Differentiate V=x³ with respect to time, substitute V=100 to find x, then solve dV/dt=3x² dx/dt.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=12", "qpPages": [ 2, 2 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q04a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Eliminate the parameter to obtain the stated Cartesian curve equation.", "answerGuide": "Solve one parametric equation for t, substitute into the other, and simplify to the required integer-coefficient form.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q04b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "proof" ], "primaryTopicId": "coordinate-geometry", "summary": "Prove by contradiction that the line and curve do not intersect.", "answerGuide": "Assume an intersection, substitute the line into the Cartesian curve, and show the resulting quadratic has a negative discriminant.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q05", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": null, "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the given substitution to evaluate the integral and identify both constants.", "answerGuide": "Set u to the stated expression in x, convert dx and the integrand, integrate in u, then rewrite the result in x.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=15", "qpPages": [ 3, 3 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q06a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate the curve to find dy/dx.", "answerGuide": "Differentiate each term with respect to x, include dy/dx on y terms, and isolate the derivative.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=16", "qpPages": [ 3, 3 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q06b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation at the specified point.", "answerGuide": "Evaluate the tangent gradient at P, take its negative reciprocal, and use point-slope form.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=16", "qpPages": [ 3, 3 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q06c", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact second intersection of the normal and the curve.", "answerGuide": "Substitute the normal equation into the curve, solve for the second root after removing the known contact point, and recover both coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=16", "qpPages": [ 3, 3 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q07a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Write a vector equation for the line through A and B.", "answerGuide": "Use A as the position vector and B−A as a direction vector.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=18", "qpPages": [ 4, 4 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q07b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine α and β from the condition that C lies on the line.", "answerGuide": "Equate the three coordinates of the line and C, then solve the parameter equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=18", "qpPages": [ 4, 4 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q07c", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the acute angle between OC and AB.", "answerGuide": "Form the two direction vectors, use the dot-product formula, and choose the acute angle.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=18", "qpPages": [ 4, 4 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q07d", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact position vector of the point on the line nearest the origin.", "answerGuide": "Set the line position vector perpendicular to its direction vector and solve for the line parameter.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=18", "qpPages": [ 4, 4 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q08a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Integrate the product of cosine and sine terms using an identity.", "answerGuide": "Use a product-to-sum identity, integrate each cosine term, and determine the requested coefficients.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=20", "qpPages": [ 5, 5 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q08b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact volume generated by rotating the bounded region about the x-axis.", "answerGuide": "Use V=π∫y²dx with the stated bounds, expand the trigonometric square, integrate, and simplify to the required exact form.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=20", "qpPages": [ 5, 5 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q09a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Solve the cooling differential equation and determine its constants from the initial temperature.", "answerGuide": "Separate variables, integrate, apply the initial condition, and rearrange into the requested exponential model.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=22", "qpPages": [ 6, 6 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q09b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Use the measured temperature at 20 minutes to find the rate constant.", "answerGuide": "Substitute t=20 and the observed temperature into the model, then solve with logarithms.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=22", "qpPages": [ 6, 6 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q09c", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Predict the drink temperature after one hour.", "answerGuide": "Substitute t=60 and the fitted rate constant into the model and round to one decimal place.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=22", "qpPages": [ 6, 6 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q10a", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the normal line to the parametric curve at the given parameter.", "answerGuide": "Differentiate x and y with respect to the parameter, calculate dy/dx at the point, then form and simplify the normal equation.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=24", "qpPages": [ 7, 7 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA14-01A-q10b", "paperId": "ial-2026-6-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Find the exact area bounded by the parametric curve, its normal and the x-axis.", "answerGuide": "Identify the parameter values of the intersections, integrate y dx using the parametric equations, and combine the signed areas.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/82/ea/4c/b7/0b/2a/35/ad/33/17/53/ea/9f/70/cd.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/8a/89/de/a4/36/c0/73/c3/c1/2d/72/dc/f6/67/cb/7a.pdf#page=24", "qpPages": [ 7, 7 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q01a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use momentum conservation to find the mass ratio in the direct collision.", "answerGuide": "Choose a positive direction, equate total momentum before and after, and use the reversed half-speeds.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q01b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the magnitude of the collision impulse on particle B.", "answerGuide": "Use impulse equals change in momentum for B, including its reversal of direction.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q02a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the support force at D from the stated force ratio and vertical equilibrium.", "answerGuide": "Use the ratio between support reactions, balance total vertical forces, and solve for the force at D.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q02b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the position x of the 75 kg particle using moments.", "answerGuide": "Take moments about one support, substitute the known weights and lever arms, and solve for x.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q02c", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the support reaction at D when the rod is on the point of tilting after the load change.", "answerGuide": "Use the limiting pivot condition to identify the zero support, then balance forces and moments for the remaining reaction.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q03a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the train’s deceleration from the speed-time graph.", "answerGuide": "Use the gradient of the relevant straight segment of the speed-time graph.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q03b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the acceleration-time graph over the full journey.", "answerGuide": "Show constant negative acceleration, any zero-acceleration cruise interval, and the final positive acceleration interval with correct time boundaries.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q03c", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total journey time from the speed-time graph.", "answerGuide": "Calculate distance as the area under the graph, equate it to the usual journey distance, and solve for T.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q03d", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total time for which the train travels below 30 m/s.", "answerGuide": "Use the acceleration gradients to find the threshold crossing times and add the intervals below 30 m/s.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q04a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Use the resultant force and acceleration magnitude to find the particle mass.", "answerGuide": "Add the force vectors, equate resultant magnitude to ma, and solve for m.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=19", "qpPages": [ 5, 5 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q04b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find k so the combined force is perpendicular to the given direction.", "answerGuide": "Form the combined vector and set its dot product with the specified vector to zero.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=19", "qpPages": [ 5, 5 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q05a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when car A reaches the finish using constant acceleration.", "answerGuide": "Use s=ut+½at² with the remaining distance and solve for the positive time.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=22", "qpPages": [ 5, 5 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q05b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find how long after the stated speed difference car B catches car A.", "answerGuide": "Write both velocity and displacement equations, solve for the time when the speed difference condition holds, then solve the relative-displacement equation for the catch-up interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=22", "qpPages": [ 5, 5 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q06a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension in the first section of the light washing line.", "answerGuide": "Resolve forces vertically at B and use the known mass and angle.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=25", "qpPages": [ 6, 6 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q06b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the angle of the final section by resolving forces at both garments.", "answerGuide": "Use horizontal and vertical equilibrium at B and C, eliminate the horizontal-section tension, and solve for α.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=25", "qpPages": [ 6, 6 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q06c", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Explain why the first tension cannot be determined if the line has mass.", "answerGuide": "State that a massive line has non-uniform tension, so its weight distribution introduces unknown forces.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=25", "qpPages": [ 6, 6 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q07a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the speed of the boat from its constant-velocity position vector.", "answerGuide": "Differentiate the position vector to get constant velocity components, then find their magnitude.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=29", "qpPages": [ 7, 7 ], "msPages": [ 29, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q07b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the time of day when the boat’s bearing from A is 315 degrees.", "answerGuide": "Subtract A’s position vector, impose the direction ratio corresponding to the bearing, solve for t, and convert hours after 2 pm to clock time.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=29", "qpPages": [ 7, 7 ], "msPages": [ 29, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q08a", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the friction force on particle B before release.", "answerGuide": "Use the normal reaction and coefficient of friction to calculate the limiting friction magnitude.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=31", "qpPages": [ 8, 8 ], "msPages": [ 31, 33 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q08b", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Use equations of motion to show the acceleration is 0.3g.", "answerGuide": "Resolve forces along each particle’s motion, use the common acceleration from the inextensible string, and solve the simultaneous equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=31", "qpPages": [ 8, 8 ], "msPages": [ 31, 33 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q08c", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "State how the inextensible-string condition enters the motion equations.", "answerGuide": "State that the two particles have equal acceleration magnitudes while the string remains taut.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=31", "qpPages": [ 8, 8 ], "msPages": [ 31, 33 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WME01-01A-q08d", "paperId": "ial-2026-6-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(d)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show that B stops just as it reaches the pulley after A hits the ground.", "answerGuide": "Find the speed when A reaches the ground using suvat, then use B’s remaining travel distance and its post-impact deceleration to show its stopping distance equals BP.", "qpUrl": "https://www.alevelcopilot.com/whfile/69/5f/21/2c/54/fe/72/fd/b6/12/64/18/97/36/09/7b.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/50/c5/25/7f/22/de/00/9e/ea/be/2b/91/30/d5/53/f8.pdf#page=31", "qpPages": [ 8, 8 ], "msPages": [ 31, 33 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q01a", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Draw and label the tree diagram for Jeff’s three attempts.", "answerGuide": "Show success and failure branches for each attempt, with probabilities conditional on reaching that attempt.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q01b", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use the total win probability to find p.", "answerGuide": "Add the mutually exclusive paths on which Jeff first succeeds and set the result equal to 0.74.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q01c", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that Jeff loses the game.", "answerGuide": "Use the complement of the probability that he wins.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q01d", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability that a win occurs on the second attempt.", "answerGuide": "Divide the probability of first-attempt failure followed by second-attempt success by the total win probability.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q02ai", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the median age from the ordered stem-and-leaf data.", "answerGuide": "Locate the central observation in the ordered list of 23 ages.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q02aii", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the interquartile range of the ages.", "answerGuide": "Find the lower and upper quartiles using the stated convention and subtract.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q02b", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a box plot and identify the outlier using the 1.5-IQR rule.", "answerGuide": "Calculate the outlier fences, identify any observation beyond them, and plot the quartiles, median, whiskers and outlier.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q02ci", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the mean annual time at the club from the supplied totals.", "answerGuide": "Add the extra total to the provided sum, then divide by 23.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q02cii", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the standard deviation of annual club time.", "answerGuide": "Use the supplied sum of squares and the mean to evaluate the standard deviation for the 23 members.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q03a", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the mean condition to show that r=0.18.", "answerGuide": "Form E(X) from the symmetric probability table and solve the given expectation equation.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q03b", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the variance and total probability conditions to determine s and t.", "answerGuide": "Form E(X²), use Var(X)=E(X²)−E(X)² to solve for s, then use total probability to find t.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q03ci", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)(i)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the expectation of the stated linear transformation of X.", "answerGuide": "Apply E(a+bX)=a+bE(X).", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q03cii", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the variance of the transformed random variable.", "answerGuide": "Apply Var(a+bX)=b²Var(X).", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q03d", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the stated quadratic expression in X is positive.", "answerGuide": "Solve the quadratic inequality, identify which listed X values satisfy it, and add their probabilities.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04a", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Scale the histogram bar width and height for the specified race-time class.", "answerGuide": "Use the supplied class width and frequency density to establish the horizontal and vertical scales, then calculate the target bar dimensions.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04b", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median race time by linear interpolation.", "answerGuide": "Locate the median rank within its grouped interval and interpolate proportionally.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04ci", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)(i)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean race time using the supplied totals.", "answerGuide": "Divide the provided total of fx by the number of runners.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04cii", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)(ii)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Estimate the standard deviation of race times.", "answerGuide": "Use the grouped sum of squares and estimated mean in the standard-deviation formula.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04d", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Estimate the probability that two selected race times are both between 38 and 44 minutes.", "answerGuide": "Estimate the count in the interval by interpolation, then use sampling without replacement for the pair.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04ei", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(e)(i)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "State the effect of correcting the fastest time on the median.", "answerGuide": "Explain whether changing the lowest observation affects the central ordered value.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q04eii", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(e)(ii)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "State the effect of correcting the fastest time on the interquartile range.", "answerGuide": "Explain whether the lower and upper quartiles change when only the minimum value is corrected.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q05a", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use independence and the supplied intersection probability to find x.", "answerGuide": "Set P(A∩B)=P(A)P(B), substitute the equal probabilities, and solve.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q05bi", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use the conditions for C and D to find P(D).", "answerGuide": "Translate the stated intersection and conditional probabilities into equations and solve for P(D).", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q05bii", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the conditional probability of D given that C did not occur.", "answerGuide": "Use P(D∩C′)/P(C′), obtaining the numerator from P(D)−P(C∩D).", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q06a", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability a newborn weighs more than 3500 g.", "answerGuide": "Standardise 3500 using the stated normal mean and standard deviation, then take the upper tail.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q06b", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the conditional probability that a high-classified newborn exceeds 3600 g.", "answerGuide": "Divide the tail probability above 3600 by the probability above 3500 using standard normal values.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q06c", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find w when 75.6% of high-classified newborns exceed w.", "answerGuide": "Set P(W>w|W>3500)=0.756, convert to an unconditional tail area, and invert the normal distribution.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q07a", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the exact sum of the paired products xy.", "answerGuide": "Use the regression gradient relation to recover the correlation, then use the product-moment formula and the summary statistics.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WST01-01A-q07b", "paperId": "ial-2026-6-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the least-squares regression line of y on x.", "answerGuide": "Recover r from the regression information, calculate the y-on-x gradient, then use the means to find the intercept.", "qpUrl": "https://www.alevelcopilot.com/whfile/a8/cd/63/c4/e3/6a/d5/3e/59/6d/8e/c9/e9/65/76/da.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/a8/90/ec/b6/88/d0/a5/1f/ae/e3/26/28/89/05/62/03.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q01a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the remainder when the cubic is divided by x − 2.", "answerGuide": "Use the remainder theorem: evaluate f(2) and simplify to 8a − 40.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q01b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the factor condition x − 2 to show that a = 5.", "answerGuide": "Since x − 2 is a factor, the remainder is zero; solve 8a − 40 = 0.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q01c", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factorise the cubic completely.", "answerGuide": "Substitute a = 5, take out x − 2, and factor the remaining quadratic to obtain (x − 2)(5x − 3)(x + 4).", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q02a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral of kx⁻³ + 4x^(3/2) from 1 to 4 in terms of k.", "answerGuide": "Integrate term by term to get −k/(2x²) + (8/5)x^(5/2), then apply the limits to obtain 15k/32 + 248/5.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q02b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact value of k when the definite integral equals 50.", "answerGuide": "Set 15k/32 + 248/5 = 50 and solve to get k = 64/75.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q03a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "sequences", "summary": "Show that the recurrence sequence is periodic and state its period.", "answerGuide": "Generate the terms 5, −1, 2, 5; the sequence repeats with period 3.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q03b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "sequences", "summary": "State the value of u₂₀₀.", "answerGuide": "Use the period 3 and 200 ≡ 2 (mod 3), so u₂₀₀ = u₂ = −1.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q03c", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "sequences", "proof" ], "primaryTopicId": "sequences", "summary": "Find the sum of the first 200 terms of the sequence.", "answerGuide": "Each three-term cycle sums to 6. The first 198 terms contain 66 cycles (sum 396); add u₁₉₉ + u₂₀₀ = 5 − 1 to obtain 400.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q04a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinates of the stationary points of the curve.", "answerGuide": "Differentiate, set 6x² + 7 − 3/x² = 0, and solve in x² to get x = ±1/√3.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q04b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-ranges for which the curve is increasing.", "answerGuide": "Use the sign of 6x² + 7 − 3/x²; it is positive when x < −1/√3 or x > 1/√3.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=3", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q05a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Use all table values in the trapezium rule to estimate the integral of f(x) from 3 to 4.5.", "answerGuide": "Use step width 0.25 and the seven tabulated ordinates; the estimate is 9.91 to 3 significant figures.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q05b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Find an antiderivative of √x(3 + 1/x²).", "answerGuide": "Rewrite as 3x^(1/2) + x^(−3/2) and integrate to obtain 2x^(3/2) − 2x^(−1/2) + C.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q05c", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Estimate the shaded area between the curves from x = 3 to x = 4.5.", "answerGuide": "Subtract the comparison-curve integral over [3, 4.5] from the trapezium estimate in part (a); give the result to 2 significant figures (about 1.0).", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q06i", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve 7^(x+3) = 5 using logarithms, to 3 significant figures.", "answerGuide": "Take logarithms to obtain x = log₇5 − 3 ≈ −2.17.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=13", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q06iia", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(ii)(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Rewrite the logarithmic equation as 6(log₈ y)² + 5log₈ y − 21 = 0.", "answerGuide": "Use log₈(64y³) = 2 + 3log₈y, substitute into the original equation, and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=13", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q06iib", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(ii)(b)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the exact values of y that satisfy the logarithmic equation.", "answerGuide": "Solve (3log₈y + 7)(2log₈y − 3) = 0; the exact values are y = 1/128 and y = 16√2.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=13", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q07a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "State the centre of the circle.", "answerGuide": "Read the centre from (x − 3)² + (y − 6)² = k: (3, 6).", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q07b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation y = mx of the line through the origin and the circle centre.", "answerGuide": "Its gradient is 6/3 = 2, so the equation is y = 2x.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q07c", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the two possible values of k from the closest point distance condition.", "answerGuide": "The centre is 3√5 from the origin and the nearest point is √5 away, so the radius is 2√5 or 4√5; square to get k = 20 or 80.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=5", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q08a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the first four terms in ascending powers of x for (k + 4x)¹².", "answerGuide": "Use the binomial theorem to obtain k¹² + 48k¹¹x + 1056k¹⁰x² + 14080k⁹x³.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q08b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the x³ coefficient to find k exactly.", "answerGuide": "Set 14080k⁹ = 55/2; then k⁹ = 1/512, giving k = 1/2.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q08c", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the y³ coefficient in (y + 2/y)(k + 4y²)¹².", "answerGuide": "Using k = 1/2, add the y·y² and (2/y)·y⁴ contributions; the coefficient is 267/128.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q09a", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Show that the City 1 model gives approximately 21.3°C at noon.", "answerGuide": "At noon t = 4, so θ = 17 + 5sin(π/3) ≈ 21.3°C.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=18", "qpPages": [ 7, 7 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q09b", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Show that equal temperatures at t = T lead to the stated quadratic in sin(πT/12).", "answerGuide": "Set the two models equal, use cos²u = 1 − sin²u, and simplify to 3sin²(πT/12) − sin(πT/12) − 2 = 0.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=18", "qpPages": [ 7, 7 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q09c", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find all values of T in the 24-hour period when the city temperatures are equal.", "answerGuide": "Solve for sin(πT/12) = −2/3 or 1 and retain values in 0 ≤ T ≤ 24: T = 6, 14.8, and 21.2 hours (3 s.f. where needed).", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=7", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=18", "qpPages": [ 7, 7 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q10i", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(i)", "marks": 3, "topicIds": [ "proof", "binomial" ], "primaryTopicId": "proof", "summary": "Use proof by exhaustion to show f(n) is a perfect square for positive integers n < 7.", "answerGuide": "Evaluate f(n) for n = 1,…,6 to get 25, 121, 169, 169, 121, 25; each is a perfect square.", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=20", "qpPages": [ 8, 8 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-6-WMA12-01A-q10ii", "paperId": "ial-2026-6-WMA12-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01A", "variant": "Paper A · China regional", "questionNumber": 10, "questionPart": "10(ii)", "marks": 4, "topicIds": [ "proof", "binomial" ], "primaryTopicId": "proof", "summary": "Prove the stated identity for consecutive binomial coefficients from the factorial formula.", "answerGuide": "Write both coefficients as factorials, combine over a common denominator, and simplify to (n + 1)!/((r + 1)!(n − r)!) = C(n + 1, r + 1).", "qpUrl": "https://www.alevelcopilot.com/whfile/15/3e/f4/58/ab/60/62/55/50/67/24/1c/8c/2a/65/4c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/4b/2d/8f/cd/fa/d8/fe/2c/25/ca/0c/71/29/7e/47/b2.pdf#page=20", "qpPages": [ 8, 8 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2023-w-11-q01a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-w-11-q01b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1(b): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-w-11-q02", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-11-q03", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the rate of change of a cube’s volume when its edge is 20 cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this rates-of-change question challenged most candidates; many bypassed differentiation and the chain rule.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Use V=x³, differentiate, then apply the chain rule with dx/dt=0.01." }, { "id": "cie-2023-w-11-q04a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(a): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2023-w-11-q04b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(b): Use the progression rule to find the requested term or sum.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2023-w-11-q04c", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-11-q05a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-w-11-q05b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-11-q06a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-w-11-q06b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "6(b): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2023-w-11-q06c", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the shaded motif area formed by two circular arcs, in terms of π, r and √3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A fully correct area solution was rare; candidates struggled to plan a sector-and-segment method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Express the motif as a major sector minus the equal circular segments, using sector, triangle and segment areas." }, { "id": "cie-2023-w-11-q07", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "7: Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2023-w-11-q08a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-11-q08b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-11-q09a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-11-q09b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the interval of x-values that permits the composite function gf.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=18", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part challenged most candidates; many tried to form gf directly and made no progress.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Complete the quadratic for f(x), impose g’s input-domain condition, and solve the resulting quadratic inequality." }, { "id": "cie-2023-w-11-q09c", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "9(c): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=18", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-11-q09d", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of the restricted quadratic function h.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=18", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few obtained the full inverse because they did not use the restricted domain to select the negative branch.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 3 ], "tagConfidence": "reviewed", "answerGuide": "Complete the square, solve for the inverse and choose the branch consistent with x<0." }, { "id": "cie-2023-w-11-q10a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Differentiate the given gradient function once more to obtain the second derivative.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given gradient expression with respect to x once more and simplify to obtain d²y/dx²." }, { "id": "cie-2023-w-11-q10b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-11-q10c", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(c): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=20", "qpPages": [ 16, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-11-q10d", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(d)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(d): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=20", "qpPages": [ 16, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-11-q11a", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Apply the tangency condition to determine the line and curve parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-11-q11b", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b): Apply the tangency condition to determine the line and curve parameters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=21", "qpPages": [ 18, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-11-q11c", "paperId": "cie-2023-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(c): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_11.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_11.pdf#page=21", "qpPages": [ 18, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-w-12-q01", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "1: Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-w-12-q02", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-12-q03a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-12-q03b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3(b): Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2023-w-12-q04a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "4(a): Use the sector angle and tangent lengths to find the shaded boundary length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2023-w-12-q04b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-12-q05a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-12-q05b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Use the angle from part (a) to find the first ten terms’ sum in the requested surd form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "About one fifth omitted this part; others confused the term index or did not give the requested exact form.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Evaluate the first term and common ratio, then apply the finite geometric-series formula for ten terms." }, { "id": "cie-2023-w-12-q06a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-12-q06b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-12-q06c", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the equation of the quadratic after the stated translation and reflection sequence.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Parts (b) and (c) were more challenging; candidates often used the wrong order, and 24% omitted (c).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 6 ], "tagConfidence": "reviewed", "answerGuide": "Apply transformations in the stated order, track horizontal and vertical changes, then expand." }, { "id": "cie-2023-w-12-q07a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-12-q07b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-12-q07c", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(c): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-12-q08a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-12-q08bi", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "8(b)(i): Apply the function definitions to find the requested expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2023-w-12-q08bii", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "8(b)(ii): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2023-w-12-q08c", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-12-q09a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the coordinates of the two intersections of curves containing fractional powers of x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many could not handle the fractional powers to find the intersections; 22% then missed the dependent area part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Set the expressions equal, substitute u=√x, solve the quadratic and recover both coordinates." }, { "id": "cie-2023-w-12-q09b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-12-q10a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=20", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-12-q10b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 1, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "State the intervals on which the given function is increasing.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=21", "qpPages": [ 16, 17 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says 36% omitted this linked part and few correct answers were seen.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 7 ], "tagConfidence": "reviewed", "answerGuide": "Use the stationary x-values from part (a) to identify where the derivative is positive." }, { "id": "cie-2023-w-12-q11a", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=22", "qpPages": [ 18, 18 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-12-q11b", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=23", "qpPages": [ 18, 19 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-12-q11c", "paperId": "cie-2023-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(c): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_12.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_12.pdf#page=23", "qpPages": [ 18, 19 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-w-13-q01", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "1: Integrate the gradient and use the given point to determine the curve equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate term by term, use any given point to determine the constant, and simplify the requested expression." }, { "id": "cie-2023-w-13-q02a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "2(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-13-q02b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "2(b): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-13-q03a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-w-13-q03b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-13-q04ai", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a)(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-13-q04aii", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "4(a)(ii): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-w-13-q04b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-13-q05a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-13-q05b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "5(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2023-w-13-q06", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-w-13-q07a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "7(a): Determine the ranges from the stated function definitions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2023-w-13-q07b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "7(b): Apply the function definitions to find the requested expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2023-w-13-q07c", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(c): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=11", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-13-q08a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Read the sine-curve diagram to state its scale factor and one phase shift.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Only a minority gave both parameters correctly; common errors halved a incorrectly or omitted π from b.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 10 ], "tagConfidence": "reviewed", "answerGuide": "Read the period for a, then use a corresponding point or intercept to determine a valid b." }, { "id": "cie-2023-w-13-q08b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "8(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-13-q09a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-13-q09b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "9(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-13-q09c", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(c): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=12", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-13-q10a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(a): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-w-13-q10b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 1, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(b): Use the circle and sector measurements to find the requested length.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the circle or sector formula using the given angle and arc length, keeping the angle in radians." }, { "id": "cie-2023-w-13-q10c", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)", "marks": 7, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "10(c): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2023-w-13-q11a", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-13-q11b", "paperId": "cie-2023-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2023, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "11(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_13.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-21-q01", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find an exact compound-angle sine value from an acute-angle sine ratio.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few solutions were correct; many used a calculator instead of the exact ratio and compound-angle formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the acute-angle condition for the positive cosine, then apply the compound-angle identity without a calculator." }, { "id": "cie-2023-w-21-q02", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2023-w-21-q03a", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-w-21-q03b", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-21-q04a", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-w-21-q04b", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-21-q04c", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-21-q05a", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-w-21-q05b", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-w-21-q06a", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-w-21-q06b", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on the specified negative-radian interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few solutions were fully correct; candidates often tried invalid factorisation and struggled with negative radian angles.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 13 ], "tagConfidence": "reviewed", "answerGuide": "Use part (a), solve the quadratic in cos θ with the quadratic formula, then select roots in the interval." }, { "id": "cie-2023-w-21-q06c", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact integral of the trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few used the required double-angle identity correctly before integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 13 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the squared trig term using a double-angle identity before integrating and applying limits." }, { "id": "cie-2023-w-21-q07a", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-21-q07b", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show the stated relation for q at the stationary point of the implicit curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates showed the given result after substituting p and q into the original equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Substitute p=ln 3 and y=q into the original equation, simplify and isolate the exponential term in q." }, { "id": "cie-2023-w-21-q07c", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-21-q07d", "paperId": "cie-2023-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_21.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-w-22-q01", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-w-22-q02", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-22-q03a", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2023-w-22-q03b", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "3(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-22-q04a", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=7", "qpPages": [ 7, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-w-22-q04b", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the inequality involving modulus and linear graphs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few chose the correct inequality region after finding both critical values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 16 ], "tagConfidence": "reviewed", "answerGuide": "Use the sketch to choose the valid intersection boundary, reject the extraneous critical value and state the region." }, { "id": "cie-2023-w-22-q04c", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "4(c): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-w-22-q04d", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "List the integers satisfying both inequalities from the preceding parts.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few correct solutions; candidates often carried an invalid critical value from the earlier part.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 16 ], "tagConfidence": "reviewed", "answerGuide": "Intersect the integer sets from parts (b) and (c), using the corrected boundary from the sketch." }, { "id": "cie-2023-w-22-q05a", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-22-q05b", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the division result from part (a) to evaluate the exact integral in logarithmic form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few earned marks; many ignored ‘Hence’ and tried to integrate the original rational expression directly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 16 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand as a polynomial plus a simple rational term using the quotient and remainder, then integrate." }, { "id": "cie-2023-w-22-q06a", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2023-w-22-q06b", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Show that the parameter at the second point satisfies the stated equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few correct solutions; most lacked the correct parametric derivative from part (a).", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the parametric gradient from part (a), set it equal to 12 and rearrange to an equation in t." }, { "id": "cie-2023-w-22-q06c", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-w-22-q07a", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-w-22-q07b", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-22-q07c", "paperId": "cie-2023-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral between the specified points on the trigonometric curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_22.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_22.pdf#page=11", "qpPages": [ 13, 13 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few obtained the given result because they used an incorrect upper limit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Use the coordinates from earlier parts, integrate the cosine expression and substitute the correct limits." }, { "id": "cie-2023-w-23-q01", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find an exact compound-angle sine value from an acute-angle sine ratio.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few solutions were correct; many used a calculator instead of the exact ratio and compound-angle formula.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Use the acute-angle condition for the positive cosine, then apply the compound-angle identity without a calculator." }, { "id": "cie-2023-w-23-q02", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2023-w-23-q03a", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "3(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-w-23-q03b", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-23-q04a", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=7", "qpPages": [ 5, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-w-23-q04b", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-23-q04c", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(c): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=8", "qpPages": [ 5, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-23-q05a", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-w-23-q05b", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-w-23-q06a", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-w-23-q06b", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on the specified negative-radian interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Few solutions were fully correct; candidates often tried invalid factorisation and struggled with negative radian angles.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Use part (a), solve the quadratic in cos θ with the quadratic formula, then select roots in the interval." }, { "id": "cie-2023-w-23-q06c", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact integral of the trigonometric expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=10", "qpPages": [ 9, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Few used the required double-angle identity correctly before integration.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite the squared trig term using a double-angle identity before integrating and applying limits." }, { "id": "cie-2023-w-23-q07a", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-23-q07b", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show the stated relation for q at the stationary point of the implicit curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Few candidates showed the given result after substituting p and q into the original equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Substitute p=ln 3 and y=q into the original equation, simplify and isolate the exponential term in q." }, { "id": "cie-2023-w-23-q07c", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(c): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2023-w-23-q07d", "paperId": "cie-2023-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2023, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "7(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_23.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-w-31-q01", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "1: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-w-31-q02", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Shade an Argand-plane region defined by a modulus comparison and an argument restriction.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "A high proportion did not attempt the complex-number questions, including Question 2.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 21 ], "tagConfidence": "reviewed", "answerGuide": "Convert the modulus comparison to a half-plane and the argument condition to a wedge; shade their intersection." }, { "id": "cie-2023-w-31-q03", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use two plotted (x, ln y) points to determine a and b in y=abˣ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Very few used the expected log-linear method; most treated the plotted ln y values as y values.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 21, 22 ], "tagConfidence": "reviewed", "answerGuide": "Take logs to get ln y=ln a+x ln b; use intercept and gradient to solve for a and b." }, { "id": "cie-2023-w-31-q04a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2023-w-31-q04b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2023-w-31-q05a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-31-q05b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=10", "qpPages": [ 7, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-31-q06a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-31-q06b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact point where the normal to a parametric curve has gradient −2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "This part was more challenging; several confused the gradient condition for the normal.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 22 ], "tagConfidence": "reviewed", "answerGuide": "Find dy/dx=(dy/dt)/(dx/dt), use the normal-gradient condition to solve for t, then substitute." }, { "id": "cie-2023-w-31-q07", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=12", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-31-q08a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "8(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-w-31-q08b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-31-q08c", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "8(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=13", "qpPages": [ 11, 11 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-w-31-q08d", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "8(d): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=14", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-w-31-q09a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2023-w-31-q09b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-31-q10a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Rewrite the expression in the required form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-31-q10b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the partial fractions to expand f(x) through the x² term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=17", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The expansion was more challenging; only a few successfully used the binomial method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Expand each factor by the binomial series to the required order, multiply by its numerator and collect powers." }, { "id": "cie-2023-w-31-q10c", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "10(c): Identify the requested term in a binomial expansion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2023-w-31-q11a", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 1, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=18", "qpPages": [ 16, 16 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-w-31-q11b", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-w-31-q11c", "paperId": "cie-2023-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 5, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(c): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_31.pdf#page=19", "qpPages": [ 16, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2023-w-32-q01a", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-w-32-q01b", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2023-w-32-q02", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2: Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2023-w-32-q03", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2023-w-32-q04a", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "4(a): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2023-w-32-q04b", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-32-q05", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate an exact integral with an improper rational integrand.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Many made little progress because they did not simplify the improper fraction first; few split all terms correctly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 25, 26 ], "tagConfidence": "reviewed", "answerGuide": "Simplify the improper fraction by division or partial fractions before integrating each term." }, { "id": "cie-2023-w-32-q06a", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Sketch two functions to show their equation has exactly one root in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Many sketches were unrecognisable and few used the graphs analytically to establish the root.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 26 ], "tagConfidence": "reviewed", "answerGuide": "Sketch cot x and 2−cos x with correct intercepts and behaviour, showing their single crossing." }, { "id": "cie-2023-w-32-q06b", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-32-q06c", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2023-w-32-q07a", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2023-w-32-q07b", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=16", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-32-q08a", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "8(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=17", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-32-q08b", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "8(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_32.pdf#page=17", "qpPages": [ 12, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2023-w-32-q09a", "paperId": "cie-2023-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(a): 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"https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-33-q07b", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Use the 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through x³." }, { "id": "cie-2023-w-33-q09c", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "State the exact interval of x-values for which the expansion is valid.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=19", "qpPages": [ 15, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "This part was very challenging; some gave |x|<2 without applying the stricter condition.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 32 ], "tagConfidence": "reviewed", "answerGuide": "Intersect the convergence intervals for both binomial expansions; the stricter condition gives the answer." }, { "id": "cie-2023-w-33-q10a", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2023-w-33-q10b", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "10(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=20", "qpPages": [ 16, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2023-w-33-q11a", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a unit vector parallel to the given three-dimensional line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "This short part was unexpectedly difficult; some found squared magnitude 6 but did not normalize the vector.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 32 ], "tagConfidence": "reviewed", "answerGuide": "Read the direction vector and divide it by its magnitude, taking the square root of its squared length." }, { "id": "cie-2023-w-33-q11b", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "11(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=21", "qpPages": [ 18, 18 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2023-w-33-q11c", "paperId": "cie-2023-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2023, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)", "marks": 5, "topicIds": [ "vectors", "coordinate-geometry", "algebra" ], "primaryTopicId": "vectors", "summary": "Determine whether two three-dimensional lines are parallel, intersecting or skew.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_33.pdf#page=21", "qpPages": [ 18, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Question 11(c) caused serious issues; candidates often ignored parallel lines or mishandled vector signs.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 32 ], "tagConfidence": "reviewed", "answerGuide": "Compare direction vectors; if not parallel, solve for a common point and test consistency across coordinates." }, { "id": "cie-2023-w-41-q01", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-41-q02", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2023-w-41-q03a", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2023-w-41-q03b", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2023-w-41-q03c", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(c): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2023-w-41-q04a", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-41-q04b", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2023-w-41-q04c", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find how far the joined particles travel after collision before friction stops them.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Very few earned full marks and many left it blank; candidates missed the post-collision speed or chose an inefficient method.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 34 ], "tagConfidence": "reviewed", "answerGuide": "Use the post-collision speed from momentum, then apply constant-acceleration motion with friction retarding motion." }, { "id": "cie-2023-w-41-q05", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction coefficient for two connected particles in limiting equilibrium on different slopes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Many found this extended response challenging and were unsure which principle to apply to each particle.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 34, 35 ], "tagConfidence": "reviewed", "answerGuide": "Draw force diagrams, resolve parallel and perpendicular to each plane, and solve equilibrium equations with correct friction directions." }, { "id": "cie-2023-w-41-q06ai", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a)(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-w-41-q06aii", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a)(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-w-41-q06b", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find a car’s constant speed downhill with power and speed-dependent resistance given.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=12", "qpPages": [ 10, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "One of the most challenging parts; candidates omitted the weight component or could not solve the resulting equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 35 ], "tagConfidence": "reviewed", "answerGuide": "Use P=Fv, then apply Newton’s second law at constant speed with gravity and resistance included." }, { "id": "cie-2023-w-41-q07", "paperId": "cie-2023-w-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 9, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find total distance for a particle with piecewise velocity and continuity at t=8.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_41.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_41.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Responses ranged from complete solutions to blank or no progress; candidates sometimes differentiated velocity instead of integrating.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 35 ], "tagConfidence": "reviewed", "answerGuide": "Integrate each velocity segment, use continuity and the rest endpoint to find the constant, and combine distances." }, { "id": "cie-2023-w-42-q01", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-42-q02", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2023-w-42-q03", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-42-q04a", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(a): Use the contact forces and friction limit to find the coefficient or threshold force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-w-42-q04b", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled during the third second under a resultant force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Very few earned full marks; many found displacement for the first three seconds instead of the third second.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Find acceleration from Newton’s law, calculate displacement at 3 s and 2 s, then subtract." }, { "id": "cie-2023-w-42-q05a", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-42-q05b", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-42-q05c", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time difference between two particles reaching the ground after collision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=13", "qpPages": [ 6, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Listed among the most challenging parts; candidates used inconsistent signs or the pre-collision speed.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Use post-collision velocity and height in each vertical-motion equation, solve both times, then subtract." }, { "id": "cie-2023-w-42-q06a", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2023-w-42-q06b", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the slope angle when a train travels at constant speed under specified power.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=15", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "One of the most challenging parts; candidates often omitted weight components or resistances.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Use P=Fv, then balance driving force with unchanged resistances and the weight component along the slope." }, { "id": "cie-2023-w-42-q07a", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2023-w-42-q07b", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time curve using stationary velocities from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "A correct cubic sketch was uncommon; many drew straight segments or placed the zero-velocity endpoint incorrectly.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 37, 38 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity to locate turning points, evaluate their velocities, and sketch the cubic with correct endpoints." }, { "id": "cie-2023-w-42-q07c", "paperId": "cie-2023-w-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find total distance travelled from t=0 to t=15.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_42.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_42.pdf#page=18", "qpPages": [ 10, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "One of the most challenging parts; many integrated the full interval to get displacement instead of distance.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 37 ], "tagConfidence": "reviewed", "answerGuide": "Locate every zero of velocity, integrate on constant-sign intervals and add absolute displacements." }, { "id": "cie-2023-w-43-q01", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-43-q02a", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-43-q02b", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the constant resistive force on a nail after a hammer collision and 4 cm motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Rather demanding, with few fully correct responses; errors included wrong speed, distance units or combined mass.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Use momentum conservation for post-impact speed, then work-energy over 0.04 m with the combined mass and gravity." }, { "id": "cie-2023-w-43-q03a", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(a): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-w-43-q03b", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(b): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-w-43-q03c", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(c): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-43-q04a", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2023-w-43-q04b", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2023-w-43-q05a", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2023-w-43-q05b", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(b): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=14", "qpPages": [ 6, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2023-w-43-q06a", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=15", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2023-w-43-q06b", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find total distance in the first 12 seconds from the acceleration function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=16", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "More challenging than part (a); many integrated 0 to 12 without splitting at instantaneous rest times.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 40 ], "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration using the initial condition, locate rest times, and add absolute displacements between them." }, { "id": "cie-2023-w-43-q07a", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(a): Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2023-w-43-q07b", "paperId": "cie-2023-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2023, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 8, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find distance travelled before rest in two-stage connected-particle motion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_43.pdf#page=18", "qpPages": [ 10, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Much more difficult than part (a), especially because acceleration changes when B reaches the ground.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s law or energy before B hits the ground, recalculate after acceleration changes, then combine distances." }, { "id": "cie-2023-w-51-q01a", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-w-51-q01b", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2023-w-51-q02a", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=6", "qpPages": [ 4, 4 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-51-q02b", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-51-q02c", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "For eight dice throws, find the probability of at least 8 on fewer than three occasions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The context challenged many and this binomial part was often omitted; success criteria and accuracy caused errors.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 43 ], "tagConfidence": "reviewed", "answerGuide": "Use Bin(8,15/36) and sum the terms for 0, 1 and 2 successes." }, { "id": "cie-2023-w-51-q03a", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-w-51-q03b", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the least weight qualifying as a large egg given 40% medium and small below 76 g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many found it challenging and made no attempt; successful work used a diagram and a normal-quantile equation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Find the small-egg cumulative probability, add 0.40 and convert that normal quantile back to a weight." }, { "id": "cie-2023-w-51-q03c", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability that more than 68 of 150 eggs are medium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "A significant number made little or no progress; errors included wrong parameters, continuity correction or using exact binomial probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 43, 44 ], "tagConfidence": "reviewed", "answerGuide": "Approximate Bin(150,0.4) by N(60,36), use boundary 68.5 and evaluate the upper tail." }, { "id": "cie-2023-w-51-q04a", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "4(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2023-w-51-q04b", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate grouped-data estimates of mean and standard deviation for athletes’ times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=12", "qpPages": [ 8, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Finding the standard deviation was considerably more challenging and few solutions were fully accurate.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 44 ], "tagConfidence": "reviewed", "answerGuide": "Use class midpoints and frequencies for the grouped mean and variance, then take the square root." }, { "id": "cie-2023-w-51-q05a", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-51-q05b", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-51-q05c", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(product of three spinner scores ≤4 | red score is odd).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report identifies this as the paper’s most challenging question; few used conditional probability and a scenario list.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 44 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate qualifying triples with red score 1 or 3, find the joint probability and divide by P(X odd)." }, { "id": "cie-2023-w-51-q06a", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-51-q06b", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-51-q06c", "paperId": "cie-2023-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_51.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_51.pdf#page=17", "qpPages": [ 12, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2023-w-52-q01a", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-52-q01b", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-52-q02a", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-52-q02b", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "For ten spins, find the probability of scoring 5 more than four but fewer than eight times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The context challenged many; candidates used the wrong distribution or misread the strict success criteria.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 47 ], "tagConfidence": "reviewed", "answerGuide": "Use Bin(10,1/5) and add the terms for 5, 6 and 7 successes." }, { "id": "cie-2023-w-52-q03", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-52-q04a", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-52-q04b", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-w-52-q04c", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-52-q05ai", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a)(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-52-q05aii", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find h when 40% of club members have a height greater than h.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "This inverse-normal part was more challenging; many used the wrong tail or treated 0.4 as a z-value.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 48 ], "tagConfidence": "reviewed", "answerGuide": "Convert the upper tail to a z-value, use h=166+10z and retain precision before rounding to 2 d.p." }, { "id": "cie-2023-w-52-q05b", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find P(X>0) when the normal variable’s mean and standard deviation satisfy the stated relation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many omitted this part or made little progress; successful solutions formed a one-variable equation and standardised 0.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 48 ], "tagConfidence": "reviewed", "answerGuide": "Standardise 0 using the relation between σ and μ, solve the resulting equation for z, then use the table." }, { "id": "cie-2023-w-52-q06a", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-52-q06b", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=15", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-52-q06c", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(bag Y chosen | the removed marbles have different colours).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=16", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Very few were correct and nearly a fifth omitted it; candidates often did not identify conditional probability.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 49 ], "tagConfidence": "reviewed", "answerGuide": "Find P(Y and different colours) from the tree, then divide by P(different colours), the complement of part (b)." }, { "id": "cie-2023-w-52-q07a", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-52-q07b", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=18", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-52-q07c", "paperId": "cie-2023-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability a four-letter selection contains at least one D and exactly one A.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_52.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_52.pdf#page=19", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "This challenged most candidates; successful solutions listed admissible cases and counted selections rather than arrangements.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 49, 50 ], "tagConfidence": "reviewed", "answerGuide": "Count valid selection cases by number of Ds, then divide by the total number of four-letter selections." }, { "id": "cie-2023-w-53-q01a", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-53-q01b", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-53-q02a", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-w-53-q02b", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-w-53-q03a", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-53-q03b", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(both marbles came from bag B | one white and one red).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This conditional-probability part was challenging; errors often omitted one of the two possible orders.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 52 ], "tagConfidence": "reviewed", "answerGuide": "Include both RW and WR outcomes from bag B in the numerator, divided by the total W-R probability from either bag." }, { "id": "cie-2023-w-53-q04a", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "data-representation" ], "primaryTopicId": "data-representation", "summary": "4(a): Represent the grouped data graphically and estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries or use the requested statistical graph, then read the estimate from the corresponding frequency level." }, { "id": "cie-2023-w-53-q04b", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2023-w-53-q04c", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=11", "qpPages": [ 6, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-53-q05ai", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a)(i): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-53-q05aii", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a)(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-53-q05b", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-53-q05c", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation for the probability more than 20 of 75 test-takers pass.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=14", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Choosing the correct probability area was identified as the biggest challenge in this approximation question.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 52 ], "tagConfidence": "reviewed", "answerGuide": "Approximate Bin(75,0.3), use continuity boundary 20.5, standardise and choose the upper tail from a sketch." }, { "id": "cie-2023-w-53-q06ai", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a)(i): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-53-q06aii", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a)(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-53-q06b", "paperId": "cie-2023-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2023, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability Jai and Kaz share a group when 12 people are split into groups of 5, 4 and 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_53.pdf#page=16", "qpPages": [ 10, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This was challenging; many did not correctly count the three separate group-size scenarios.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 53 ], "tagConfidence": "reviewed", "answerGuide": "Count the three cases in which the pair is together in each group size, then divide by the total groupings." }, { "id": "cie-2023-w-61-q01", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-61-q02", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-61-q03ai", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a)(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-61-q03aii", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a)(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-w-61-q03bi", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b)(i): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-w-61-q03bii", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b)(ii): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-w-61-q04a", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-61-q04b", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-61-q05", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5: Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-w-61-q06a", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-61-q06b", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2023-w-61-q06c", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-61-q07a", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-w-61-q07b", "paperId": "cie-2023-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_61.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_61.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2023-w-62-q01a", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-w-62-q01b", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-w-62-q02a", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-62-q02b", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-62-q03ai", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 1, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Explain why P(X=8)=0.0439 alone does not justify rejecting the null hypothesis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "This was not well attempted; candidates compared a point probability with 5% rather than the relevant tail.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 56 ], "tagConfidence": "reviewed", "answerGuide": "Use the one-sided p-value P(X≥8), not the point probability P(X=8), for comparison with 5%." }, { "id": "cie-2023-w-62-q03aii", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "3(a)(ii): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-w-62-q03b", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-62-q03c", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type I error for the later test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "This was not well attempted; candidates often used p=0.8 or omitted the relevant upper-tail terms.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 56 ], "tagConfidence": "reviewed", "answerGuide": "Use the rejection region under the true null model p=0.5 and add its binomial tail probability." }, { "id": "cie-2023-w-62-q04a", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "4(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-w-62-q04b", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-w-62-q04c", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(c): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=8", "qpPages": [ 7, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2023-w-62-q05a", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "5(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2023-w-62-q05b", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-62-q05c", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2023-w-62-q06", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6", "marks": 7, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that batch profit exceeds $40 from independent normal ingredient quantities and costs.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Few were fully successful; errors included not squaring standard deviations and confusing cost, revenue and profit.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Calculate means and variances of each ingredient cost, combine independent costs, derive profit’s normal distribution and standardise." }, { "id": "cie-2023-w-62-q07a", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-62-q07b", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-62-q07ci", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(c)(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-62-q07cii", "paperId": "cie-2023-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(c)(ii): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_62.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_62.pdf#page=12", "qpPages": [ 12, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-63-q01", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-63-q02", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-63-q03ai", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a)(i): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-63-q03aii", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a)(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2023-w-63-q03bi", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b)(i): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-w-63-q03bii", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b)(ii): Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-w-63-q04a", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2023-w-63-q04b", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2023-w-63-q05", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5: Use the poisson model to calculate the requested probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=9", "qpPages": [ 9, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the Poisson parameter from the stated rate and interval, then use the probability formula or cumulative probabilities for the event." }, { "id": "cie-2023-w-63-q06a", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2023-w-63-q06b", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2023-w-63-q06c", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(c): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2023-w-63-q07a", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=12", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2023-w-63-q07b", "paperId": "cie-2023-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2023, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the Type II error probability for the later test when the true mean is 0.4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_qp_63.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_ms_63.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says many candidates omitted the first step in this probability calculation.", "difficultySourceUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w23_er.pdf", "sourceEvidencePages": [ 60 ], "tagConfidence": "reviewed", "answerGuide": "Use the prior test’s non-rejection boundary, model the sample under mean 0.4, then calculate the probability in that region." }, { "id": "cie-2025-s-11-q01", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "1: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-11-q02a", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "2(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2025-s-11-q02b", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-11-q03a", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the tenth term of a geometric progression with negative common ratio from its third term and the sum of its first three terms.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Cambridge's examiner report says this part was difficult for most candidates. Successful solutions used the given third term and three-term sum to form an equation for the common ratio before finding the tenth term.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 2 ], "tagConfidence": "reviewed", "answerGuide": "Use ar² = 18 and a + ar + ar² = 26 to determine the negative common ratio and first term, then evaluate ar⁹." }, { "id": "cie-2025-s-11-q03b", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "3(b): Apply the progression sum formula to find the requested total.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the first term and common ratio, then apply the relevant nth-term or sum formula; check any convergence condition for an infinite sum." }, { "id": "cie-2025-s-11-q04", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4: Integrate between the stated bounds to find the area.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2025-s-11-q05ai", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a)(i): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-11-q05aii", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a)(ii): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-11-q05b", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "5(b): Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2025-s-11-q06a", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2025-s-11-q06b", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 8, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-11-q07a", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2025-s-11-q07b", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 10, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-11-q08", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 8, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "8: Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 12, 12 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the radius and tangent length, then subtract the sector area from the adjoining triangle area to obtain the shaded region." }, { "id": "cie-2025-s-11-q09a", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "geometry" ], "primaryTopicId": "geometry", "summary": "9(a): Combine the triangle and sector areas to find the shaded area.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 14, 14 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes this part as challenging for most candidates and notes that many found only the radius.", 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"algebra" ], "primaryTopicId": "algebra", "summary": "10(c): Rewrite the expression in the required form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 16, 17 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-11-q10d", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(d)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "10(d): Determine the ranges from the stated function definitions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 16, 17 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the output of the inner function into the outer function, respect the given domains, and simplify." }, { "id": "cie-2025-s-11-q10e", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(e)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(e): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 16, 17 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-11-q10f", "paperId": "cie-2025-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(f)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "10(f): Compose the two defined functions in the stated order.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_11.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_11.pdf", "qpPages": [ 16, 17 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part was challenging for most candidates and that few answers were correct.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 4 ], "tagConfidence": "reviewed", "answerGuide": "Substitute in the defined function rules in the correct order, then simplify the resulting expression." }, { "id": "cie-2025-s-12-q01", "paperId": "cie-2025-s-12", "board": "CIE", "qualification": 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"Identify the common difference or ratio, apply the appropriate progression formula, and round only at the requested final stage." }, { "id": "cie-2025-s-12-q02", "paperId": "cie-2025-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "2: Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_12.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2025-s-12-q03", "paperId": "cie-2025-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "3: Determine the requested coefficient in the binomial expansion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_12.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2025-s-12-q04b", "paperId": "cie-2025-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_12.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-12-q05a", "paperId": "cie-2025-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_12.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-12-q05b", "paperId": "cie-2025-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2025-s-13-q02", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": 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9, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the arc length to find the radius, then use the right triangle formed by the tangents to calculate their lengths and the full perimeter." }, { "id": "cie-2025-s-13-q08c", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(c): Integrate between the stated bounds to find the area.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=9", "msUrl": 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-13-q09b", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "9(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 10, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2025-s-13-q10a", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(a): Find the stationary coordinates and classify the point.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 12, 12 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-13-q10b", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "10(b): Find the stationary coordinates and classify the point.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 12, 12 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-13-q10ci", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure 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coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-13-q10cii", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(c)(ii)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Write the equation of a translated and reflected curve in the specified rational-plus-linear form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 12, 13 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge’s examiner report says 10(c)(ii) was the most frequently omitted part of this question, following a poorly answered 10(c)(i).", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 10 ], "tagConfidence": "reviewed", "answerGuide": "Apply the stated translation and x-axis reflection to the original curve, then rearrange to match the required form." }, { "id": "cie-2025-s-13-q11a", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "11(a): Determine the ranges from the stated function definitions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 14, 14 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated domain restrictions to the function rule and express the resulting set of possible values." }, { "id": "cie-2025-s-13-q11b", "paperId": "cie-2025-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_13.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_13.pdf", "qpPages": [ 14, 15 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-15-q01", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the given derivative and use (2,4) to find the curve equation.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each term and use the point to determine the constant of integration." }, { "id": "cie-2025-s-15-q02", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use binomial expansions whose x² coefficients are related to find the constant a.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the relevant terms in both expansions, equate coefficients and solve the quadratic." }, { "id": "cie-2025-s-15-q03a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a quartic equation exactly by completing the square.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite it as a squared quadratic and solve for the exact roots." }, { "id": "cie-2025-s-15-q03b", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the roots from 3(a) to solve the tangent equation for 0°<θ<180°.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Some candidates ignored the obtuse solution or mishandled the negative tangent for it.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 11, 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the exact tangent values, include both acute and obtuse angles in range, and check signs." }, { "id": "cie-2025-s-15-q04", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the volume from rotating the bounded region about the x-axis.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the x-axis intersection, integrate πy² between the correct limits, and round as requested." }, { "id": "cie-2025-s-15-q05a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the perimeter of the shaded region in the sector diagram.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the central angle and chord, then combine the straight sides and arc length." }, { "id": "cie-2025-s-15-q05b", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the area of the shaded region in the sector diagram.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 6, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the appropriate triangle area from the sector area." }, { "id": "cie-2025-s-15-q06a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Compare the amounts Ananya receives from her parents on her eleventh birthday.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the eleventh terms of the arithmetic and geometric progressions and compare them." }, { "id": "cie-2025-s-15-q06b", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the total received through the eighteenth birthday from both parents.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Add the arithmetic-series and geometric-series sums for 18 terms." }, { "id": "cie-2025-s-15-q07a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 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"questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find C's coordinates in the parallelogram.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Only half of candidates who completed 7(a) correctly completed this part; the report notes vector-orientation errors without a diagram.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 12 ], "tagConfidence": "reviewed", "answerGuide": "Use the opposite-side vector equality or parallelogram coordinate relation." }, { "id": "cie-2025-s-15-q07c", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the area of the triangle formed by the perpendicular bisector of AB and the axes.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the perpendicular bisector, its axis intercepts and the triangle area." }, { "id": "cie-2025-s-15-q08a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find a and b from the curve passing through a stationary point at (1,9).", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the point condition and set the derivative to zero at x=1, then solve simultaneously." }, { "id": "cie-2025-s-15-q08b", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates of the other stationary point.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using a and b, solve dy/dx=0 and substitute into the curve." }, { "id": "cie-2025-s-15-q08c", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find dx/dt when dy/dt is 6 and x=5.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 12, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use dy/dt=(dy/dx)(dx/dt) and evaluate the curve gradient at x=5." }, { "id": "cie-2025-s-15-q09a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the transformations from f to g.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Read the vertical stretch and translation from the relationship between the functions." }, { "id": "cie-2025-s-15-q09b", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch f and g on the same axes with their domains and ranges.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 14, 14 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the transformed cosine curves with the stated domains, ranges and turning-point gradients." }, { "id": "cie-2025-s-15-q09c", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate the stated composition involving the inverse of g and f.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 14, 15 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the inverse mapping, evaluate f at the given input, then substitute into g inverse." }, { "id": "cie-2025-s-15-q09d", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(d)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Explain why the composite function fg cannot be formed.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 14, 15 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says the required depth of knowledge was seldom evident; many confused domain with range.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 13 ], "tagConfidence": "reviewed", "answerGuide": "Compare the range of g with the domain of f and state the incompatibility." }, { "id": "cie-2025-s-15-q10a", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent to the circle at (2,8) in the required linear form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 16, 16 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the centre, determine the radius gradient and use the negative reciprocal for the tangent." }, { "id": "cie-2025-s-15-q10b", "paperId": "cie-2025-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that a line missing the circle satisfies the stated quadratic inequality in k.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_15.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_15.pdf", "qpPages": [ 16, 17 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the circle, obtain the quadratic intersection condition and require no real intersections." }, { "id": "cie-2025-s-21-q01", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "1: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-21-q02a", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "2(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2025-s-21-q02b", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2025-s-21-q02c", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(c): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says very few candidates found a correct answer to this exact integer-condition part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2025-s-21-q03a", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few completely correct sketches were seen for this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 14 ], "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-21-q03b", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(b): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2025-s-21-q03c", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "3(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2025-s-21-q04a", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-21-q04b", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4(b): Integrate between the stated bounds to find the area.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 6, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few correct area answers were seen, often because candidates used the wrong upper limit.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 15 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2025-s-21-q05a", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-21-q05b", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-21-q05c", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(c): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 8, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-21-q06a", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-21-q06b", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says errors in differentiation and substitution meant that fewer candidates gave a fully correct tangent.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 15 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the stated function, apply the given point or tangency condition, and substitute back to obtain the requested values." }, { "id": "cie-2025-s-21-q06c", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(c): Use the derivative sign to find the least integer boundary for an increasing function.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor the derivative, determine where it is positive, and choose the smallest integer n that makes the whole domain x > n lie in that interval." }, { "id": "cie-2025-s-21-q07a", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2025-s-21-q07b", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-21-q07c", "paperId": "cie-2025-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "integration", "summary": "Evaluate an exact trigonometric integral using an identity from the earlier parts.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_21.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_21.pdf", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report identifies this exact integral as a part with very few fully correct solutions.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 14, 16 ], "tagConfidence": "reviewed", "answerGuide": "Use the identity established earlier in the question to rewrite the integrand, then integrate and substitute the exact limits. Keep the algebraic working visible." }, { "id": "cie-2025-s-22-q01", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "1: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2025-s-22-q02a", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "2(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-22-q02b", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report notes that many candidates included the false critical value in this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2025-s-22-q03", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "3: Find the stationary coordinates and classify the point.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report describes mixed success in finding both stationary points, with arithmetic errors common.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 17 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-22-q04a", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2025-s-22-q04b", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "4(b): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2025-s-22-q04c", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4(c): Integrate between the stated bounds to find the area.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 6, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report notes common coefficient and sign errors in this exact part, with earlier errors limiting its available marks.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the bounding curves over the stated interval and evaluate the limits." }, { "id": "cie-2025-s-22-q05a", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-22-q05b", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5(b): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-22-q05c", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(c): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-22-q06a", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "6(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2025-s-22-q06b", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says simplification and substitution errors commonly prevented a correct tangent.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2025-s-22-q07a", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says few candidates continued with the double-angle identities to reach the requested cosine form.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-22-q07b", "paperId": "cie-2025-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the given trigonometric equation over the stated interval after using the cosine form established in the previous part.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_22.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_22.pdf", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says very few candidates gave fully correct solutions; many did not obtain both angles in the required interval.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 18 ], "tagConfidence": "reviewed", "answerGuide": "Use the result from part (a) to express the equation as a shifted cosine equation, then find both solutions in the interval." }, { "id": "cie-2025-s-23-q01", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the definite logarithmic integral and identify the required integer.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", 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4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Too many candidates included the false critical solution x<−2.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Solve across sign intervals or use critical values, then reject values inconsistent with the graph." }, { "id": "cie-2025-s-23-q03", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find both stationary points of the given rational curve.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Success at obtaining both stationary points was mixed because of arithmetic errors.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 19 ], "tagConfidence": "reviewed", "answerGuide": "Differentiate using the quotient rule, solve the derivative equation and substitute both x-values." }, { "id": "cie-2025-s-23-q04a", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "exponentials", "trigonometry", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show that the intersection x-coordinate satisfies the given logarithmic equation.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Equate the curve values and rearrange to the stated logarithmic form." }, { "id": "cie-2025-s-23-q04b", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use fixed-point iteration from 0.5 to find the intersection x-coordinate.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate in radian mode, recording values to six significant figures before rounding." }, { "id": "cie-2025-s-23-q04c", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "integration", "exponentials", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the area bounded by the curves and y-axis.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 6, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Common errors were the coefficient or sign of cos(3x); an incorrect 4(b) limited the available marks.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the curves to the intersection found in part (b), keeping radians." }, { "id": "cie-2025-s-23-q05a", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find unknown constants given two linear factors of a polynomial.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem at the two roots to form simultaneous equations." }, { "id": "cie-2025-s-23-q05b", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factorise the polynomial using the constants from part (a) and the stated factors.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Divide out the known factors and factorise the remaining quadratic." }, { "id": "cie-2025-s-23-q05c", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the least positive radian value satisfying the polynomial equation in cot(2θ).", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor the polynomial, find the required cotangent value, and choose the least positive angle." }, { "id": "cie-2025-s-23-q06a", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show that the curve has no point with the specified y-coordinate.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given y-value and show the resulting equation has no real x solution." }, { "id": "cie-2025-s-23-q06b", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at the given point on the logarithmic curve.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Many gained the first three marks, then made simplification or substitution errors in the tangent.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Use implicit differentiation, evaluate the gradient and write the line in the required form." }, { "id": "cie-2025-s-23-q07a", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express the trigonometric expression as R cos(2θ−α)+k.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "Few candidates continued with the double-angle identities to reach the requested cosine form.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Expand the compound angle, use double-angle identities and match coefficients." }, { "id": "cie-2025-s-23-q07b", "paperId": "cie-2025-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the stated trigonometric equation over the specified interval using part (a).", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_23.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_23.pdf", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Very few candidates produced fully correct answers; many found only one of the required angles.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 20 ], "tagConfidence": "reviewed", "answerGuide": "Rewrite as a cosine equation, find both angles in range and verify the interval." }, { "id": "cie-2025-s-25-q01", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the definite logarithmic integral and identify the required integer.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the logarithmic expression, apply limits and logarithm laws." }, { "id": "cie-2025-s-25-q02a", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Sketch the modulus curve and straight line on common axes.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw the V-shaped modulus graph and place the line with the correct relative gradient." }, { "id": "cie-2025-s-25-q02b", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve the inequality involving a modulus expression and linear function.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve across sign intervals or use critical values, then reject values inconsistent with the graph." }, { "id": "cie-2025-s-25-q03", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find both stationary points of the given rational curve.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the quotient rule, solve the derivative equation and substitute both x-values." }, { "id": "cie-2025-s-25-q04a", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "exponentials", "trigonometry", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show that the intersection x-coordinate satisfies the given logarithmic equation.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Equate the curve values and rearrange to the stated logarithmic form." }, { "id": "cie-2025-s-25-q04b", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use fixed-point iteration from 0.5 to find the intersection x-coordinate.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate in radian mode, recording values to six significant figures before rounding." }, { "id": "cie-2025-s-25-q04c", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "integration", "exponentials", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the area bounded by the curves and y-axis.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 6, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the difference between the curves to the intersection found in part (b), keeping radians." }, { "id": "cie-2025-s-25-q05a", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find unknown constants given two linear factors of a polynomial.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the factor theorem at the two roots to form simultaneous equations." }, { "id": "cie-2025-s-25-q05b", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factorise the polynomial using the constants from part (a) and the stated factors.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Divide out the known factors and factorise the remaining quadratic." }, { "id": "cie-2025-s-25-q05c", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the least positive radian value satisfying the polynomial equation in cot(2θ).", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor the polynomial, find the required cotangent value, and choose the least positive angle." }, { "id": "cie-2025-s-25-q06a", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show that the curve has no point with the specified y-coordinate.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given y-value and show the resulting equation has no real x solution." }, { "id": "cie-2025-s-25-q06b", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at the given point on the logarithmic curve.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use implicit differentiation, evaluate the gradient and write the line in the required form." }, { "id": "cie-2025-s-25-q07a", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express the trigonometric expression as R cos(2θ−α)+k.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Expand the compound angle, use double-angle identities and match coefficients." }, { "id": "cie-2025-s-25-q07b", "paperId": "cie-2025-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the stated trigonometric equation over the specified interval using part (a).", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_25.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_25.pdf", "qpPages": [ 12, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite as a cosine equation, find both angles in range and verify the interval." }, { "id": "cie-2025-s-31-q01a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-31-q01b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2025-s-31-q02", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "2: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2025-s-31-q03", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3: Solve the complex-number equation and express the result in cartesian form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the complex number in Cartesian form, use its conjugate as defined, and equate real and imaginary parts to solve." }, { "id": "cie-2025-s-31-q04", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "4: Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2025-s-31-q05", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5: Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-31-q06a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(a): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=9", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes that finding the argument of the product was more difficult than finding its modulus.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-31-q06b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(b): Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=9", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-31-q06c", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(c): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=9", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-31-q07a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says the majority did not attempt the required expansion and scored no marks.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 23 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-31-q07b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-31-q08a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "8(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-31-q08b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "8(b): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-31-q08c", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "8(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-31-q09a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "9(a): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2025-s-31-q09b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "9(b): Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 14, 15 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2025-s-31-q09c", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "9(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 14, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2025-s-31-q10a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-31-q10b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(b): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 16, 16 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says only a minority made progress, often because they did not separate the variables.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 24 ], "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-31-q11a", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(a): Differentiate the stated function to find the requested gradient or parameter.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=18", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 18, 18 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, solve where the first derivative is zero, find the corresponding coordinates, then use the second derivative or sign change to classify the point." }, { "id": "cie-2025-s-31-q11b", "paperId": "cie-2025-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "integration", "summary": "Set up a volume-of-revolution integral and evaluate it exactly.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_31.pdf#page=18", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_31.pdf", "qpPages": [ 18, 19 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report notes that many candidates left this part blank and that only a minority reached the correct integral method.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 24 ], "tagConfidence": "reviewed", "answerGuide": "Identify the radius and limits from the diagram, set up the volume-of-revolution integral, and evaluate it exactly." }, { "id": "cie-2025-s-32-q01", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-32-q02a", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "2(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-32-q02b", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "2(b): Identify the requested term in a binomial expansion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2025-s-32-q03", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "3: Represent the complex numbers or loci on an argand diagram.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-32-q04", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-32-q05", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5: Rewrite the expression in the required form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-32-q06a", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "6(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-32-q06b", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-32-q06c", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "6(c): Iterate from the stated starting value to approximate the root to the required accuracy.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the current estimate into the stated iteration or approximation rule, retain adequate precision, and report the requested accuracy." }, { "id": "cie-2025-s-32-q07a", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(a): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-32-q07b", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "7(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 10, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-32-q08", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8", "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8: Solve the trigonometric equation over the stated interval.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 12, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates did not recognise the integral form required in this whole question.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 27, 28 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-32-q09a", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-32-q09b", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-32-q09c", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(c): Use the position vectors to calculate the area of the quadrilateral.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 14, 15 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the required side lengths and perpendicular height from the position vectors, then use the appropriate quadrilateral area formula." }, { "id": "cie-2025-s-32-q10a", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Use the factor or remainder condition to determine the polynomial coefficients.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor or remainder theorem by substituting the stated root or divisor, then solve the resulting equations for the coefficients." }, { "id": "cie-2025-s-32-q10b", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(b): Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says only a few candidates obtained the required term correctly in the integration-by-parts work.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 28, 29 ], "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-32-q11a", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Use the geometric conditions to determine the requested equation or coordinates.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=18", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 18, 18 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report identifies differentiating cos²x as the main obstacle in this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 25, 29 ], "tagConfidence": "reviewed", "answerGuide": "Translate the geometric condition into an equation using gradients, distances or tangency as appropriate, then solve and verify the requested coordinates or parameters." }, { "id": "cie-2025-s-32-q11b", "paperId": "cie-2025-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Use a stated substitution to find the area under a trigonometric curve over a fixed interval.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_32.pdf#page=18", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_32.pdf", "qpPages": [ 18, 19 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s report lists repeated errors on this exact substituted integral: candidates used the wrong derivative for u, lost a coefficient, misapplied a double-angle identity, or changed the limits incorrectly.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 29 ], "tagConfidence": "reviewed", "answerGuide": "Set u = cos x, transform the integrand and limits, then integrate and evaluate the definite integral." }, { "id": "cie-2025-s-33-q01a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "graphs" ], "primaryTopicId": "graphs", "summary": "1(a): Sketch or interpret the stated graph using its key points and features.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Mark intercepts, asymptotes and interval restrictions first, then sketch the required branches or loci accurately." }, { "id": "cie-2025-s-33-q01b", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Solve the inequality by splitting at its critical values and checking the resulting intervals.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the points where the expressions change sign, solve in each interval, and combine the intervals satisfying the original inequality." }, { "id": "cie-2025-s-33-q02", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "2: Use logarithms or exponential laws to solve the stated equation or model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take logarithms or exponentiate both sides as appropriate, and apply any stated domain restriction." }, { "id": "cie-2025-s-33-q03", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3: Apply trigonometric identities to determine the requested value or solutions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-33-q04a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-33-q04b", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the given algebraic conditions to determine the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says few completely correct solutions were seen for this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 30, 31 ], "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-33-q05a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "5(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-33-q05b", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "5(b): Use the tangent gradient and point conditions to determine the unknown constants.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve to obtain the tangent gradient at the given point, then use the point-on-line and point-on-curve conditions to solve for the constants." }, { "id": "cie-2025-s-33-q06", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the complex numbers satisfying a real-valued quotient condition and a specified modulus. The answer must be written in Cartesian form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says very few solutions were completely correct; many used the wrong conjugate and did not remove the imaginary part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 31 ], "tagConfidence": "reviewed", "answerGuide": "Write z = x + iy, multiply by the conjugate of the denominator, set the imaginary part to zero, and use the modulus condition." }, { "id": "cie-2025-s-33-q07a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7(a): Rewrite the expression in the required form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-33-q07b", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "7(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 8, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-33-q07c", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "7(c): Identify the requested term in a binomial expansion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 8, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few correct responses were seen for the range of validity in this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 32 ], "tagConfidence": "reviewed", "answerGuide": "Write the general binomial term, choose the power that matches the requested term or coefficient, then simplify and state the result." }, { "id": "cie-2025-s-33-q08a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(a): Use a trigonometric identity to establish the stated value.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the equation to sine and cosine terms, use cos²x = 1 − sin²x to form a quadratic in sin x, and select the value consistent with the stated equation." }, { "id": "cie-2025-s-33-q08b", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(b): Solve the trigonometric equation over the stated interval.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 10, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange with the relevant trigonometric identity, solve for the principal angle, then include every solution in the stated interval." }, { "id": "cie-2025-s-33-q09a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 12, 12 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-33-q09b", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(b): Use a dot product to verify the stated perpendicular relationship.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 12, 12 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the relevant direction vectors and show their scalar product is zero." }, { "id": "cie-2025-s-33-q09c", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "9(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 12, 13 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-33-q10a", "paperId": "cie-2025-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 8, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "10(a): Solve the stated equation and retain the admissible solution.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_33.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_33.pdf", "qpPages": [ 14, 14 ], "msPages": [ 22, 22 ], "hardestPart": 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"studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-35-q09b", "paperId": "cie-2025-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "9(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_35.pdf#page=14", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_35.pdf", "qpPages": [ 14, 15 ], "msPages": [ 27, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-35-q10a", "paperId": "cie-2025-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(a): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_35.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_35.pdf", "qpPages": [ 16, 16 ], "msPages": [ 28, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-35-q10b", "paperId": "cie-2025-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use the vertex order of a rhombus and the given position vectors to find the fourth vertex.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_35.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_35.pdf", "qpPages": [ 16, 17 ], "msPages": [ 29, 29 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Cambridge’s examiner report says full marks were uncommon and this part was often omitted; a common error was using vector CD where DC was required.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 34, 37 ], "tagConfidence": "reviewed", "answerGuide": "Apply the parallelogram vector relation with the vertices in cyclic order, then solve for the fourth position vector." }, { "id": "cie-2025-s-35-q10c", "paperId": "cie-2025-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "10(c): Use the given vectors to find the requested geometric result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_35.pdf#page=16", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_35.pdf", "qpPages": [ 16, 17 ], "msPages": [ 30, 30 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report says very few candidates obtained the correct answer to this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 34, 37 ], "tagConfidence": "reviewed", "answerGuide": "Form the required displacement vectors and apply scalar products, magnitudes or vector products as the geometry requires." }, { "id": "cie-2025-s-35-q11", "paperId": "cie-2025-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11", "marks": 8, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "11: Solve the stated equation and retain the admissible solution.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_35.pdf#page=18", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_35.pdf", "qpPages": [ 18, 18 ], "msPages": [ 31, 31 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-41-q01", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-41-q02", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "2: Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2025-s-41-q03a", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2025-s-41-q03b", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 4, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2025-s-41-q04ai", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a)(i): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-41-q04aii", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(a)(ii): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says this exact acceleration part proved more challenging than the preceding power calculation.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 39 ], "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-41-q04b", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "4(b): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-41-q05a", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-41-q05b", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says this collision part was much more challenging, with fewer candidates earning full marks.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 38, 39 ], "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-41-q06a", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-41-q06b", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy to analyze connected particles moving on inclined planes with friction.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report calls the connected-particle energy work the most challenging question on the paper; this later part caused especially little progress.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 38, 39, 40 ], "tagConfidence": "reviewed", "answerGuide": "Apply work–energy to the connected system, including the work done by gravity and the energy lost to friction, then solve for the requested quantity." }, { "id": "cie-2025-s-41-q07a", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-41-q07b", "paperId": "cie-2025-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_41.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_41.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says analysis of the two particles was a struggle for most candidates; few completed the displacement comparison.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 40 ], "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q01a", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q01b", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says only a minority were successful when resolving the force component in this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 41 ], "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2025-s-42-q01c", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(c): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-42-q02", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q03", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3: Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this force-resolution question challenged a significant number of candidates.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2025-s-42-q04a", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Sketch the velocity-time graph across the stated stages of motion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot the velocity at each key time, join constant-acceleration stages with straight segments, and show the correct slope after any collision or release." }, { "id": "cie-2025-s-42-q04b", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q04c", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(c): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 6, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q05a", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "5(a): Apply newton’s second law to the connected particles to determine the requested tension or motion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a force equation for each connected particle, use the shared acceleration and equal tension, then solve the simultaneous equations." }, { "id": "cie-2025-s-42-q05b", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q05c", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(c): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q06a", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q06b", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 10, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates struggled with the required integration and limits in this exact displacement part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 42 ], "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-42-q07", "paperId": "cie-2025-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use an energy method to find the return speed of a particle projected up and back down a rough inclined plane.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_42.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_42.pdf", "qpPages": [ 12, 12 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report calls Question 7 the paper’s most challenging for most candidates; many ignored the instruction to use energy throughout.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 42, 43 ], "tagConfidence": "reviewed", "answerGuide": "Use work-energy for the outward journey to determine the distance and friction work, then account for both journeys to find the return speed." }, { "id": "cie-2025-s-43-q01a", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1(a): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-43-q01b", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "1(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2025-s-43-q02", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2: Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-43-q03a", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(a): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says many candidates struggled to find the later speed and form the required time equation.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 44, 45 ], "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2025-s-43-q03b", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3(b): Use the velocity-time information to find the requested displacement or speed.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the graph gradient for acceleration and the signed area for displacement; apply the stated time interval." }, { "id": "cie-2025-s-43-q04", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve the tensions in a three-block equilibrium system to determine an unknown mass and an angle.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says candidates struggled with the unfamiliar force system; many were unsure which forces acted on each block, and few formed both required equations.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 45 ], "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally and vertically at the junction, express the tensions through the hanging masses, and solve the resulting equations for m and α." }, { "id": "cie-2025-s-43-q05a", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-43-q05b", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 6, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-43-q06a", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-43-q06b", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(b): Apply the work-energy relation to find the requested speed or resistance.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 8, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the relevant conservation or impulse equation, keep signs consistent with the chosen direction, and solve for the unknown." }, { "id": "cie-2025-s-43-q07a", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(a): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-43-q07bi", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b)(i): Apply the motion data to determine the requested speed, distance or time.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report notes that candidates commonly omitted the constant of integration in this exact velocity part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 45, 46 ], "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-43-q07bii", "paperId": "cie-2025-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "7(b)(ii): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_43.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_43.pdf", "qpPages": [ 10, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q01", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "1: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q02a", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q02b", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Use the motion equations to calculate the requested distance or displacement.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Fewer than half of candidates earned full marks for this exact part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 47, 48 ], "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q03", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "3: Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q04a", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(a): Resolve the forces and find the magnitude and direction of their resultant.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve each force into horizontal and vertical components, add components, then calculate the resultant magnitude and direction." }, { "id": "cie-2025-s-45-q04b", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(b): Apply the force model to find the requested tension, reaction or acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 6, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says a considerable number of candidates did not know how to start this resultant-force part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 48 ], "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the relevant directions and apply Newton’s second law or equilibrium conditions before solving." }, { "id": "cie-2025-s-45-q05a", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "5(a): Use power, force and speed relations to calculate the requested motion quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv and resolve the other forces to obtain the requested speed, driving force or power." }, { "id": "cie-2025-s-45-q05b", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "5(b): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q06a", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(a): Differentiate the displacement or apply the motion equations to find the requested velocity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Select the constant-acceleration relation containing the known and requested quantities, substitute consistently, and check the direction and units." }, { "id": "cie-2025-s-45-q06b", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(b): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2025-s-45-q06c", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "6(c): Use the velocity relation to determine the requested acceleration.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 10, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate velocity with respect to time, then substitute the requested time or solve for the stated condition." }, { "id": "cie-2025-s-45-q07a", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(a): Resolve the forces in equilibrium to determine the requested reaction or force.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 12, 12 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the forces normal and parallel to the plane, use F = μR, and apply equilibrium or Newton’s second law at the limiting condition." }, { "id": "cie-2025-s-45-q07b", "paperId": "cie-2025-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use energy for two connected particles on rough inclined planes to find their speed after moving a stated distance.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_45.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_45.pdf", "qpPages": [ 12, 13 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says this part was challenging for most candidates, with many scoring few marks or not attempting it.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 47, 49 ], "tagConfidence": "reviewed", "answerGuide": "Account for the kinetic-energy change of both particles, gravitational potential-energy changes, and friction work, then solve for speed." }, { "id": "cie-2025-s-51-q01", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says a significant number of candidates were unable to tackle this whole normal-distribution question.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 50, 51 ], "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2025-s-51-q02a", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the forbidden cases." }, { "id": "cie-2025-s-51-q02b", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level 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"board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says some candidates could not tackle this exact selection-probability part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 50, 51 ], "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-51-q03a", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-51-q03b", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 13, 13 ], 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-51-q03d", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(d): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 4, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this part challenged many candidates and produced a significant number of blank responses.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 51 ], "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-51-q04a", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-51-q04b", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 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"CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Count the arrangements that meet the stated restriction on repeated letters or positions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count all arrangements with repeated-letter factorials, then impose the positional restriction by fixing or subtracting the 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appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-51-q06a", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-51-q06b", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 10, 10 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-51-q06c", "paperId": "cie-2025-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Calculate a conditional probability for a selection made without replacement.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_51.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_51.pdf", "qpPages": [ 10, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report records very few correct solutions for this conditional-probability part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 50, 53 ], "tagConfidence": "reviewed", "answerGuide": "Use the conditional probability definition and count the relevant outcomes without replacement; update the total after the first selection." }, { "id": "cie-2025-s-52-q01", "paperId": "cie-2025-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_52.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-52-q02", "paperId": "cie-2025-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2: Combine the event 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"questionPart": "3(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(b): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_52.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says the conditional probability was more challenging because of the replacement conditions.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 55 ], "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2025-s-52-q04a", "paperId": "cie-2025-s-52", "board": "CIE", "qualification": 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"data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Construct or read a cumulative-frequency graph to estimate the requested statistic.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_52.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Plot cumulative totals at class boundaries, draw a smooth increasing curve, and read the requested quantile at its cumulative frequency." }, { "id": "cie-2025-s-52-q05c", "paperId": "cie-2025-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/52", "variant": 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"cie-2025-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count five-letter selections from AMALGAMATE that contain at least one M and at least two As.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_52.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", "qpPages": [ 8, 9 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says the permutations-and-combinations question challenged many candidates; listing valid cases or using a diagram was more successful.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 56, 57 ], "tagConfidence": "reviewed", "answerGuide": "List the valid letter-count patterns first, count the available choices for each pattern, and add the cases." }, { "id": "cie-2025-s-52-q07a", "paperId": "cie-2025-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_52.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "The report says over 10 per cent of candidates did not attempt this part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-52-q07c", "paperId": "cie-2025-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_52.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_52.pdf", "qpPages": [ 10, 11 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "The report says over 15 per cent of candidates did not attempt this part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 57 ], "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-53-q01a", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "1(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-53-q01b", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge 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"cie-2025-s-53-q02a", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-53-q02b", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-53-q03a", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, 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"studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-53-q03c", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a repeated die-throw process produces a second 3 before the sixth throw.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=5", "msUrl": 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"CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-53-q05c", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2025-s-53-q06a", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-53-q06b", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2025-s-53-q06c", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(c): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 10, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2025-s-53-q07a", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 12, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-53-q07b", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(b): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 12, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-53-q07c", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 12, 13 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-53-q07d", "paperId": "cie-2025-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(d): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_53.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_53.pdf", "qpPages": [ 12, 13 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-55-q01a", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Construct the distribution table for the difference between two dice scores.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "List possible score differences and their probabilities, checking that they sum to one." }, { "id": "cie-2025-s-55-q01b", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Find E(X) and Var(X) from the distribution table.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use expectation and variance formulas with unrounded probabilities." }, { "id": "cie-2025-s-55-q02a", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Estimate how many of 150 trees fall below the height threshold under a normal model.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the threshold, find the lower-tail probability, multiply by 150 and round to an integer." }, { "id": "cie-2025-s-55-q02b", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the height threshold above which 25% of trees lie.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the 75th percentile z-value and transform back to the original height scale." }, { "id": "cie-2025-s-55-q03a", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "distributions", "binomial", "probability" ], "primaryTopicId": "distributions", "summary": "Approximate the probability that fewer than 27 of 80 pupils have music lessons.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal approximation to the binomial with continuity correction, then standardise." }, { "id": "cie-2025-s-55-q03b", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial", "probability" ], "primaryTopicId": "binomial", "summary": "Find the probability that no more than two of ten pupils have music lessons.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the exact binomial probabilities for zero, one and two, or the complementary tail." }, { "id": "cie-2025-s-55-q04a", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Draw the probability tree for repeated acceptance, rejection or retesting.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "A surprising number drew extra branches at each stage, producing an incorrect 27-path tree.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 62 ], "tagConfidence": "reviewed", "answerGuide": "Draw only valid branches and label outcomes and probabilities." }, { "id": "cie-2025-s-55-q04b", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability an applicant is accepted on the first, second or third test.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Candidates with extra tree branches often struggled to count all three acceptance routes; the report requires their exact sum.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 62 ], "tagConfidence": "reviewed", "answerGuide": "Add the probabilities of acceptance on each attempt, retaining the exact value." }, { "id": "cie-2025-s-55-q04c", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability an accepted applicant took at least two tests, given they were accepted.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 6, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use conditional probability with acceptance on tests two or three as numerator and total acceptance as denominator." }, { "id": "cie-2025-s-55-q04d", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability all three independent friends are rejected.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 6, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 6, "difficultyEvidence": "A common error was failing to cube the rejection probability for three applicants.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 63 ], "tagConfidence": "reviewed", "answerGuide": "Find one applicant's rejection probability and cube it." }, { "id": "cie-2025-s-55-q05a", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a back-to-back stem-and-leaf diagram for two teams' completion times.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Order and align the leaves, identify both teams, and give an unambiguous key with units." }, { "id": "cie-2025-s-55-q05b", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw box-and-whisker plots for both teams on a common scale.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Disappointingly few candidates produced two accurate plots with the required scale and labels.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 63 ], "tagConfidence": "reviewed", "answerGuide": "Find five-number summaries, then plot both diagrams with a labelled scale and units." }, { "id": "cie-2025-s-55-q05c", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare the teams' completion times using centre and consistency.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 8, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report calls this one of the least well-answered questions and requires contextual speed and consistency comparisons.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 63 ], "tagConfidence": "reviewed", "answerGuide": "Make one comparison about speed and one about consistency, both referring to time." }, { "id": "cie-2025-s-55-q06a", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability", "binomial" ], "primaryTopicId": "probability", "summary": "Find the probability a four-person team contains one man and three women.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use combinations or a without-replacement product and account for the four possible positions of the man." }, { "id": "cie-2025-s-55-q06b", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability", "binomial" ], "primaryTopicId": "probability", "summary": "Count ways to choose three distinguishable teams of four for three different towns without restrictions.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose teams successively from remaining members, accounting for the different towns." }, { 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 64 ], "tagConfidence": "reviewed", "answerGuide": "Enumerate the valid gender distributions across towns, count each distribution and add the outcomes." }, { "id": "cie-2025-s-55-q06d", "paperId": "cie-2025-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "Count arrangements of seven men with exactly two between George and Harry.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_55.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_55.pdf", "qpPages": [ 10, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose valid positions for the brothers, arrange them in either order, and permute the remaining men." }, { "id": "cie-2025-s-61-q01", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1: Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2025-s-61-q02a", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-61-q02b", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Explain why the central limit theorem is needed for the probability calculation.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The examiner report notes that many candidates misunderstood which assumptions to state and few gave the requested explanation.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 65, 66 ], "tagConfidence": "reviewed", "answerGuide": "State why the sampling distribution can be approximated by a normal distribution under the central limit theorem, using the sample-size conditions in the question." }, { "id": "cie-2025-s-61-q03a", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International 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"cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "3(b): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2025-s-61-q04a", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-61-q04b", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-61-q05ai", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a)(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-61-q05aii", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 6, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a)(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-61-q05b", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 6, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the distribution parameters, translate the event into the variable’s probability statement, and evaluate it using the stated model." }, { "id": "cie-2025-s-61-q06a", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-61-q06b", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the probability of a Type II error in a binomial hypothesis test.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report flags common tail-probability and parameter errors, with many candidates omitting this part.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 65, 66 ], "tagConfidence": "reviewed", "answerGuide": "A Type II error is failing to reject the null hypothesis when the alternative is true. 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"answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-61-q07a", "paperId": "cie-2025-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_61.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report says few candidates knew how 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_61.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says many candidates recognised integration but were unable to carry it out successfully.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 65, 66 ], "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-62-q01a", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], 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"9709/62", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "2: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-62-q03a", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "3(a): Rewrite the expression in the required form.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set up the relevant equation from the given information, solve it carefully, and state the result in the requested form." }, { "id": "cie-2025-s-62-q03b", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use equal probabilities at two consecutive values of a Poisson random variable to determine n.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says many candidates could form the equation but failed when simplifying the factorials and powers to find n.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 68 ], "tagConfidence": "reviewed", "answerGuide": "Write both Poisson probabilities, cancel their common exponential factor, simplify the factorial ratio, and solve for n." }, { "id": "cie-2025-s-62-q04", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4: Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "The report says this whole question was among the more demanding questions; candidates often confused a count with a proportion.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 68 ], "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-62-q05", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5", "marks": 4, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "5: Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "The report says few candidates stated the required assumption in this whole question.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 68 ], "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2025-s-62-q06ai", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a)(i): Use the stated probability distribution to calculate the requested quantity.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2025-s-62-q06aii", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a)(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-62-q06b", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 8, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2025-s-62-q07a", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "7(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-62-q07b", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(b): Use the density function to determine the requested parameter, moment or probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability one to find the constant or parameter, then integrate the density over the requested range or moment." }, { "id": "cie-2025-s-62-q07c", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "7(c): Use the data distribution to estimate the median or interquartile range.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=10", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 10, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the ordered data or cumulative-frequency positions to locate the median and quartiles, then subtract Q1 from Q3 for the IQR." }, { "id": "cie-2025-s-62-q08a", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "8(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2025-s-62-q08b", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "8(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-62-q08c", "paperId": "cie-2025-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "8(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_62.pdf#page=12", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_62.pdf", "qpPages": [ 12, 13 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2025-s-63-q01a", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-63-q01bi", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b)(i): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2025-s-63-q01bii", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(b)(ii): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=2", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2025-s-63-q02ai", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "2(a)(i): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2025-s-63-q02aii", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "2(a)(ii): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2025-s-63-q02b", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Use a normal model or approximation to calculate the requested probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=3", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the variable using its mean and standard deviation, use the normal table or calculator, and adjust continuity limits when approximating a discrete count." }, { "id": "cie-2025-s-63-q03ai", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(a)(i): Use the sample information to calculate the requested measure of spread.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-63-q03aii", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(a)(ii): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-63-q03b", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-63-q04a", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "4(a): Show the requested result using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-63-q04b", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-63-q05", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model the combined mass of n men with a normal distribution and find the largest n consistent with a bridge load-risk limit.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says many candidates made errors deriving the distribution for the total mass before solving the sample-size condition.", "difficultySourceUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_er.pdf", "sourceEvidencePages": [ 71 ], "tagConfidence": "reviewed", "answerGuide": "Use the critical normal value and the aggregate mean and variance, form the resulting equation in n, and choose the largest valid integer." }, { "id": "cie-2025-s-63-q06a", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "6(a): Show the requested result using the supplied mathematical 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using the supplied mathematical relationships.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=7", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Start from the supplied definitions or identities, show each algebraic step, and end with the required statement." }, { "id": "cie-2025-s-63-q07a", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "7(a): Complete the stated significance test and interpret its conclusion.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "State the null and alternative hypotheses, calculate the test statistic, compare it with the critical value or p-value at the stated level, and give a contextual conclusion." }, { "id": "cie-2025-s-63-q07b", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(b): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-63-q07c", "paperId": "cie-2025-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "7(c): Use conditional probability to calculate the requested event probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_63.pdf#page=8", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_63.pdf", "qpPages": [ 8, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P(A|B) = P(A∩B)/P(B), identify the conditioned event, and calculate both probabilities consistently." }, { "id": "cie-2025-s-65-q01a", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & 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"qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "2(a): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-65-q02b", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify which step in a confidence-interval calculation relies on the Central Limit Theorem.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=4", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Cambridge’s examiner report says many candidates found this part very challenging and often gave CLT conditions or a distribution statement without identifying the needed 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"https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-65-q03bii", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "3(b)(ii): Use the sample data or summary statistics to calculate the requested measure.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied data or coded totals in the appropriate statistical formula, retaining sufficient precision until the final answer." }, { "id": "cie-2025-s-65-q03c", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "3(c): Combine the event probabilities to calculate the requested result.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=5", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-65-q04a", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Translate the stated event into a cumulative or complementary probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Define the relevant events, use the appropriate addition, multiplication or complement rule, and simplify to the requested probability." }, { "id": "cie-2025-s-65-q04b", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b): Choose a suitable approximating distribution and calculate the stated probability.", "qpUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_qp_65.pdf#page=6", "msUrl": "https://cdn.qualifiedquest.com/papers/a-level/mathematics-9709/2025/9709_s25_ms_65.pdf", "qpPages": [ 6, 7 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check the approximation conditions, set the approximating distribution parameter, then evaluate the required tail probability." }, { "id": "cie-2025-s-65-q05", "paperId": "cie-2025-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", 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Sketch the reciprocal-square branches in the correct quadrants with the axes as asymptotes, then read the number of intersections.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20221011.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-rms-20230112.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma11-01-que-20221011.pdf", "markScheme": "wma11-01-rms-20230112.pdf", "examinerReport": "wma11-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "45ba3354e26c0be74024e97fffb3b049054a224bd7c6196286f1aa4aa15430fd", "markScheme": "b67a586a161f41a1aaea71c7077a23b0dfa86f386c30f6ccf21a0309f4d46ab3", "examinerReport": "6985c839dc618b7398155b1b2864075bee5492f19352b02654767ccfd3f03c2e" } }, { "id": "ial-2022-10-WMA11-01-q6aii", "paperId": "ial-2022-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 2, "topicIds": [ "functions", "graphs" ], "primaryTopicId": "functions", "summary": "6(a)(ii): Sketch a cubic and a reciprocal-square curve from their key features, then count their intersections.", "answerGuide": "Mark the cubic's turning-point shape and required intercepts. 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Apply the specified input transformation to x-values, reflect the requested graph, track transformed turning points, then express the final solution set with the correct inequalities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-que-20221011.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma11-01-rms-20230112.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma11-01-que-20221011.pdf", "markScheme": "wma11-01-rms-20230112.pdf", "examinerReport": "wma11-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "45ba3354e26c0be74024e97fffb3b049054a224bd7c6196286f1aa4aa15430fd", "markScheme": "b67a586a161f41a1aaea71c7077a23b0dfa86f386c30f6ccf21a0309f4d46ab3", "examinerReport": "6985c839dc618b7398155b1b2864075bee5492f19352b02654767ccfd3f03c2e" } }, { "id": "ial-2022-10-WMA11-01-q7cii", "paperId": "ial-2022-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2022, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)(ii)", "marks": 1, "topicIds": [ "functions", "graphs", "transformations" ], "primaryTopicId": "functions", "summary": "7(c)(ii): Read intervals and coordinates from a graph, solve a transformed-function equation, reflect and scale the graph, and state an inequality solution.", "answerGuide": "Use the graph's labelled points and intervals. 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Use the binomial theorem and simplify the first three terms in ascending powers; differentiate that expansion and substitute x=0.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q2b", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "binomial", "differentiation", "algebra" ], "primaryTopicId": "binomial", "summary": "2(b): Use a remainder condition to find a binomial parameter, expand through the first three powers, then find the curve's gradient at the origin.", "answerGuide": "Apply the remainder theorem to determine the parameter. Use the binomial theorem and simplify the first three terms in ascending powers; differentiate that expansion and substitute x=0.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q2c", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "binomial", "differentiation", "algebra" ], "primaryTopicId": "binomial", "summary": "2(c): Use a remainder condition to find a binomial parameter, expand through the first three powers, then find the curve's gradient at the origin.", "answerGuide": "Apply the remainder theorem to determine the parameter. Use the binomial theorem and simplify the first three terms in ascending powers; differentiate that expansion and substitute x=0.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q3ai", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 1, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "3(a)(i): Evaluate the first terms of a periodic cosine-squared sequence, then find an exact finite sum using its repeating cycle.", "answerGuide": "Substitute n=1, 2 and 3 into the sequence. Identify the period and sum per cycle, count full cycles and the remainder through n=50, then simplify the exact total.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q3aii", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 1, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "3(a)(ii): Evaluate the first terms of a periodic cosine-squared sequence, then find an exact finite sum using its repeating cycle.", "answerGuide": "Substitute n=1, 2 and 3 into the sequence. 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Rearrange the logarithmic equation for t in terms of w, then substitute w=15 and round as requested.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q4c", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "4(c): Use a logarithmic growth model and one known weight to determine its base, invert the model, then find the age for a second weight.", "answerGuide": "Substitute the known age and weight and combine logarithms to solve for the base. 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Integrate the second curve exactly, subtract its area from the estimated first area, divide by the rectangle area and convert to a percentage.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q6b", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "numerical-methods", "integration", "graphs" ], "primaryTopicId": "numerical-methods", "summary": "6(b): Estimate a definite integral with the trapezium rule, then use it to estimate the shaded fraction of a logo.", "answerGuide": "Use all five table ordinates with the correct width in the trapezium formula. 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Use the radius to write the circle equation; find the radius gradient and perpendicular tangent. Integrate the difference between the curve and line over the y-axis-bounded interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q9b", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry", "integration" ], "primaryTopicId": "differentiation", "summary": "9(b): Find the minimum point of a cubic, use it as the centre of a circle through a specified curve point, then find its tangent and a bounded exact area.", "answerGuide": "Differentiate and solve for the minimum turning point. 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Convert all terms in the equation to powers of 2, collect the exponents and isolate x in terms of a.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20221015.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20230112.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma12-01-que-20221015.pdf", "markScheme": "wma12-01-rms-20230112.pdf", "examinerReport": "wma12-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "da78976c74753e5887d1d3d6b5e1be14824d40094de6cdcc7fa09dfe073f888b", "markScheme": "f22d2ce29ecb01f900ed2efe7d4211265ce388444d87543a7ead8ed8fcc31863", "examinerReport": "720c07023b9f4fe3f3fa777f6a6569f0164a0201e7289124da7919a9192c160f" } }, { "id": "ial-2022-10-WMA12-01-q10ib", "paperId": "ial-2022-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2022, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(i)(b)", "marks": 2, "topicIds": [ "exponentials", "indices", "algebra" ], "primaryTopicId": "exponentials", "summary": "10(i)(b): Express two logarithms using a defined parameter and solve an exponential equation in terms of that parameter.", "answerGuide": "Apply logarithm power and quotient laws to rewrite each expression. 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Substitute g(−π) into f.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q2c", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "functions", "trigonometry", "transformations" ], "primaryTopicId": "functions", "summary": "2(c): Find the range and inverse of a restricted function, state the inverse domain and evaluate a function composition.", "answerGuide": "Use the restricted domain to determine the range. Swap x and y and solve for the inverse; its domain is the range of the original function. Substitute g(−π) into f.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=8", "qpPages": [ 6, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q3a", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "differentiation", "exponentials", "functions" ], "primaryTopicId": "differentiation", "summary": "3(a): Find the exact maximum point of an exponential-quadratic curve and state parameter values giving at least two distinct roots.", "answerGuide": "Differentiate by the product and chain rules, solve for the non-minimum stationary point and substitute for its y-coordinate. Use the curve's maximum and minimum values to state the horizontal-level range with at least two intersections.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q3b", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "differentiation", "exponentials", "functions" ], "primaryTopicId": "differentiation", "summary": "3(b): Find the exact maximum point of an exponential-quadratic curve and state parameter values giving at least two distinct roots.", "answerGuide": "Differentiate by the product and chain rules, solve for the non-minimum stationary point and substitute for its y-coordinate. Use the curve's maximum and minimum values to state the horizontal-level range with at least two intersections.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q4a", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "4(a): Rearrange a base-ten logarithmic equation to express x in terms of y, then find dy/dx.", "answerGuide": "Convert the logarithmic equation to exponential form and isolate x. Differentiate the relation in the convenient direction, invert dx/dy and substitute the expression for y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q4b", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Differentiate a logarithmic relation implicitly and express the derivative in terms of x.", "answerGuide": "Rearrange to express x as a function of y, differentiate dx/dy, then invert and substitute for y using the original relation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson notes that this part was discriminating; candidates needed to differentiate in the more convenient direction before converting to dy/dx.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" }, "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2022-10-WMA13-01-q5a", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "5(a): Use a profit model to verify an early loss, bracket a zero, derive an iteration, iterate to the requested terms and convert the break-even time to months.", "answerGuide": "Substitute t=1 and evaluate the model. Evaluate the signs at 6 and 7. Rearrange P=0 to the given fixed-point form, iterate from t0=6 through t6 to three decimals, then use the converged year value to find total months.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q5b", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "5(b): Use a profit model to verify an early loss, bracket a zero, derive an iteration, iterate to the requested terms and convert the break-even time to months.", "answerGuide": "Substitute t=1 and evaluate the model. Evaluate the signs at 6 and 7. Rearrange P=0 to the given fixed-point form, iterate from t0=6 through t6 to three decimals, then use the converged year value to find total months.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q5c", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "5(c): Use a profit model to verify an early loss, bracket a zero, derive an iteration, iterate to the requested terms and convert the break-even time to months.", "answerGuide": "Substitute t=1 and evaluate the model. Evaluate the signs at 6 and 7. Rearrange P=0 to the given fixed-point form, iterate from t0=6 through t6 to three decimals, then use the converged year value to find total months.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q5d", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "5(d): Use a profit model to verify an early loss, bracket a zero, derive an iteration, iterate to the requested terms and convert the break-even time to months.", "answerGuide": "Substitute t=1 and evaluate the model. Evaluate the signs at 6 and 7. Rearrange P=0 to the given fixed-point form, iterate from t0=6 through t6 to three decimals, then use the converged year value to find total months.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q5e", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials", "algebra" ], "primaryTopicId": "numerical-methods", "summary": "5(e): Use a profit model to verify an early loss, bracket a zero, derive an iteration, iterate to the requested terms and convert the break-even time to months.", "answerGuide": "Substitute t=1 and evaluate the model. Evaluate the signs at 6 and 7. Rearrange P=0 to the given fixed-point form, iterate from t0=6 through t6 to three decimals, then use the converged year value to find total months.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q6", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Show a derivative formula for a rational function of a trigonometric expression.", "answerGuide": "Apply the quotient rule, then use trigonometric identities to simplify the numerator and denominator to the requested form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=16", "msUrl": 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Read the minimum and its earliest positive x, adjust the y-value for the vertical transformation, then reflect the minimum to a maximum and halve its x-coordinate for f(2x).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-que-20221020.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma13-01-rms-20230112.pdf#page=16", "qpPages": [ 24, 27 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma13-01-que-20221020.pdf", "markScheme": "wma13-01-rms-20230112.pdf", "examinerReport": "wma13-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "d394fd48e94e65e88f1f0177a6fe0dd13b9fcfa8a2a2471c7c1809261be12197", "markScheme": "79bdd5540552a60a64a74b68e8de92ddd0b7f41640fc8a915d1ca4880f1b7a9b", "examinerReport": "3fb2f0b71d2146951bfa16181f27f3f9c4599b8b0921133403be7e8adcd800a6" } }, { "id": "ial-2022-10-WMA13-01-q8d", "paperId": "ial-2022-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2022, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 1, "topicIds": [ "trigonometry", "functions", "transformations" ], "primaryTopicId": "trigonometry", "summary": "Identify the first maximum point after a function transformation.", "answerGuide": "Use the earlier minimum location; 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Use the parameter interval and monotonicity/endpoints to obtain the exact domain constant and range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q6b", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "6(b): Find a tangent to a parametric curve at its x-axis crossing, then determine the domain endpoint and range of the corresponding function.", "answerGuide": "Find the parameter value at the crossing, calculate dy/dx as (dy/dt)/(dx/dt), and use point-gradient form. Use the parameter interval and monotonicity/endpoints to obtain the exact domain constant and range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q6c", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "6(c): Find a tangent to a parametric curve at its x-axis crossing, then determine the domain endpoint and range of the corresponding function.", "answerGuide": "Find the parameter value at the crossing, calculate dy/dx as (dy/dt)/(dx/dt), and use point-gradient form. Use the parameter interval and monotonicity/endpoints to obtain the exact domain constant and range.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q7i", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 7, "topicIds": [ "integration", "exponentials", "trigonometry" ], "primaryTopicId": "integration", "summary": "7(i): Evaluate an exponential rational integral by substitution and partial fractions, then determine constants in an exponential-cosine antiderivative.", "answerGuide": "Substitute u=e^x−3, convert dx and the integrand to rational form, decompose and integrate logarithmically. 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"43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q7ii", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "integration", "exponentials", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find constants in an antiderivative involving an exponential and cosine factor.", "answerGuide": "Integrate by parts twice, collect the resulting copy of the original integral, and solve for the unknown coefficients.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=14", "qpPages": [ 18, 21 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson notes that the repeated-integration method in this part was not handled well by many candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" }, "sourceEvidencePages": { "examinerReport": [ 5, 5 ] } }, { "id": "ial-2022-10-WMA14-01-q8", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8", "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Complete a proof by contradiction that two positive integers cannot satisfy the given quadratic Diophantine equation.", "answerGuide": "Factor the expression into integer factors, list the possible factor pairs of 25, solve the resulting simultaneous equations, and show each case contradicts the required integer condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=22", "msUrl": 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"session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9", "marks": 5, "topicIds": [ "vectors", "proof" ], "primaryTopicId": "vectors", "summary": "Prove that two three-dimensional vector lines are skew.", "answerGuide": "Show their direction vectors are not parallel. Equate their coordinate components to test for an intersection, derive inconsistent parameter values and conclude the lines are non-parallel and do not intersect.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=18", "qpPages": [ 24, 25 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q10a", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "differentiation", "integration", "algebra" ], "primaryTopicId": "differentiation", "summary": "Model the radius of a shrinking sphere from an inverse-square rate and initial conditions.", "answerGuide": "Write dr/dt as a negative constant divided by r squared, separate variables, integrate, and use both radius-time conditions to determine the relation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that many candidates struggled to start by relating the stated rate to the radius change over time.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" }, "sourceEvidencePages": { "examinerReport": [ 3, 6 ] } }, { "id": "ial-2022-10-WMA14-01-q10b", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "differentiation", "integration", "algebra" ], "primaryTopicId": "differentiation", "summary": "10(b): Model an ice sphere whose radius decreases at a rate inversely proportional to its squared radius; find its radius-time equation, melting time and graph.", "answerGuide": "Write dr/dt=−k/r², separate and integrate, then use both initial and later radius data to determine the constants. Set r=0 for the melt time and sketch the decreasing concave curve with correct endpoints.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q10c", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 1, "topicIds": [ "differentiation", "integration", "algebra" ], "primaryTopicId": "differentiation", "summary": "10(c): Model an ice sphere whose radius decreases at a rate inversely proportional to its squared radius; find its radius-time equation, melting time and graph.", "answerGuide": "Write dr/dt=−k/r², separate and integrate, then use both initial and later radius data to determine the constants. Set r=0 for the melt time and sketch the decreasing concave curve with correct endpoints.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=19", "qpPages": [ 26, 29 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q11a", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "11(a): Implicitly differentiate a closed-curve equation and find how far its southernmost point lies from the origin.", "answerGuide": "Differentiate both sides with respect to x using the chain and product rules and isolate dy/dx. For the southern extreme set dy/dx=0, solve with the curve equation and round the negative y-coordinate's distance to the nearest 0.1 km.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-que-20221025.pdf#page=30", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma14-01-rms-20230112.pdf#page=21", "qpPages": [ 30, 32 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wma14-01-que-20221025.pdf", "markScheme": "wma14-01-rms-20230112.pdf", "examinerReport": "wma14-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "780689e6751ab4bdd05999b2bd4956b601bef3990d7e6befd290e1cad1788215", "markScheme": "37172ceb8712b36b7b52bc070f15c9feae9f43f2866de5681814c75a231dca37", "examinerReport": "43530a2f6d76fa4eaec9bc7a1dcf894a4a06bb29e8c34b8022e3275e6ffe66f6" } }, { "id": "ial-2022-10-WMA14-01-q11b", "paperId": "ial-2022-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2022, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "11(b): Implicitly differentiate a closed-curve equation and find how far its southernmost point lies from the origin.", "answerGuide": "Differentiate both sides with respect to x using the chain and product rules and isolate dy/dx. 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Calculate the change in momentum of truck S and state impulse units.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q1b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "1(b): Use conservation of momentum for two railway trucks that join in a collision, then find the impulse on one truck.", "answerGuide": "Apply momentum conservation to the combined trucks to obtain their common speed. 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For the stated particle mass, apply vertical force balance and moments about a support to determine the tension at C.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q3a", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(a): Resolve forces on a particle held on a rough inclined plane by a horizontal force; first verify zero friction and then find the minimum force.", "answerGuide": "Resolve weight and horizontal force parallel and perpendicular to the plane to verify the no-friction value. For the minimum, use limiting friction with the correct direction, relate the normal reaction to the forces and solve the equilibrium equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q3b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "3(b): Resolve forces on a particle held on a rough inclined plane by a horizontal force; first verify zero friction and then find the minimum force.", "answerGuide": "Resolve weight and horizontal force parallel and perpendicular to the plane to verify the no-friction value. For the minimum, use limiting friction with the correct direction, relate the normal reaction to the forces and solve the equilibrium equations.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q4a", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(a): Use a lift's cable tension to find its upward acceleration, then calculate a passenger's mass from the normal force.", "answerGuide": "Apply Newton's second law to the lift and its mass to determine acceleration. For Alan, use the floor's normal reaction and weight in the accelerating frame equation to solve for mass.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q4b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "4(b): Use a lift's cable tension to find its upward acceleration, then calculate a passenger's mass from the normal force.", "answerGuide": "Apply Newton's second law to the lift and its mass to determine acceleration. For Alan, use the floor's normal reaction and weight in the accelerating frame equation to solve for mass.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q5a", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Find how far a vertically projected ball travels while its speed is below a stated threshold.", "answerGuide": "Use the speed threshold to find the height below the apex with constant-acceleration equations, then double that height by symmetry.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes Question 5 as the paper's next most demanding question and discusses the speed-and-height setup.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" }, "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2022-10-WME01-01-q5b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "Find the time for which the ball's downward speed exceeds its launch speed.", "answerGuide": "Locate the descending instant at which the ball again has its launch-speed magnitude, then use constant acceleration from there to the ground.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson discusses this follow-on time interval in its account of the challenging vertical-motion question.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" }, "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2022-10-WME01-01-q5c", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "kinematics", "graphs" ], "primaryTopicId": "kinematics", "summary": "5(c): Analyze vertical motion of a projected ball to find a speed-threshold distance and duration, then sketch its speed-time graph.", "answerGuide": "Use constant-acceleration equations to find the heights where speed is 14.7 m/s and combine the relevant distances. Find when the descending speed reaches 29.4 m/s and subtract from impact time; sketch speed as a V-shaped piecewise linear graph with the correct intercepts and apex time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q6a", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics", "vectors", "kinematics" ], "primaryTopicId": "mechanics", "summary": "6(a): Use the direction of a particle's acceleration to prove a relation between two force components, then find its speed after four seconds.", "answerGuide": "Combine the applied force vectors and use the stated motion direction to set the acceleration component ratio. With p=5, find q, calculate the resultant force and acceleration, then use v=at from rest.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q6b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics", "vectors", "kinematics" ], "primaryTopicId": "mechanics", "summary": "6(b): Use the direction of a particle's acceleration to prove a relation between two force components, then find its speed after four seconds.", "answerGuide": "Combine the applied force vectors and use the stated motion direction to set the acceleration component ratio. With p=5, find q, calculate the resultant force and acceleration, then use v=at from rest.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q7a", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "7(a): Analyze two equal masses connected over a pulley on a rough table; find the tension factor, travel time and behavior when friction is at least one.", "answerGuide": "Write Newton's second-law equations for each mass and use the friction law to find the common acceleration and k. Apply s=1/2 at² for the distance d. Compare the driving weight with limiting friction when μ≥1 to determine whether motion starts.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q7b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the time for two connected particles to reach a pulley when resistance acts.", "answerGuide": "Combine the two equations of motion to obtain the shared acceleration, then use the given travel distance in s equals one half at squared.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson notes that candidates often found this connected-particle time calculation more difficult.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" }, "sourceEvidencePages": { "examinerReport": [ 5, 5 ] } }, { "id": "ial-2022-10-WME01-01-q7c", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "7(c): Analyze two equal masses connected over a pulley on a rough table; find the tension factor, travel time and behavior when friction is at least one.", "answerGuide": "Write Newton's second-law equations for each mass and use the friction law to find the common acceleration and k. Apply s=1/2 at² for the distance d. Compare the driving weight with limiting friction when μ≥1 to determine whether motion starts.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q8a", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "vectors", "kinematics", "trigonometry" ], "primaryTopicId": "vectors", "summary": "8(a): Use constant vector velocities and relative position to find a ship's speed, the second ship's velocity, and a bearing at a specified separation.", "answerGuide": "Find A's speed from the magnitude of its velocity. Differentiate the relative displacement with respect to time and compare components to obtain p and q. Set the separation magnitude to 15, solve for time, determine the relative position and convert its direction into a bearing.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME01-01-q8b", "paperId": "ial-2022-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2022, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 7, "topicIds": [ "vectors", "kinematics", "trigonometry" ], "primaryTopicId": "vectors", "summary": "8(b): Use constant vector velocities and relative position to find a ship's speed, the second ship's velocity, and a bearing at a specified separation.", "answerGuide": "Find A's speed from the magnitude of its velocity. Differentiate the relative displacement with respect to time and compare components to obtain p and q. 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Differentiate the relative displacement with respect to time and compare components to obtain p and q. Set the separation magnitude to 15, solve for time, determine the relative position and convert its direction into a bearing.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-que-20221018.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme01-01-rms-20230112.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme01-01-que-20221018.pdf", "markScheme": "wme01-01-rms-20230112.pdf", "examinerReport": "wme01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "475baf1109ab2f0a1ca9422c1cdb71fcadb5d5f380e1e52d468411898f799890", "markScheme": "2d5b4693042b08d32cbfe6a73221f830e484f2d5924d20000c1666bd9872012a", "examinerReport": "8a7916075d73b17c1dad6ba886330fb3c757ffa884edbc66bcf59f7b4849e300" } }, { "id": "ial-2022-10-WME02-01-q1a", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "1(a): Find the centre of mass of three point masses, then use its position on a given line to determine a parameter.", "answerGuide": "Use the weighted coordinate averages to obtain the centre of mass. Substitute the coordinates into the line equation and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q1b", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "1(b): Find the centre of mass of three point masses, then use its position on a given line to determine a parameter.", "answerGuide": "Use the weighted coordinate averages to obtain the centre of mass. Substitute the coordinates into the line equation and solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q2", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find a car's instantaneous acceleration from its engine power, slope, mass and constant resistance.", "answerGuide": "Convert the engine power to a driving force using P=Fv. Resolve the car's weight along the slope, subtract the resistance and apply Newton's second law.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q3", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "energy-momentum", "vectors" ], "primaryTopicId": "energy-momentum", "summary": "Find an impulse parameter from a vector velocity change and a specified final speed.", "answerGuide": "Apply impulse-momentum componentwise, write the final velocity components in terms of the parameter, set their squared magnitudes to the stated speed squared, and solve.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports a common misconception in this question: treating a speed as if it were a vector.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" }, "sourceEvidencePages": { "examinerReport": [ 3, 3 ] } }, { "id": "ial-2022-10-WME02-01-q4a", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "vectors", "kinematics", "integration" ], "primaryTopicId": "vectors", "summary": "4(a): Find the acceleration when a particle's velocity is parallel to a vector, then integrate velocity to obtain its position at time T.", "answerGuide": "Use the component ratio to solve for λ, differentiate velocity to get acceleration and substitute t=λ. Integrate each velocity component, use the position at t=1 to find the constants and evaluate the vector at T.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q4b", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "vectors", "kinematics", "integration" ], "primaryTopicId": "vectors", "summary": "4(b): Find the acceleration when a particle's velocity is parallel to a vector, then integrate velocity to obtain its position at time T.", "answerGuide": "Use the component ratio to solve for λ, differentiate velocity to get acceleration and substitute t=λ. Integrate each velocity component, use the position at t=1 to find the constants and evaluate the vector at T.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q5a", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "5(a): Use rod-block geometry and moments to find a contact reaction, then resolve the hinge force into magnitude and direction.", "answerGuide": "Apply the right-triangle geometry to show AD. Take moments about A to determine the normal reaction at C. Resolve horizontal and vertical forces to find the hinge reaction components, then calculate k and tan α.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q5b", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "5(b): Use rod-block geometry and moments to find a contact reaction, then resolve the hinge force into magnitude and direction.", "answerGuide": "Apply the right-triangle geometry to show AD. Take moments about A to determine the normal reaction at C. Resolve horizontal and vertical forces to find the hinge reaction components, then calculate k and tan α.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q5c", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 8, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Find the exact magnitude and direction of the resultant hinge force on a uniform rod.", "answerGuide": "Resolve the hinge-force components using equilibrium and earlier geometry, combine components for the magnitude, and use their ratio to determine direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson notes that many candidates had difficulty finding and combining the required force components in this part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" }, "sourceEvidencePages": { "examinerReport": [ 5, 5 ] } }, { "id": "ial-2022-10-WME02-01-q6a", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics", "geometry", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "6(a): Find the centre of mass of a composite lamina, then determine its shape parameter from the equilibrium orientation when suspended.", "answerGuide": "Combine the centres of the two equal rectangles to locate the composite centre of mass relative to PV. Use the line from suspension point P to the centre of mass as the downward vertical and apply the given tangent ratio to solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q6b", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics", "geometry", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "6(b): Find the centre of mass of a composite lamina, then determine its shape parameter from the equilibrium orientation when suspended.", "answerGuide": "Combine the centres of the two equal rectangles to locate the composite centre of mass relative to PV. Use the line from suspension point P to the centre of mass as the downward vertical and apply the given tangent ratio to solve for k.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q7a", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "energy-momentum", "mechanics", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "7(a): Apply momentum and restitution to a direct collision, then analyze a rebound from a wall and the time until a second collision.", "answerGuide": "Use momentum conservation and the coefficient of restitution to solve for B's post-collision speed and then A's speed. Apply restitution at the wall, use relative velocities and the distance d to determine the interval until the next collision.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q7b", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "energy-momentum", "mechanics", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the time between successive collisions after a particle rebounds between surfaces.", "answerGuide": "Apply the coefficient of restitution at each impact, determine the post-impact speeds, and add the travel times using the relevant separation and relative speed.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson identifies difficulty choosing a strategy for the rebound-time calculation and carrying the successive impact speeds through it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" }, "sourceEvidencePages": { "examinerReport": [ 5, 5 ] } }, { "id": "ial-2022-10-WME02-01-q8a", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics", "kinematics", "trigonometry" ], "primaryTopicId": "energy-momentum", "summary": "8(a): Find work against friction on a ramp, use work-energy to determine the initial speed, then analyze the projectile's flight and impact direction.", "answerGuide": "Calculate friction from the normal reaction and distance. Apply work-energy between A and B to find U. Resolve the velocity at B and use vertical motion to find flight time; then combine horizontal and vertical velocity components before impact to find θ.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q8b", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics", "kinematics", "trigonometry" ], "primaryTopicId": "energy-momentum", "summary": "8(b): Find work against friction on a ramp, use work-energy to determine the initial speed, then analyze the projectile's flight and impact direction.", "answerGuide": "Calculate friction from the normal reaction and distance. Apply work-energy between A and B to find U. Resolve the velocity at B and use vertical motion to find flight time; then combine horizontal and vertical velocity components before impact to find θ.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q8c", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics", "kinematics", "trigonometry" ], "primaryTopicId": "energy-momentum", "summary": "8(c): Find work against friction on a ramp, use work-energy to determine the initial speed, then analyze the projectile's flight and impact direction.", "answerGuide": "Calculate friction from the normal reaction and distance. Apply work-energy between A and B to find U. Resolve the velocity at B and use vertical motion to find flight time; then combine horizontal and vertical velocity components before impact to find θ.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WME02-01-q8d", "paperId": "ial-2022-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2022, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics", "kinematics", "trigonometry" ], "primaryTopicId": "energy-momentum", "summary": "8(d): Find work against friction on a ramp, use work-energy to determine the initial speed, then analyze the projectile's flight and impact direction.", "answerGuide": "Calculate friction from the normal reaction and distance. Apply work-energy between A and B to find U. Resolve the velocity at B and use vertical motion to find flight time; then combine horizontal and vertical velocity components before impact to find θ.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20221028.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20230112.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wme02-01-que-20221028.pdf", "markScheme": "wme02-01-rms-20230112.pdf", "examinerReport": "wme02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "9b15db46f8cdc8032a772d195ba31c64ae5bba6e2aa6a43cc5762b267e8f3b91", "markScheme": "df3e9280b7463b32dfbfabdb1b06729a9861e0dabc26a9982ac536284711432c", "examinerReport": "30f1b7dbd39e60f43b4bb0cbfdf6f22d6304733bc9b95a233a3b1581ba59c0f8" } }, { "id": "ial-2022-10-WST01-01-q1a", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "1(a): Use a grouped histogram to calculate bar areas and heights, interpolate quartiles and use them to describe skewness.", "answerGuide": "Scale bar areas by class width and frequency. Interpolate Q2 and Q3 from the cumulative frequencies, calculate the interquartile range, then substitute the quartiles into the stated skewness measure and interpret its sign.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q1b", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "1(b): Use a grouped histogram to calculate bar areas and heights, interpolate quartiles and use them to describe skewness.", "answerGuide": "Scale bar areas by class width and frequency. Interpolate Q2 and Q3 from the cumulative frequencies, calculate the interquartile range, then substitute the quartiles into the stated skewness measure and interpret its sign.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q1ci", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(i)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "1(c)(i): Use a grouped histogram to calculate bar areas and heights, interpolate quartiles and use them to describe skewness.", "answerGuide": "Scale bar areas by class width and frequency. Interpolate Q2 and Q3 from the cumulative frequencies, calculate the interquartile range, then substitute the quartiles into the stated skewness measure and interpret its sign.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q1cii", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)(ii)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "1(c)(ii): Use a grouped histogram to calculate bar areas and heights, interpolate quartiles and use them to describe skewness.", "answerGuide": "Scale bar areas by class width and frequency. Interpolate Q2 and Q3 from the cumulative frequencies, calculate the interquartile range, then substitute the quartiles into the stated skewness measure and interpret its sign.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q1d", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "1(d): Use a grouped histogram to calculate bar areas and heights, interpolate quartiles and use them to describe skewness.", "answerGuide": "Scale bar areas by class width and frequency. Interpolate Q2 and Q3 from the cumulative frequencies, calculate the interquartile range, then substitute the quartiles into the stated skewness measure and interpret its sign.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q2a", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "2(a): Calculate summary products and correlation for film costs and sales, derive the regression line, make a prediction and solve an inequality for cost.", "answerGuide": "Use the summary-statistic formulae for S_tt and S_ct, then calculate r and interpret its direction. Find the regression gradient and intercept, predict sales at c=90, and solve the model inequality t<0.8c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q2b", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "2(b): Calculate summary products and correlation for film costs and sales, derive the regression line, make a prediction and solve an inequality for cost.", "answerGuide": "Use the summary-statistic formulae for S_tt and S_ct, then calculate r and interpret its direction. Find the regression gradient and intercept, predict sales at c=90, and solve the model inequality t<0.8c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q2c", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "2(c): Calculate summary products and correlation for film costs and sales, derive the regression line, make a prediction and solve an inequality for cost.", "answerGuide": "Use the summary-statistic formulae for S_tt and S_ct, then calculate r and interpret its direction. Find the regression gradient and intercept, predict sales at c=90, and solve the model inequality t<0.8c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q2d", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "2(d): Calculate summary products and correlation for film costs and sales, derive the regression line, make a prediction and solve an inequality for cost.", "answerGuide": "Use the summary-statistic formulae for S_tt and S_ct, then calculate r and interpret its direction. Find the regression gradient and intercept, predict sales at c=90, and solve the model inequality t<0.8c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q2e", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "2(e): Calculate summary products and correlation for film costs and sales, derive the regression line, make a prediction and solve an inequality for cost.", "answerGuide": "Use the summary-statistic formulae for S_tt and S_ct, then calculate r and interpret its direction. Find the regression gradient and intercept, predict sales at c=90, and solve the model inequality t<0.8c.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q2f", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(f)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Use a regression model to find when sales fall below a chosen fraction of production cost.", "answerGuide": "Substitute the regression equation into the sales-below-cost inequality, collect the cost terms, and solve while preserving the inequality direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson's question-specific feedback identifies this inequality interpretation as challenging; its general overview has an inconsistent part label.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" }, "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2022-10-WST01-01-q3a", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(a): Reverse a coding rule to find the sample mean and standard deviation of body lengths, then update the statistics after adding one squirrel.", "answerGuide": "Transform the coded mean and standard deviation back to b using b=21+2x. Use the new total coded sum to find the ninth length, update the sum of squares and calculate the nine-value standard deviation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q3b", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(b): Reverse a coding rule to find the sample mean and standard deviation of body lengths, then update the statistics after adding one squirrel.", "answerGuide": "Transform the coded mean and standard deviation back to b using b=21+2x. Use the new total coded sum to find the ninth length, update the sum of squares and calculate the nine-value standard deviation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q3ci", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)(i)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(c)(i): Reverse a coding rule to find the sample mean and standard deviation of body lengths, then update the statistics after adding one squirrel.", "answerGuide": "Transform the coded mean and standard deviation back to b using b=21+2x. Use the new total coded sum to find the ninth length, update the sum of squares and calculate the nine-value standard deviation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q3cii", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)(ii)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "3(c)(ii): Reverse a coding rule to find the sample mean and standard deviation of body lengths, then update the statistics after adding one squirrel.", "answerGuide": "Transform the coded mean and standard deviation back to b using b=21+2x. Use the new total coded sum to find the ninth length, update the sum of squares and calculate the nine-value standard deviation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q4", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Recover the probabilities from a discrete cumulative distribution and calculate the expected value.", "answerGuide": "Evaluate the cumulative function at each support value, take successive differences to obtain point probabilities, then sum wP(W=w).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q5a", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(a): Use a normal model to find a class probability, determine a threshold from a jumbo proportion and solve a conditional percentile for tiny fruit.", "answerGuide": "Standardize the large-fruit interval to find the probability for (a). For (b), equate the upper-tail probability to 0.35 and invert the normal distribution. For (c), express the conditional probability among tiny fruit, solve for the required cumulative probability and find y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q5b", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(b): Use a normal model to find a class probability, determine a threshold from a jumbo proportion and solve a conditional percentile for tiny fruit.", "answerGuide": "Standardize the large-fruit interval to find the probability for (a). For (b), equate the upper-tail probability to 0.35 and invert the normal distribution. For (c), express the conditional probability among tiny fruit, solve for the required cumulative probability and find y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q5c", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find a weight threshold exceeded by three quarters of a conditional normal distribution.", "answerGuide": "Express the required probability as a conditional ratio, then standardize the normal variable and solve for the threshold using the relevant normal quantile.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies the conditional-probability interpretation in this part as a challenging step.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" }, "sourceEvidencePages": { "examinerReport": [ 5, 5 ] } }, { "id": "ial-2022-10-WST01-01-q6ai", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a)(i): Read marginal and conditional probabilities from a four-event Venn diagram, solve for two region probabilities and determine a final unknown region.", "answerGuide": "Sum the correct Venn regions for P(A) and form conditional probabilities with the appropriate event denominator. Use the given conditional and complement probabilities plus total probability to solve for q and r, then apply the union condition to find s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q6aii", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a)(ii): Read marginal and conditional probabilities from a four-event Venn diagram, solve for two region probabilities and determine a final unknown region.", "answerGuide": "Sum the correct Venn regions for P(A) and form conditional probabilities with the appropriate event denominator. Use the given conditional and complement probabilities plus total probability to solve for q and r, then apply the union condition to find s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q6aiii", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(iii)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(a)(iii): Read marginal and conditional probabilities from a four-event Venn diagram, solve for two region probabilities and determine a final unknown region.", "answerGuide": "Sum the correct Venn regions for P(A) and form conditional probabilities with the appropriate event denominator. Use the given conditional and complement probabilities plus total probability to solve for q and r, then apply the union condition to find s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q6b", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(b): Read marginal and conditional probabilities from a four-event Venn diagram, solve for two region probabilities and determine a final unknown region.", "answerGuide": "Sum the correct Venn regions for P(A) and form conditional probabilities with the appropriate event denominator. Use the given conditional and complement probabilities plus total probability to solve for q and r, then apply the union condition to find s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q6c", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "6(c): Read marginal and conditional probabilities from a four-event Venn diagram, solve for two region probabilities and determine a final unknown region.", "answerGuide": "Sum the correct Venn regions for P(A) and form conditional probabilities with the appropriate event denominator. Use the given conditional and complement probabilities plus total probability to solve for q and r, then apply the union condition to find s.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q7a", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "probability", "distributions", "data-analysis" ], "primaryTopicId": "probability", "summary": "7(a): Use a discrete probability model to find a conditional probability, moments and transformed variance, compare two independent selections and form a product distribution.", "answerGuide": "Renormalize the probabilities after excluding 5. Calculate E(X²) and Var(X), use Var(a+bX)=b²Var(X), sum joint probabilities for Bruno's number exceeding Charlie's, then group ordered pairs by product value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q7b", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "probability", "distributions", "data-analysis" ], "primaryTopicId": "probability", "summary": "7(b): Use a discrete probability model to find a conditional probability, moments and transformed variance, compare two independent selections and form a product distribution.", "answerGuide": "Renormalize the probabilities after excluding 5. Calculate E(X²) and Var(X), use Var(a+bX)=b²Var(X), sum joint probabilities for Bruno's number exceeding Charlie's, then group ordered pairs by product value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q7c", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability", "distributions", "data-analysis" ], "primaryTopicId": "probability", "summary": "7(c): Use a discrete probability model to find a conditional probability, moments and transformed variance, compare two independent selections and form a product distribution.", "answerGuide": "Renormalize the probabilities after excluding 5. Calculate E(X²) and Var(X), use Var(a+bX)=b²Var(X), sum joint probabilities for Bruno's number exceeding Charlie's, then group ordered pairs by product value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q7d", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "probability", "distributions", "data-analysis" ], "primaryTopicId": "probability", "summary": "7(d): Use a discrete probability model to find a conditional probability, moments and transformed variance, compare two independent selections and form a product distribution.", "answerGuide": "Renormalize the probabilities after excluding 5. Calculate E(X²) and Var(X), use Var(a+bX)=b²Var(X), sum joint probabilities for Bruno's number exceeding Charlie's, then group ordered pairs by product value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q7e", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 3, "topicIds": [ "probability", "distributions", "data-analysis" ], "primaryTopicId": "probability", "summary": "7(e): Use a discrete probability model to find a conditional probability, moments and transformed variance, compare two independent selections and form a product distribution.", "answerGuide": "Renormalize the probabilities after excluding 5. Calculate E(X²) and Var(X), use Var(a+bX)=b²Var(X), sum joint probabilities for Bruno's number exceeding Charlie's, then group ordered pairs by product value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" } }, { "id": "ial-2022-10-WST01-01-q7f", "paperId": "ial-2022-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2022, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(f)", "marks": 4, "topicIds": [ "probability", "distributions", "data-analysis" ], "primaryTopicId": "probability", "summary": "Construct the probability distribution of the product of two independent selections.", "answerGuide": "List ordered pairs of outcomes, group pairs that give the same product, and add their independent probabilities for each product value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20221014.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20230112.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson calls this distribution-building part challenging, particularly the need to combine outcomes that produce the same product.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst01-01-que-20221014.pdf", "markScheme": "wst01-01-rms-20230112.pdf", "examinerReport": "wst01-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "13f237910922628ad78e850af474b0f4b34bae9277c943ef565cec7e6f50e61c", "markScheme": "a96db93aaee1cd74ab51713cbcad267c165ae0ade8346ef76fd89079f3a2f632", "examinerReport": "f003a8bc49539e8981ffd97405a5f2ea26d792f32afabb10b8f921fecf470a87" }, "sourceEvidencePages": { "examinerReport": [ 3, 5 ] } }, { "id": "ial-2022-10-WST02-01-q1ai", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(i)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(a)(i): Apply Poisson and binomial models to fault counts in cloth rolls, then calculate expected profit per piece.", "answerGuide": "Use a Poisson mean of 1.5 for the two fault-count events. Find the per-roll probability for fewer than 13 faults, then use a binomial model for exactly ten successful rolls. For profit, use a Poisson mean of 0.4 faults per piece and weight the profit outcomes by their probabilities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q1aii", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(a)(ii): Apply Poisson and binomial models to fault counts in cloth rolls, then calculate expected profit per piece.", "answerGuide": "Use a Poisson mean of 1.5 for the two fault-count events. Find the per-roll probability for fewer than 13 faults, then use a binomial model for exactly ten successful rolls. For profit, use a Poisson mean of 0.4 faults per piece and weight the profit outcomes by their probabilities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q1b", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(b): Apply Poisson and binomial models to fault counts in cloth rolls, then calculate expected profit per piece.", "answerGuide": "Use a Poisson mean of 1.5 for the two fault-count events. Find the per-roll probability for fewer than 13 faults, then use a binomial model for exactly ten successful rolls. For profit, use a Poisson mean of 0.4 faults per piece and weight the profit outcomes by their probabilities.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q1c", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find expected profit from one segment when the fault count follows a Poisson model.", "answerGuide": "Find the probability of no faults using the Poisson distribution, account for the profit outcomes with and without faults, and take their weighted average.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson describes this as the most challenging part of Question 1, with the expected-value setup driving errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" }, "sourceEvidencePages": { "examinerReport": [ 3, 3 ] } }, { "id": "ial-2022-10-WST02-01-q2a", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions", "integration", "data-representation" ], "primaryTopicId": "distributions", "summary": "2(a): Sketch a piecewise density, determine its constant from total probability, find the mean and calculate its interquartile range.", "answerGuide": "Sketch the given branches with support endpoints. Integrate across the support and set the total area to one to solve for k. Integrate xf(x) for E(X). Find Q1 and Q3 from the cumulative area, solving the relevant quadratic and subtracting.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q2b", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "distributions", "integration", "data-representation" ], "primaryTopicId": "distributions", "summary": "2(b): Sketch a piecewise density, determine its constant from total probability, find the mean and calculate its interquartile range.", "answerGuide": "Sketch the given branches with support endpoints. Integrate across the support and set the total area to one to solve for k. Integrate xf(x) for E(X). Find Q1 and Q3 from the cumulative area, solving the relevant quadratic and subtracting.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q2c", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "distributions", "integration", "data-representation" ], "primaryTopicId": "distributions", "summary": "2(c): Sketch a piecewise density, determine its constant from total probability, find the mean and calculate its interquartile range.", "answerGuide": "Sketch the given branches with support endpoints. Integrate across the support and set the total area to one to solve for k. Integrate xf(x) for E(X). Find Q1 and Q3 from the cumulative area, solving the relevant quadratic and subtracting.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q2d", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 5, "topicIds": [ "distributions", "integration", "data-representation" ], "primaryTopicId": "distributions", "summary": "Find the interquartile range from a continuous distribution with a piecewise density.", "answerGuide": "Integrate the density to form the cumulative distribution function, solve for the 25th and 75th percentiles, and subtract the quartiles.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes this as the most difficult part of Question 2 and notes problems constructing and applying the cumulative distribution function.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20230112.pdf", "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" }, "sourceEvidencePages": { "examinerReport": [ 3, 4 ] } }, { "id": "ial-2022-10-WST02-01-q3a", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "statistical-testing", "probability", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(a): Carry out a two-tailed binomial test for a claimed seed failure rate, then an upper-tailed approximation test for a new germination outcome.", "answerGuide": "State H0 and H1 and find lower and upper tail critical values with probabilities as close as possible to 2.5%. Add the tail probabilities for the actual significance level. For the second test, state hypotheses, use a suitable Poisson approximation and compare the observed count with the 5% critical region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q3b", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "statistical-testing", "probability", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(b): Carry out a two-tailed binomial test for a claimed seed failure rate, then an upper-tailed approximation test for a new germination outcome.", "answerGuide": "State H0 and H1 and find lower and upper tail critical values with probabilities as close as possible to 2.5%. Add the tail probabilities for the actual significance level. For the second test, state hypotheses, use a suitable Poisson approximation and compare the observed count with the 5% critical region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q3c", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "statistical-testing", "probability", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(c): Carry out a two-tailed binomial test for a claimed seed failure rate, then an upper-tailed approximation test for a new germination outcome.", "answerGuide": "State H0 and H1 and find lower and upper tail critical values with probabilities as close as possible to 2.5%. Add the tail probabilities for the actual significance level. For the second test, state hypotheses, use a suitable Poisson approximation and compare the observed count with the 5% critical region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q4a", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(a): Use a binomial model for a completion count, then apply a normal approximation with continuity correction to find an integer threshold.", "answerGuide": "Calculate the required binomial probability by subtracting cumulative probabilities. For the sample of 140, state its mean and variance, standardize the given lower-tail probability with the appropriate continuity correction, and solve for n.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q4b", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(b): Use a binomial model for a completion count, then apply a normal approximation with continuity correction to find an integer threshold.", "answerGuide": "Calculate the required binomial probability by subtracting cumulative probabilities. For the sample of 140, state its mean and variance, standardize the given lower-tail probability with the appropriate continuity correction, and solve for n.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20221021.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230112.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "wst02-01-que-20221021.pdf", "markScheme": "wst02-01-rms-20230112.pdf", "examinerReport": "wst02-01-pef-20230112.pdf" }, "sourceSha256": { "questionPaper": "987c5f815ed2f7151ae6bcdf4b5e153ef1ba395701deeb94b0a53e0d782edd94", "markScheme": "855bce946b388c139a133b30fdf9499030f4dbda7fa6761eee0db2eac33b5495", "examinerReport": "c0fae2571dbc46f3f59cf883faa8e9a03b6f984843a10521c4ba42f7c5194927" } }, { "id": "ial-2022-10-WST02-01-q5a", "paperId": "ial-2022-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2022, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "5(a): Use continuity of a piecewise cumulative distribution to determine a constant, then find a tail probability and a conditional probability.", "answerGuide": "Match the adjacent formulae at the boundary to determine c. Evaluate 1−F(3.5) for the tail probability. For the conditional probability, divide the probability over the required interval by P(3.50.", "answerGuide": "Use the limiting value as x increases and the value approached as x tends to 0 from above; check which endpoints are excluded.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q02b", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the inverse function f⁻¹.", "answerGuide": "Set y=4/(3x+5), rearrange to make x the subject, then replace y with x and state the valid domain.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q02c", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the composite function fg(x).", "answerGuide": "Substitute g(x)=1/x into f and simplify the resulting rational expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q02d", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(d)", "marks": 4, "topicIds": [ "functions", "proof" ], "primaryTopicId": "proof", "summary": "Show that fg(x)=gf(x) has no real solutions.", "answerGuide": "Equate the two composites, simplify to a quadratic, then show its discriminant is negative or complete the square.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q03ia", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(ia)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide x²−3x+6 by x+2.", "answerGuide": "Use polynomial division or a quotient-and-remainder identity to obtain the linear quotient and constant remainder.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q03ib", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(ib)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use the division result to integrate (x²−3x+6)/(x+2).", "answerGuide": "Rewrite the rational expression as the quotient plus a constant over x+2, then integrate each term and include a constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q03ii", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(ii)", "marks": 5, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral of (cos θ−sin θ)² from 0 to π/6.", "answerGuide": "Expand the square, use double-angle identities to integrate the trigonometric terms, then apply both exact limits.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q04", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": null, "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "For y=8 tan(2x), express dx/dy in the form A/(B+y²).", "answerGuide": "Differentiate y with respect to x, take the reciprocal, substitute tan(2x)=y/8, and simplify to identify A and B.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=12", "qpPages": [ 3, 3 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q05a", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove cot²x/(1+cot²x)=cos²x.", "answerGuide": "Rewrite cot x as cos x/sin x and use sin²x+cos²x=1 to simplify the denominator.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=13", "qpPages": [ 4, 4 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q05b", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the identity from part (a) to solve the given trigonometric equation for 0≤x<360°.", "answerGuide": "Substitute the identity, rearrange to a quadratic in cos x, solve for admissible cosine values, and list all angles in the interval to 1 decimal place.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=13", "qpPages": [ 4, 4 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q06a", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "State the range of the car-value model V for t≥0.", "answerGuide": "Evaluate the initial value at t=0 and use the horizontal limiting value as t tends to infinity; state the correct open and closed endpoints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=15", "qpPages": [ 5, 5 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q06b", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the rate at which the modelled car value is decreasing when t=10.", "answerGuide": "Differentiate V(t), substitute t=10, and report the signed derivative in pounds per year as requested.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=15", "qpPages": [ 5, 5 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q06c", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the exact age t when the modelled value is £15,000.", "answerGuide": "Set V=15000, substitute a suitable exponential variable to obtain a quadratic, solve it, and use logarithms to isolate t.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=15", "qpPages": [ 5, 5 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q07a", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express √5 cos θ−2 sin θ in phase-amplitude form R cos(θ+α).", "answerGuide": "Match the cosine and sine coefficients to R cos(θ+α), then calculate R and α in the required range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q07b", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve √5 cos θ−2 sin θ=0.5 for −π<θ<π.", "answerGuide": "Use the phase-amplitude form from part (a), solve the cosine equation, and retain all solutions in the open interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q07c", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Use the stated range of f to find B and the possible values of A.", "answerGuide": "Use the midpoint of the range to determine the vertical shift B and its half-width to determine the possible signed scale factors A.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q08a", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch y=a−|x| and y=|3x−2a| on separate diagrams, labelling all axis intersections.", "answerGuide": "Use the absolute-value shapes and solve for the x- and y-intercepts in terms of positive a; keep the two diagrams separate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=17", "qpPages": [ 7, 7 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q08b", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the x-values satisfying a−|x|=|3x−2a| for positive a.", "answerGuide": "Split at the sign changes of x and 3x−2a, solve each linear case, and keep only solutions valid in that case.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=18", "qpPages": [ 7, 7 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q09a", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Sketch y=arctan x, using its range and end behaviour.", "answerGuide": "Draw an increasing curve through the origin in quadrants I and III that flattens toward y=±π/2 without vertical asymptotes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q09b", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the exact x satisfying 3 arctan(x+1)−π=0.", "answerGuide": "Isolate arctan(x+1), apply tangent, and subtract 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q09c", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Show the positive root α of arctan x−4+x/2=0 lies between 5 and 6.", "answerGuide": "Evaluate the left-hand side at 5 and 6 to show a sign change, then cite continuity to establish a root in the interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA13-01A-q09d", "paperId": "ial-2026-1-WMA13-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Use the supplied fixed-point iteration from x₀=5 to find x₁ and α to 4 decimal places.", "answerGuide": "Substitute x₀ into the iteration once for x₁, then iterate to convergence and round α to four decimal places.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3A_MS.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2023-10-WMA11-01-q01a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a given power function once and twice. Scored part 1(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the power rule to each term, then differentiate the first derivative again." }, { "id": "ial-2023-10-WMA11-01-q01b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a given power function once and twice. Scored part 1(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the power rule to each term, then differentiate the first derivative again." }, { "id": "ial-2023-10-WMA11-01-q02a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Rewrite expressions in simplified index form. Scored part 2(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given powers and combine indices carefully." }, { "id": "ial-2023-10-WMA11-01-q02b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Rewrite expressions in simplified index form. Scored part 2(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given powers and combine indices carefully." }, { "id": "ial-2023-10-WMA11-01-q02c", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Rewrite expressions in simplified index form. Scored part 2(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given powers and combine indices carefully." }, { "id": "ial-2023-10-WMA11-01-q03a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "indices" ], "primaryTopicId": "algebra", "summary": "Simplify a surd expression and solve the resulting equation. Scored part 3(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rationalise or rewrite the surds first, then solve algebraically and simplify." }, { "id": "ial-2023-10-WMA11-01-q03b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "indices" ], "primaryTopicId": "algebra", "summary": "Simplify a surd expression and solve the resulting equation. Scored part 3(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rationalise or rewrite the surds first, then solve algebraically and simplify." }, { "id": "ial-2023-10-WMA11-01-q04a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "functions", "graphs", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve using its asymptote, factorisation, sketch and intersections. Scored part 4(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read each requested property from the equation or sketch; use the factorisation and intersections to justify the solution count." }, { "id": "ial-2023-10-WMA11-01-q04b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "functions", "graphs", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve using its asymptote, factorisation, sketch and intersections. Scored part 4(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read each requested property from the equation or sketch; use the factorisation and intersections to justify the solution count." }, { "id": "ial-2023-10-WMA11-01-q04c", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "functions", "graphs", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve using its asymptote, factorisation, sketch and intersections. Scored part 4(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read each requested property from the equation or sketch; use the factorisation and intersections to justify the solution count." }, { "id": "ial-2023-10-WMA11-01-q04d", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "functions", "graphs", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve using its asymptote, factorisation, sketch and intersections. Scored part 4(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=11", "qpPages": [ 8, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read each requested property from the equation or sketch; use the factorisation and intersections to justify the solution count." }, { "id": "ial-2023-10-WMA11-01-q05a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "trigonometry", "geometry" ], "primaryTopicId": "trigonometry", "summary": "Use triangle trigonometry to find the area of a roof frame. Scored part 5(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=13", "qpPages": [ 10, 13 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the cosine rule to determine the angle or side lengths, then apply the triangle area formula." }, { "id": "ial-2023-10-WMA11-01-q05b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "trigonometry", "geometry" ], "primaryTopicId": "trigonometry", "summary": "Use triangle trigonometry to find the area of a roof frame. Scored part 5(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=13", "qpPages": [ 10, 13 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the cosine rule to determine the angle or side lengths, then apply the triangle area formula." }, { "id": "ial-2023-10-WMA11-01-q06a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Use a repeated-root condition to determine a parameter range. Scored part 6(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the discriminant condition for two distinct real roots and solve the resulting inequality." }, { "id": "ial-2023-10-WMA11-01-q06b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Use a repeated-root condition to determine a parameter range. Scored part 6(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=15", "qpPages": [ 14, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the discriminant condition for two distinct real roots and solve the resulting inequality." }, { "id": "ial-2023-10-WMA11-01-q07a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "graphs" ], "primaryTopicId": "differentiation", "summary": "Use derivative information to investigate a curve. Scored part 7(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=17", "qpPages": [ 16, 19 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate or use the supplied derivative, then connect stationary points and curve behaviour to the stated condition." }, { "id": "ial-2023-10-WMA11-01-q07b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "differentiation", "graphs" ], "primaryTopicId": "differentiation", "summary": "Use derivative information to investigate a curve. Scored part 7(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=17", "qpPages": [ 16, 19 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate or use the supplied derivative, then connect stationary points and curve behaviour to the stated condition." }, { "id": "ial-2023-10-WMA11-01-q08a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "algebra", "functions", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find two curve intersections and their exact separation. Scored part 8(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the intersection relation exactly, substitute to recover coordinates, and use the distance formula." }, { "id": "ial-2023-10-WMA11-01-q08b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "algebra", "functions", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find the exact distance between the two curve-intersection points obtained from the quartic relation.", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies Question 8 as one of the paper's most challenging questions. It notes that exact y-coordinates and the distance calculation were particularly difficult, with many candidates resorting to decimals or omitting the y-difference.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Treat the quartic as a quadratic in x squared, retain both signed x roots, substitute into the curve for exact y-coordinates, then use the distance formula." }, { "id": "ial-2023-10-WMA11-01-q09a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Use sector geometry and an area condition to find a target perimeter. Scored part 9(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use arc length and sector angle to find the radius, then combine the given area with the straight and curved boundaries." }, { "id": "ial-2023-10-WMA11-01-q09b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Use sector geometry and an area condition to find a target perimeter. Scored part 9(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use arc length and sector angle to find the radius, then combine the given area with the straight and curved boundaries." }, { "id": "ial-2023-10-WMA11-01-q09c", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Use sector geometry and an area condition to find a target perimeter. Scored part 9(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use arc length and sector angle to find the radius, then combine the given area with the straight and curved boundaries." }, { "id": "ial-2023-10-WMA11-01-q10a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Analyse trigonometric curves using intercepts, period and shared points. Scored part 10(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=23", "qpPages": [ 28, 29 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first intercept and period to identify the trigonometric parameters; substitute the given points into the second curve." }, { "id": "ial-2023-10-WMA11-01-q10b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Analyse trigonometric curves using intercepts, period and shared points. Scored part 10(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=23", "qpPages": [ 28, 29 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first intercept and period to identify the trigonometric parameters; substitute the given points into the second curve." }, { "id": "ial-2023-10-WMA11-01-q10c", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Determine the unknown parameter from two given points on a trigonometric curve.", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=23", "qpPages": [ 28, 29 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Question 10 as one of the most challenging questions. In part (c), very few candidates correctly solved for the parameter; many stopped after setting up simultaneous equations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute both coordinates to form equations in the amplitude and parameter, use the odd symmetry of sine to simplify, and solve for the parameter." }, { "id": "ial-2023-10-WMA11-01-q11a", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "quadratics", "graphs", "algebra" ], "primaryTopicId": "quadratics", "summary": "Complete a square, find a minimum and describe a bounded region. Scored part 11(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to locate the minimum, determine the line from the given points, then write inequalities for each boundary." }, { "id": "ial-2023-10-WMA11-01-q11b", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(b)", "marks": 1, "topicIds": [ "quadratics", "graphs", "algebra" ], "primaryTopicId": "quadratics", "summary": "Complete a square, find a minimum and describe a bounded region. Scored part 11(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to locate the minimum, determine the line from the given points, then write inequalities for each boundary." }, { "id": "ial-2023-10-WMA11-01-q11c", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "quadratics", "graphs", "algebra" ], "primaryTopicId": "quadratics", "summary": "Complete a square, find a minimum and describe a bounded region. Scored part 11(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to locate the minimum, determine the line from the given points, then write inequalities for each boundary." }, { "id": "ial-2023-10-WMA11-01-q11d", "paperId": "ial-2023-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2023, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 11, "questionPart": "11(d)", "marks": 3, "topicIds": [ "quadratics", "graphs", "algebra" ], "primaryTopicId": "quadratics", "summary": "Complete a square, find a minimum and describe a bounded region. Scored part 11(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/23/be/d5/66/dc/be/8b/11/b9/a6/1e/ec/a7/39/7d/09.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/5f/a6/c9/7e/80/4b/7f/8e/65/b8/38/4f/d4/82/8b/b2.pdf#page=24", "qpPages": [ 30, 32 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to locate the minimum, determine the line from the given points, then write inequalities for each boundary." }, { "id": "ial-2023-10-WMA12-01-q01", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove by exhaustion that a product of positive integers is divisible by four. Scored part 1.", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the admissible integer cases from the constraints and check the product in each case." }, { "id": "ial-2023-10-WMA12-01-q02a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Generate terms of a recurrence and find a sum of its terms. Scored part 2(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the recurrence successively, then identify and sum the resulting pattern where possible." }, { "id": "ial-2023-10-WMA12-01-q02b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum of the first 100 terms generated by a repeating iteration.", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson describes this part as unexpectedly difficult. Many candidates applied ordinary arithmetic- or geometric-series formulae even though the generated values repeat.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Work out the recurring cycle of sequence values, count complete cycles among the first 100 terms, and include the remaining terms before summing." }, { "id": "ial-2023-10-WMA12-01-q03a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve trigonometric equations in the stated interval. Scored part 3(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to a standard trigonometric equation and check every solution against the given interval." }, { "id": "ial-2023-10-WMA12-01-q03b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve trigonometric equations in the stated interval. Scored part 3(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange to a standard trigonometric equation and check every solution against the given interval." }, { "id": "ial-2023-10-WMA12-01-q04a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a factor and a remainder to form equations for unknown polynomial coefficients. Scored part 4(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the known factor root and use the remainder theorem to form and solve simultaneous equations." }, { "id": "ial-2023-10-WMA12-01-q04b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a factor and a remainder to form equations for unknown polynomial coefficients. Scored part 4(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the known factor root and use the remainder theorem to form and solve simultaneous equations." }, { "id": "ial-2023-10-WMA12-01-q04c", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a factor and a remainder to form equations for unknown polynomial coefficients. Scored part 4(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the known factor root and use the remainder theorem to form and solve simultaneous equations." }, { "id": "ial-2023-10-WMA12-01-q05i", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Solve an exponential equation and a logarithmic equation. Scored part 5(i).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite both equations using consistent exponential or logarithmic rules, then solve exactly where requested." }, { "id": "ial-2023-10-WMA12-01-q05ii", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "indices", "algebra" ], "primaryTopicId": "indices", "summary": "Solve an exponential equation and a logarithmic equation. Scored part 5(ii).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=13", "qpPages": [ 12, 13 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite both equations using consistent exponential or logarithmic rules, then solve exactly where requested." }, { "id": "ial-2023-10-WMA12-01-q06a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Estimate a river cross-section using the trapezium rule and assess the estimate. Scored part 6(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the trapezium rule with the given equally spaced depths; compare the sampled curve with its shape to decide whether the estimate is an under- or overestimate." }, { "id": "ial-2023-10-WMA12-01-q06b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Estimate a river cross-section using the trapezium rule and assess the estimate. Scored part 6(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the trapezium rule with the given equally spaced depths; compare the sampled curve with its shape to decide whether the estimate is an under- or overestimate." }, { "id": "ial-2023-10-WMA12-01-q06c", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Explain why the trapezium-rule area estimate is an underestimate.", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=14", "qpPages": [ 14, 15 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls this the most difficult mark for candidates on the paper. Responses often stated only that the estimate was too small without explaining the geometric comparison.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare each trapezium with the curved cross-section and state that the trapezia lie below the shaded area, so their summed area is smaller." }, { "id": "ial-2023-10-WMA12-01-q07a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use a tangent to a circle to find related line and point information. Scored part 7(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=15", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use perpendicular radii and line gradients for the tangent, then substitute into the circle or line equations." }, { "id": "ial-2023-10-WMA12-01-q07b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use a tangent to a circle to find related line and point information. Scored part 7(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=15", "qpPages": [ 16, 19 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use perpendicular radii and line gradients for the tangent, then substitute into the circle or line equations." }, { "id": "ial-2023-10-WMA12-01-q08a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model rows of theatre seats as an arithmetic sequence and solve for the row count. Scored part 8(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=17", "qpPages": [ 20, 23 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic-series sum formula and the given totals; reject roots that cannot represent a positive integer number of rows." }, { "id": "ial-2023-10-WMA12-01-q08b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model rows of theatre seats as an arithmetic sequence and solve for the row count. Scored part 8(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=17", "qpPages": [ 20, 23 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic-series sum formula and the given totals; reject roots that cannot represent a positive integer number of rows." }, { "id": "ial-2023-10-WMA12-01-q08c", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Model rows of theatre seats as an arithmetic sequence and solve for the row count. Scored part 8(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=17", "qpPages": [ 20, 23 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic-series sum formula and the given totals; reject roots that cannot represent a positive integer number of rows." }, { "id": "ial-2023-10-WMA12-01-q09a", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Use calculus and a tangent to find properties and an area bounded by a curve. Scored part 9(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=18", "qpPages": [ 24, 27 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate to locate where the curve increases, form the tangent at the given point, then integrate the vertical difference for the finite area." }, { "id": "ial-2023-10-WMA12-01-q09b", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 8, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the area of a region formed by the curve and a tangent at a specified point.", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=18", "qpPages": [ 24, 27 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says Question 9 strongly separated candidates. Part (b) combines differentiation, integration and a multi-step area strategy; the final marks were mainly earned by the strongest candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Integrate the curve over the stated interval, differentiate to obtain the tangent gradient, find where the tangent meets the axis, and subtract the resulting triangle area." }, { "id": "ial-2023-10-WMA12-01-q10ia", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(i)(a)", "marks": 3, "topicIds": [ "binomial", "sequences", "trigonometry" ], "primaryTopicId": "binomial", "summary": "Connect binomial-expansion terms to a geometric series, then use an infinite trigonometric series. Scored part 10(i)(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=21", "qpPages": [ 28, 32 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand only the requested binomial terms, compare ratios for a geometric progression, and use the convergence formula for the infinite series." }, { "id": "ial-2023-10-WMA12-01-q10ib", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(i)(b)", "marks": 3, "topicIds": [ "binomial", "sequences", "trigonometry" ], "primaryTopicId": "binomial", "summary": "Find the value of x that makes selected binomial-expansion terms consecutive terms of a geometric progression.", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=21", "qpPages": [ 28, 32 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson specifically identifies part (i)(b) as discriminating. Many candidates selected powers x squared, x cubed and x to the fifth instead of the requested expansion terms, so they could not form the required equation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Identify the second, third and fifth terms of the expansion, equate the square of the middle term to the product of the outer terms, and solve for x." }, { "id": "ial-2023-10-WMA12-01-q10iia", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(ii)(a)", "marks": 3, "topicIds": [ "binomial", "sequences", "trigonometry" ], "primaryTopicId": "binomial", "summary": "Connect binomial-expansion terms to a geometric series, then use an infinite trigonometric series. Scored part 10(ii)(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=21", "qpPages": [ 28, 32 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand only the requested binomial terms, compare ratios for a geometric progression, and use the convergence formula for the infinite series." }, { "id": "ial-2023-10-WMA12-01-q10iib", "paperId": "ial-2023-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2023, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(ii)(b)", "marks": 3, "topicIds": [ "binomial", "sequences", "trigonometry" ], "primaryTopicId": "binomial", "summary": "Use the trigonometric geometric-series conditions to find the exact value of a later term.", "qpUrl": "https://www.alevelcopilot.com/whfile/8c/63/1a/00/86/75/89/28/ad/3a/d9/8d/45/84/ec/c0.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/ad/7d/03/0b/5c/ab/93/f2/18/6f/7e/8e/03/34/24/d7.pdf#page=21", "qpPages": [ 28, 32 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson specifically identifies part (ii)(b) as discriminating. Candidates who found cos theta still needed to substitute it exactly into the expression for the second term.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the sum-to-infinity relation to determine cos theta, obtain the needed exact sine and cosine values, then substitute into the expression for the requested term." }, { "id": "ial-2023-10-WMA13-01-q01a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "numerical-methods", "graphs" ], "primaryTopicId": "numerical-methods", "summary": "Show a root exists in a given interval and use an iterative formula. Scored part 1(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check the function values at the interval endpoints, then iterate with the stated formula to the required accuracy." }, { "id": "ial-2023-10-WMA13-01-q01b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "numerical-methods", "graphs" ], "primaryTopicId": "numerical-methods", "summary": "Show a root exists in a given interval and use an iterative formula. Scored part 1(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check the function values at the interval endpoints, then iterate with the stated formula to the required accuracy." }, { "id": "ial-2023-10-WMA13-01-q02a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate composite functions and find an inverse function. Scored part 2(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute into the function in the correct order; reverse the input-output relationship to obtain the inverse." }, { "id": "ial-2023-10-WMA13-01-q02b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate composite functions and find an inverse function. Scored part 2(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute into the function in the correct order; reverse the input-output relationship to obtain the inverse." }, { "id": "ial-2023-10-WMA13-01-q02c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate composite functions and find an inverse function. Scored part 2(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute into the function in the correct order; reverse the input-output relationship to obtain the inverse." }, { "id": "ial-2023-10-WMA13-01-q03a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Prove a double-angle identity and use it in an exact integral. Scored part 3(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand the compound-angle identity, simplify with the Pythagorean identity, then integrate the transformed expression exactly." }, { "id": "ial-2023-10-WMA13-01-q03b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Use the double-angle identity from the preceding part to evaluate a definite trigonometric integral exactly.", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=9", "qpPages": [ 8, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says Question 3 was more challenging than expected because many candidates failed to connect parts (a) and (b), preventing progress on the exact integral.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Rewrite the cosine term using the proved identity, integrate each term algebraically, substitute the limits and keep the exact result." }, { "id": "ial-2023-10-WMA13-01-q04a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use an exponential sales model to find its constants and rates of change. Scored part 4(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the initial condition to find the constant, then differentiate or solve the exponential equation for the requested rate or time." }, { "id": "ial-2023-10-WMA13-01-q04b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use an exponential sales model to find its constants and rates of change. Scored part 4(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the initial condition to find the constant, then differentiate or solve the exponential equation for the requested rate or time." }, { "id": "ial-2023-10-WMA13-01-q04c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the modelled sales growth rate at a specified time and express it in context.", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson lists the contextual interpretation of Question 4(c) among the other marks most commonly lost; candidates often left a rate in thousands without converting it to individual sales.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the exponential sales model, substitute the specified time and the fitted parameter, then state the rate in thousands per month or convert to individual sales per month." }, { "id": "ial-2023-10-WMA13-01-q04d", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use an exponential sales model to find its constants and rates of change. Scored part 4(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the initial condition to find the constant, then differentiate or solve the exponential equation for the requested rate or time." }, { "id": "ial-2023-10-WMA13-01-q05a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate a logarithmic curve and use its tangent or normal conditions. Scored part 5(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use logarithm laws before differentiating implicitly or explicitly; apply the tangent or normal gradient condition at the stated point." }, { "id": "ial-2023-10-WMA13-01-q05b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate a logarithmic curve and use its tangent or normal conditions. Scored part 5(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use logarithm laws before differentiating implicitly or explicitly; apply the tangent or normal gradient condition at the stated point." }, { "id": "ial-2023-10-WMA13-01-q05c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate a logarithmic curve and use its tangent or normal conditions. Scored part 5(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=11", "qpPages": [ 14, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use logarithm laws before differentiating implicitly or explicitly; apply the tangent or normal gradient condition at the stated point." }, { "id": "ial-2023-10-WMA13-01-q06a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Apply an exponential model for reef area and interpret its parameters. Scored part 6(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the logarithmic model to exponential form, evaluate at the specified time, and interpret the multiplier in context." }, { "id": "ial-2023-10-WMA13-01-q06b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Apply an exponential model for reef area and interpret its parameters. Scored part 6(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the logarithmic model to exponential form, evaluate at the specified time, and interpret the multiplier in context." }, { "id": "ial-2023-10-WMA13-01-q06c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Interpret the exponential model parameter that multiplies the reef area each year.", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=12", "qpPages": [ 16, 17 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that interpreting q in Question 6 was the least-scored mark on the paper and remained a weakness for most candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Explain q as the yearly multiplier for the modelled area, or convert it to the corresponding percentage change per year in context." }, { "id": "ial-2023-10-WMA13-01-q07a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve range and determine parameter values giving two intersections. Scored part 7(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the function into the requested form, use its range or turning value, and apply the two-intersection condition." }, { "id": "ial-2023-10-WMA13-01-q07b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve range and determine parameter values giving two intersections. Scored part 7(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the function into the requested form, use its range or turning value, and apply the two-intersection condition." }, { "id": "ial-2023-10-WMA13-01-q07c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Analyse a curve range and determine parameter values giving two intersections. Scored part 7(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=13", "qpPages": [ 18, 21 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the function into the requested form, use its range or turning value, and apply the two-intersection condition." }, { "id": "ial-2023-10-WMA13-01-q08a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity by rewriting the cosecant and double-angle terms.", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson calls Question 8 a strong discriminator across grades and says the third method mark in the proof was considerably harder to earn because candidates needed correct identities and enough simplification.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert cosecant to a reciprocal sine expression, apply a suitable double-angle identity, simplify to secant squared, then use the standard tangent identity." }, { "id": "ial-2023-10-WMA13-01-q08b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity and solve the resulting equation. Scored part 8(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=14", "qpPages": [ 22, 25 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Transform both sides using identities, then solve in the stated interval while excluding undefined values." }, { "id": "ial-2023-10-WMA13-01-q09a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a logarithmic inequality subject to the model domain. Scored part 9(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=16", "qpPages": [ 26, 29 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Respect the domain restriction, isolate the logarithmic expression and solve the resulting inequality." }, { "id": "ial-2023-10-WMA13-01-q09b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a logarithmic inequality subject to the model domain. Scored part 9(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=16", "qpPages": [ 26, 29 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Respect the domain restriction, isolate the logarithmic expression and solve the resulting inequality." }, { "id": "ial-2023-10-WMA13-01-q09c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a logarithmic inequality containing an absolute value and apply the domain restriction.", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=16", "qpPages": [ 26, 29 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies part (c) as the most challenging and discriminating part of Question 9. Removing the modulus correctly and retaining the domain constraint were common obstacles.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Split the absolute-value inequality into both branches, solve each logarithmic inequality exactly, combine the valid intervals, and enforce x greater than negative one." }, { "id": "ial-2023-10-WMA13-01-q10a", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Differentiate a trigonometric curve and use its stationary-point conditions. Scored part 10(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=17", "qpPages": [ 30, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the product or composite expression, substitute the given coordinate and solve the stationary-point conditions." }, { "id": "ial-2023-10-WMA13-01-q10b", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 2, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Differentiate a trigonometric curve and use its stationary-point conditions. Scored part 10(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=17", "qpPages": [ 30, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the product or composite expression, substitute the given coordinate and solve the stationary-point conditions." }, { "id": "ial-2023-10-WMA13-01-q10c", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Differentiate a trigonometric curve and use its stationary-point conditions. Scored part 10(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=17", "qpPages": [ 30, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the product or composite expression, substitute the given coordinate and solve the stationary-point conditions." }, { "id": "ial-2023-10-WMA13-01-q10d", "paperId": "ial-2023-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2023, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(d)", "marks": 2, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Differentiate a trigonometric curve and use its stationary-point conditions. Scored part 10(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/13/bd/e1/08/a2/83/60/40/e0/b1/4d/29/d6/12/0a/96.pdf#page=30", "msUrl": "https://www.alevelcopilot.com/whfile/91/03/1f/cb/af/da/6d/20/eb/ed/c6/1c/33/03/18/11.pdf#page=17", "qpPages": [ 30, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the product or composite expression, substitute the given coordinate and solve the stationary-point conditions." }, { "id": "ial-2023-10-WMA14-01-q01a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find terms in a binomial expansion and state its interval of validity. Scored part 1(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the generalized binomial expansion, simplify the requested terms, then apply the convergence condition." }, { "id": "ial-2023-10-WMA14-01-q01b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find terms in a binomial expansion and state its interval of validity. Scored part 1(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the generalized binomial expansion, simplify the requested terms, then apply the convergence condition." }, { "id": "ial-2023-10-WMA14-01-q02a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Relate the changing edge length of a cube to its surface area and volume. Scored part 2(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the surface-area and volume formulae with respect to time and substitute the given rate." }, { "id": "ial-2023-10-WMA14-01-q02b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Relate the changing edge length of a cube to its surface area and volume. Scored part 2(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the surface-area and volume formulae with respect to time and substitute the given rate." }, { "id": "ial-2023-10-WMA14-01-q03i", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate integrals using integration by parts and substitution. Scored part 3(i).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=12", "qpPages": [ 6, 9 ], "msPages": [ 12, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the stated integration method, show the substitution or parts steps, and apply the limits exactly." }, { "id": "ial-2023-10-WMA14-01-q03ii", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Evaluate integrals using integration by parts and substitution. Scored part 3(ii).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=12", "qpPages": [ 6, 9 ], "msPages": [ 12, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose the stated integration method, show the substitution or parts steps, and apply the limits exactly." }, { "id": "ial-2023-10-WMA14-01-q04a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove a reciprocal inequality for positive k by contradiction.", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=17", "qpPages": [ 10, 11 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says Question 4 caused the most problems. Many candidates made an incorrect assumption at the start of part (a), which the report identifies as a difficult contradiction proof.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Assume the claimed inequality fails, multiply by the positive variable without reversing the inequality, and derive an impossible statement about a square; finish with a clear contradiction conclusion." }, { "id": "ial-2023-10-WMA14-01-q04b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove an inequality by contradiction and test its real-number extension. Scored part 4(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=17", "qpPages": [ 10, 11 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assume the opposite inequality, transform it into an impossible statement for positive values, then provide a counterexample for the broader claim." }, { "id": "ial-2023-10-WMA14-01-q05a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation and a normal line to find a curve parameter. Scored part 5(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=20", "qpPages": [ 12, 15 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve implicitly, use the reciprocal negative gradient for the normal, then substitute the given point or line." }, { "id": "ial-2023-10-WMA14-01-q05b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Use the curve and the stated normal line to determine its positive parameter.", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=20", "qpPages": [ 12, 15 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson explicitly lists part 5(b) among the most challenging parts of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Implicitly differentiate the curve, use the normal-line gradient to obtain the tangent gradient, then use the point-on-curve and line conditions to solve for the parameter." }, { "id": "ial-2023-10-WMA14-01-q06a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Use two vector lines to find their intersection and the angle between them. Scored part 6(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=22", "qpPages": [ 16, 19 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the line parameter equations for the meeting point, then use the dot product and magnitudes for the angle." }, { "id": "ial-2023-10-WMA14-01-q06b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Use two vector lines to find their intersection and the angle between them. Scored part 6(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=22", "qpPages": [ 16, 19 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the line parameter equations for the meeting point, then use the dot product and magnitudes for the angle." }, { "id": "ial-2023-10-WMA14-01-q06c", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Use two vector lines to find their intersection and the angle between them. Scored part 6(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=22", "qpPages": [ 16, 19 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the line parameter equations for the meeting point, then use the dot product and magnitudes for the angle." }, { "id": "ial-2023-10-WMA14-01-q06d", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Find both possible coordinates of a point on one line that forms an isosceles triangle with two given points.", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=22", "qpPages": [ 16, 19 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson explicitly lists part 6(d) among the most challenging parts of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Represent the point on the second line by its parameter, equate its distance from P to the known length PQ, solve the resulting equation and substitute both parameter values." }, { "id": "ial-2023-10-WMA14-01-q07a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Compare two population models and solve the differential equation model. Scored part 7(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=25", "qpPages": [ 20, 23 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the initial condition to identify constants, solve by separation or partial fractions as required, and evaluate long-term limits." }, { "id": "ial-2023-10-WMA14-01-q07b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Compare two population models and solve the differential equation model. Scored part 7(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=25", "qpPages": [ 20, 23 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the initial condition to identify constants, solve by separation or partial fractions as required, and evaluate long-term limits." }, { "id": "ial-2023-10-WMA14-01-q07c", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Compare two population models and solve the differential equation model. Scored part 7(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=25", "qpPages": [ 20, 23 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the initial condition to identify constants, solve by separation or partial fractions as required, and evaluate long-term limits." }, { "id": "ial-2023-10-WMA14-01-q07d", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Compare two population models and solve the differential equation model. Scored part 7(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=25", "qpPages": [ 20, 23 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the initial condition to identify constants, solve by separation or partial fractions as required, and evaluate long-term limits." }, { "id": "ial-2023-10-WMA14-01-q07e", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 1, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Compare two population models and solve the differential equation model. Scored part 7(e).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=25", "qpPages": [ 20, 23 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the initial condition to identify constants, solve by separation or partial fractions as required, and evaluate long-term limits." }, { "id": "ial-2023-10-WMA14-01-q08a", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Analyse a parametric curve and set up a volume of revolution. Scored part 8(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=28", "qpPages": [ 24, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the parametric relations to find intercepts and dy/dx, then substitute into the volume formula and integrate over the parameter interval." }, { "id": "ial-2023-10-WMA14-01-q08b", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Analyse a parametric curve and set up a volume of revolution. Scored part 8(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=28", "qpPages": [ 24, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the parametric relations to find intercepts and dy/dx, then substitute into the volume formula and integrate over the parameter interval." }, { "id": "ial-2023-10-WMA14-01-q08c", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Analyse a parametric curve and set up a volume of revolution. Scored part 8(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=28", "qpPages": [ 24, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the parametric relations to find intercepts and dy/dx, then substitute into the volume formula and integrate over the parameter interval." }, { "id": "ial-2023-10-WMA14-01-q08di", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)(i)", "marks": 4, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Show that the parametric curve gives the stated integral for the solid-of-revolution volume.", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=28", "qpPages": [ 24, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson explicitly lists part 8(d)(i) among the most challenging parts of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the disk formula pi times the integral of y squared with respect to x, substitute the parametric equations, and convert dx to (dx/dt) dt with the correct parameter limits." }, { "id": "ial-2023-10-WMA14-01-q08dii", "paperId": "ial-2023-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2023, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)(ii)", "marks": 2, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Analyse a parametric curve and set up a volume of revolution. Scored part 8(d)(ii).", "qpUrl": "https://www.alevelcopilot.com/whfile/fb/10/26/c5/07/f1/d9/0b/41/3b/6a/27/a9/20/60/a0.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/9f/1e/c4/de/05/23/22/5b/ce/c6/5d/90/14/06/43/58.pdf#page=28", "qpPages": [ 24, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the parametric relations to find intercepts and dy/dx, then substitute into the volume formula and integrate over the parameter interval." }, { "id": "ial-2023-10-WME01-01-q01", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use moments to analyse a beam suspended by vertical ropes. Scored part 1.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about a convenient point, use vertical force balance, then solve for the unknown rope position or tension." }, { "id": "ial-2023-10-WME01-01-q02a", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph to determine motion quantities. Scored part 2(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read slopes and signed areas from the graph, keeping acceleration and velocity signs consistent." }, { "id": "ial-2023-10-WME01-01-q02b", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph to determine motion quantities. Scored part 2(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read slopes and signed areas from the graph, keeping acceleration and velocity signs consistent." }, { "id": "ial-2023-10-WME01-01-q02c", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use a velocity-time graph to determine motion quantities. Scored part 2(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read slopes and signed areas from the graph, keeping acceleration and velocity signs consistent." }, { "id": "ial-2023-10-WME01-01-q03a", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply momentum and Newton’s second law to a hammer and tent peg after impact. Scored part 3(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Conserve momentum through the impact, then use the stated motion after impact to find the force or resistance." }, { "id": "ial-2023-10-WME01-01-q03b", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply momentum and Newton’s second law to a hammer and tent peg after impact. Scored part 3(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Conserve momentum through the impact, then use the stated motion after impact to find the force or resistance." }, { "id": "ial-2023-10-WME01-01-q03c", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the ground resistance after a hammer and peg move together and decelerate to rest.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says the final part of Question 3 was much more challenging. Common errors included using 12 m instead of 0.12 m, inconsistent signs, or omitting the weight.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the post-collision speed and the 0.12 m stopping distance in suvat to find acceleration, then apply Newton's second law with both the resistance and weight in the consistent direction." }, { "id": "ial-2023-10-WME01-01-q04a", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Model a particle moving with constant vector acceleration. Scored part 4(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration to obtain velocity and position, applying the initial conditions in each vector component." }, { "id": "ial-2023-10-WME01-01-q04b", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Model a particle moving with constant vector acceleration. Scored part 4(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration to obtain velocity and position, applying the initial conditions in each vector component." }, { "id": "ial-2023-10-WME01-01-q04c", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Model a particle moving with constant vector acceleration. Scored part 4(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate acceleration to obtain velocity and position, applying the initial conditions in each vector component." }, { "id": "ial-2023-10-WME01-01-q05a", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse equilibrium and friction for a ring on a rough vertical rod. Scored part 5(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the rod; apply equilibrium and the limiting-friction relation only when stated." }, { "id": "ial-2023-10-WME01-01-q05b", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse equilibrium and friction for a ring on a rough vertical rod. Scored part 5(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the rod; apply equilibrium and the limiting-friction relation only when stated." }, { "id": "ial-2023-10-WME01-01-q05c", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Analyse equilibrium and friction for a ring on a rough vertical rod. Scored part 5(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces parallel and perpendicular to the rod; apply equilibrium and the limiting-friction relation only when stated." }, { "id": "ial-2023-10-WME01-01-q06a", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Use constant vector velocity to compare the positions of two ships. Scored part 6(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find velocity from displacement over elapsed time, then use the relative position or displacement requested." }, { "id": "ial-2023-10-WME01-01-q06b", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Use constant vector velocity to compare the positions of two ships. Scored part 6(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find velocity from displacement over elapsed time, then use the relative position or displacement requested." }, { "id": "ial-2023-10-WME01-01-q06c", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Use constant vector velocity to compare the positions of two ships. Scored part 6(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find velocity from displacement over elapsed time, then use the relative position or displacement requested." }, { "id": "ial-2023-10-WME01-01-q06d", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 7, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Find the exact separation of two ships after one changes course and speed.", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says the final part of Question 6 was more challenging. Candidates needed to model the changed velocity of Q with vectors and then apply Pythagoras at 14:30.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Find both ships' position vectors at 12:30, continue P with its original velocity, move Q due north for the remaining two hours, subtract the vectors and use Pythagoras for the exact distance." }, { "id": "ial-2023-10-WME01-01-q07a", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 9, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Model connected particles on a rough inclined plane with a pulley. Scored part 7(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw separate force equations for each body and the pulley; use the common acceleration, string tension and friction direction consistently." }, { "id": "ial-2023-10-WME01-01-q07b", "paperId": "ial-2023-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2023, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Model connected particles on a rough inclined plane with a pulley. Scored part 7(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/0f/88/1b/d4/08/ac/c2/6b/c7/34/b4/27/bf/a5/0e/20.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/8c/1a/7a/9a/2b/0d/bb/00/d7/0f/d8/ca/bd/6f/85/87.pdf#page=15", "qpPages": [ 24, 28 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Draw separate force equations for each body and the pulley; use the common acceleration, string tension and friction direction consistently." }, { "id": "ial-2023-10-WME02-01-q01a", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Differentiate a particle’s position vector to find velocity and acceleration. Scored part 1(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each position component; apply the direction condition to the velocity and the specified time to the acceleration." }, { "id": "ial-2023-10-WME02-01-q01b", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Differentiate a particle’s position vector to find velocity and acceleration. Scored part 1(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each position component; apply the direction condition to the velocity and the specified time to the acceleration." }, { "id": "ial-2023-10-WME02-01-q02a", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Use centre of mass and moments to analyse a composite shape in equilibrium. Scored part 2(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of each component, then take moments about the pivot and resolve forces." }, { "id": "ial-2023-10-WME02-01-q02b", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 6, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Use centre of mass and moments to analyse a composite shape in equilibrium. Scored part 2(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the centre of mass of each component, then take moments about the pivot and resolve forces." }, { "id": "ial-2023-10-WME02-01-q02c", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal force needed to hold a composite lamina in equilibrium about its pivot.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes part (c) as the least successful part of Question 2. Candidates often used incorrect moment distances or resolved forces even though the distances were already perpendicular.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the pivot and use the perpendicular lever arms of the weight and applied horizontal force; solve the resulting balance without resolving forces unnecessarily." }, { "id": "ial-2023-10-WME02-01-q03", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply impulse and momentum to a two-particle collision. Scored part 3.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=9", "qpPages": [ 8, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve velocity along the line of motion and use impulse-momentum with consistent signs." }, { "id": "ial-2023-10-WME02-01-q04a", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Model a projectile and use velocity components at a later time. Scored part 4(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write horizontal and vertical velocity components separately; use the perpendicular condition or trajectory equation as stated." }, { "id": "ial-2023-10-WME02-01-q04b", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Model a projectile and use velocity components at a later time. Scored part 4(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write horizontal and vertical velocity components separately; use the perpendicular condition or trajectory equation as stated." }, { "id": "ial-2023-10-WME02-01-q04c", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Model a projectile and use velocity components at a later time. Scored part 4(c).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write horizontal and vertical velocity components separately; use the perpendicular condition or trajectory equation as stated." }, { "id": "ial-2023-10-WME02-01-q04d", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the later velocity of a projectile is perpendicular to its initial velocity.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls part (d) the most challenging part of Question 4; successful methods used velocity vectors, while sign errors and using a distance formula were common.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the velocity components at time t, set their dot product with the initial velocity vector to zero, and solve for the positive time." }, { "id": "ial-2023-10-WME02-01-q05a", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use power, resistance and Newton’s second law for a cyclist. Scored part 5(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv to find the driving force, subtract resistance, then apply F = ma." }, { "id": "ial-2023-10-WME02-01-q05b", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use power, resistance and Newton’s second law for a cyclist. Scored part 5(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv to find the driving force, subtract resistance, then apply F = ma." }, { "id": "ial-2023-10-WME02-01-q05c", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use power, resistance and Newton’s second law for a cyclist. Scored part 5(c).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use P = Fv to find the driving force, subtract resistance, then apply F = ma." }, { "id": "ial-2023-10-WME02-01-q06a", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use moments to determine forces on a hinged uniform rod. Scored part 6(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the hinge and resolve the forces in perpendicular directions." }, { "id": "ial-2023-10-WME02-01-q06b", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use moments to determine forces on a hinged uniform rod. Scored part 6(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the hinge and resolve the forces in perpendicular directions." }, { "id": "ial-2023-10-WME02-01-q07a", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply conservation of momentum and the coefficient of restitution to a collision. Scored part 7(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, conserve momentum, and combine with the restitution or post-impact condition." }, { "id": "ial-2023-10-WME02-01-q07b", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply conservation of momentum and the coefficient of restitution to a collision. Scored part 7(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, conserve momentum, and combine with the restitution or post-impact condition." }, { "id": "ial-2023-10-WME02-01-q07c", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply conservation of momentum and the coefficient of restitution to a collision. Scored part 7(c).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, conserve momentum, and combine with the restitution or post-impact condition." }, { "id": "ial-2023-10-WME02-01-q07d", "paperId": "ial-2023-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2023, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply conservation of momentum and the coefficient of restitution to a collision. Scored part 7(d).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20231027.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240118.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose a positive direction, conserve momentum, and combine with the restitution or post-impact condition." }, { "id": "ial-2023-10-WST01-01-q01a", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use conditional probabilities to complete a game tree and find turn probabilities. Scored part 1(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply along branches and add mutually exclusive paths; use the stopping rule to enumerate valid turn outcomes." }, { "id": "ial-2023-10-WST01-01-q01b", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use conditional probabilities to complete a game tree and find turn probabilities. Scored part 1(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply along branches and add mutually exclusive paths; use the stopping rule to enumerate valid turn outcomes." }, { "id": "ial-2023-10-WST01-01-q01c", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use conditional probabilities to complete a game tree and find turn probabilities. Scored part 1(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply along branches and add mutually exclusive paths; use the stopping rule to enumerate valid turn outcomes." }, { "id": "ial-2023-10-WST01-01-q01d", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use conditional probabilities to complete a game tree and find turn probabilities. Scored part 1(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply along branches and add mutually exclusive paths; use the stopping rule to enumerate valid turn outcomes." }, { "id": "ial-2023-10-WST01-01-q01e", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of exactly two wins given that the first game was a win.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 5, "difficultyEvidence": "Pearson notes that part (e) caused candidates more difficulty because they needed a conditional-probability ratio with a numerator calculated from the tree.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the first-game condition in the denominator and add the appropriate tree paths for exactly two total wins in the numerator." }, { "id": "ial-2023-10-WST01-01-q02a", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Summarise a sample of kangaroo weights and compare distributions with a box plot. Scored part 2(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the ordered data from the stem-and-leaf display, calculate the requested summaries, and compare spread and median in context." }, { "id": "ial-2023-10-WST01-01-q02b", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Summarise a sample of kangaroo weights and compare distributions with a box plot. Scored part 2(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the ordered data from the stem-and-leaf display, calculate the requested summaries, and compare spread and median in context." }, { "id": "ial-2023-10-WST01-01-q02c", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Summarise a sample of kangaroo weights and compare distributions with a box plot. Scored part 2(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the ordered data from the stem-and-leaf display, calculate the requested summaries, and compare spread and median in context." }, { "id": "ial-2023-10-WST01-01-q02d", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare two box-plot distributions of kangaroo weights in context.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says part (d) caused the most difficulty in Question 2. Candidates often quoted statistics without making comparisons or interpreting them in context.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Compare the centres and spreads, refer to any outliers where relevant, and state what the differences imply about the weights in the two months." }, { "id": "ial-2023-10-WST01-01-q03ia", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a Venn diagram to find probabilities for two shopping events. Scored part 3(i)(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign the given probabilities to disjoint Venn regions, then use complements and addition as required." }, { "id": "ial-2023-10-WST01-01-q03ib", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(b)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a Venn diagram to find probabilities for two shopping events. Scored part 3(i)(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign the given probabilities to disjoint Venn regions, then use complements and addition as required." }, { "id": "ial-2023-10-WST01-01-q03ic", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a Venn diagram to find probabilities for two shopping events. Scored part 3(i)(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign the given probabilities to disjoint Venn regions, then use complements and addition as required." }, { "id": "ial-2023-10-WST01-01-q03iia", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a Venn diagram to find probabilities for two shopping events. Scored part 3(ii)(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign the given probabilities to disjoint Venn regions, then use complements and addition as required." }, { "id": "ial-2023-10-WST01-01-q03iib", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a Venn diagram to find probabilities for two shopping events. Scored part 3(ii)(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign the given probabilities to disjoint Venn regions, then use complements and addition as required." }, { "id": "ial-2023-10-WST01-01-q03iic", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use a Venn diagram to find probabilities for two shopping events. Scored part 3(ii)(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign the given probabilities to disjoint Venn regions, then use complements and addition as required." }, { "id": "ial-2023-10-WST01-01-q04a", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate expectation and variance for a discrete random variable and a scaled variable. Scored part 4(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probability table for E(X) and Var(X), then apply the scaling laws for expectation and variance." }, { "id": "ial-2023-10-WST01-01-q04b", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate expectation and variance for a discrete random variable and a scaled variable. Scored part 4(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probability table for E(X) and Var(X), then apply the scaling laws for expectation and variance." }, { "id": "ial-2023-10-WST01-01-q04c", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Calculate expectation and variance for a discrete random variable and a scaled variable. Scored part 4(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probability table for E(X) and Var(X), then apply the scaling laws for expectation and variance." }, { "id": "ial-2023-10-WST01-01-q04d", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a conditional probability involving a discrete variable and a scaled version of that variable.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says part (d) caused the most problems and only the strongest candidates obtained all marks. Candidates often failed to translate the Y condition back to X or selected the wrong conditional event.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Translate the condition on Y using the transformation defined in the question, identify the permitted X values and calculate the joint and conditioning probabilities from the distribution." }, { "id": "ial-2023-10-WST01-01-q05a", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to calculate probabilities for fruit weights. Scored part 5(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the boundary values using the stated mean and standard deviation; use symmetry or subtract cumulative probabilities." }, { "id": "ial-2023-10-WST01-01-q05b", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to calculate probabilities for fruit weights. Scored part 5(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the boundary values using the stated mean and standard deviation; use symmetry or subtract cumulative probabilities." }, { "id": "ial-2023-10-WST01-01-q05c", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to calculate probabilities for fruit weights. Scored part 5(c).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the boundary values using the stated mean and standard deviation; use symmetry or subtract cumulative probabilities." }, { "id": "ial-2023-10-WST01-01-q05d", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to calculate probabilities for fruit weights. Scored part 5(d).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the boundary values using the stated mean and standard deviation; use symmetry or subtract cumulative probabilities." }, { "id": "ial-2023-10-WST01-01-q05e", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal model to calculate probabilities for fruit weights. Scored part 5(e).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise the boundary values using the stated mean and standard deviation; use symmetry or subtract cumulative probabilities." }, { "id": "ial-2023-10-WST01-01-q06a", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use regression equations and summary statistics to analyse correlation. Scored part 6(a).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the regression equations at their intersection, then use the regression and summary-statistic relationships for Sxy or r." }, { "id": "ial-2023-10-WST01-01-q06b", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use regression equations and summary statistics to analyse correlation. Scored part 6(b).", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the regression equations at their intersection, then use the regression and summary-statistic relationships for Sxy or r." }, { "id": "ial-2023-10-WST01-01-q06c", "paperId": "ial-2023-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2023, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation coefficient from regression equations and summary statistics.", "qpUrl": "https://www.alevelcopilot.com/whfile/e3/66/14/ae/ef/fc/c1/f6/93/c1/f6/d9/3b/0b/21/3b.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/01/66/8a/86/de/cf/83/29/55/dc/b8/14/19/10/40/78.pdf#page=11", "qpPages": [ 22, 24 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson describes part (c) as problematic and says only stronger candidates answered it. Many found Sxx but did not use the regression information to obtain Syy.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the regression gradients and the given sums to find Sxx and Syy, then substitute Sxy, Sxx and Syy into the correlation-coefficient formula." }, { "id": "ial-2023-10-WST02-01-q01a", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model independent sales using a binomial distribution. Scored part 1(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify n and p, then use the binomial probability or cumulative formula for each requested event." }, { "id": "ial-2023-10-WST02-01-q01bi", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model independent sales using a binomial distribution. Scored part 1(b)(i).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify n and p, then use the binomial probability or cumulative formula for each requested event." }, { "id": "ial-2023-10-WST02-01-q01bii", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the minimum daily number of calls for the probability of at least one sale to exceed a threshold.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson describes part (b)(ii) as slightly more problematic; candidates often translated the at-least-one-sale condition into the wrong complement inequality.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the complement probability 1 minus the chance of no sales, set it greater than 0.95, solve the exponential inequality and choose the smallest integer n." }, { "id": "ial-2023-10-WST02-01-q02a", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a piecewise probability density function to find distribution properties. Scored part 2(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use continuity, total probability and the given density pieces to solve for constants and requested probabilities or summaries." }, { "id": "ial-2023-10-WST02-01-q02b", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a piecewise probability density function to find distribution properties. Scored part 2(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use continuity, total probability and the given density pieces to solve for constants and requested probabilities or summaries." }, { "id": "ial-2023-10-WST02-01-q02c", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the endpoint of a linear density segment from the median condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says part (c) caused the most problems. Candidates who set the triangular density area equal to one half often found the correct endpoint.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Use that the median is 4 so the area on either side is one half, then equate the area of the triangle under the linear segment to one half and solve for d." }, { "id": "ial-2023-10-WST02-01-q02d", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a piecewise probability density function to find distribution properties. Scored part 2(d).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use continuity, total probability and the given density pieces to solve for constants and requested probabilities or summaries." }, { "id": "ial-2023-10-WST02-01-q03a", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model a bus journey time with a continuous uniform distribution. Scored part 3(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine the minimum and maximum journey times from the schedule, then use uniform interval lengths for probabilities." }, { "id": "ial-2023-10-WST02-01-q03b", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model a bus journey time with a continuous uniform distribution. Scored part 3(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine the minimum and maximum journey times from the schedule, then use uniform interval lengths for probabilities." }, { "id": "ial-2023-10-WST02-01-q03c", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model a bus journey time with a continuous uniform distribution. Scored part 3(c).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Determine the minimum and maximum journey times from the schedule, then use uniform interval lengths for probabilities." }, { "id": "ial-2023-10-WST02-01-q03d", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that a uniform journey time is within five minutes of a target time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says part (d) proved more challenging because candidates struggled to translate the travel context into the correct interval probability.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Convert the wording to the target interval, intersect it with the model support [32,47], and divide the resulting interval length by 15." }, { "id": "ial-2023-10-WST02-01-q04a", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a discrete distribution for the value of a randomly selected T-shirt. Scored part 4(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probabilities for the three sizes to calculate expectations; for a sample, identify the relevant binomial or product model." }, { "id": "ial-2023-10-WST02-01-q04b", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a discrete distribution for the value of a randomly selected T-shirt. Scored part 4(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probabilities for the three sizes to calculate expectations; for a sample, identify the relevant binomial or product model." }, { "id": "ial-2023-10-WST02-01-q04c", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a discrete distribution for the value of a randomly selected T-shirt. Scored part 4(c).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the probabilities for the three sizes to calculate expectations; for a sample, identify the relevant binomial or product model." }, { "id": "ial-2023-10-WST02-01-q05a", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model weekly complaints with a Poisson distribution. Scored part 5(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson rate for the stated time interval and add or subtract the appropriate point probabilities." }, { "id": "ial-2023-10-WST02-01-q05b", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model weekly complaints with a Poisson distribution. Scored part 5(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson rate for the stated time interval and add or subtract the appropriate point probabilities." }, { "id": "ial-2023-10-WST02-01-q05c", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model weekly complaints with a Poisson distribution. Scored part 5(c).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson rate for the stated time interval and add or subtract the appropriate point probabilities." }, { "id": "ial-2023-10-WST02-01-q05d", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 7, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Model weekly complaints with a Poisson distribution. Scored part 5(d).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the Poisson rate for the stated time interval and add or subtract the appropriate point probabilities." }, { "id": "ial-2023-10-WST02-01-q06a", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a cumulative distribution function to find density and expectation values. Scored part 6(a).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each CDF segment to obtain the density, enforce continuity and total probability, then integrate for expectations." }, { "id": "ial-2023-10-WST02-01-q06b", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a cumulative distribution function to find density and expectation values. Scored part 6(b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each CDF segment to obtain the density, enforce continuity and total probability, then integrate for expectations." }, { "id": "ial-2023-10-WST02-01-q06c", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a piecewise cumulative distribution to obtain the density and calculate an expectation.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes part (c) as more challenging. Some candidates failed to differentiate the CDF to obtain the pdf and instead used the CDF in the expectation integral.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20240118.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the relevant CDF pieces to form the pdf, then integrate y times the pdf over each interval and add the results." }, { "id": "ial-2023-10-WST02-01-q07", "paperId": "ial-2023-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2023, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7", "marks": 10, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to a binomial distribution to determine its parameters. Scored part 7.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20231020.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240118.pdf#page=12", "qpPages": [ 22, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match the two stated binomial tail probabilities to continuity-corrected normal z-scores and solve for n and p." }, { "id": "ial-2026-1-WDM11-01A-q01a", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound for the ten-foot planks needed.", "answerGuide": "Add lengths, divide by capacity, and round up.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q01b", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Pack the listed pieces in order using first fit.", "answerGuide": "Place each length in the first existing bin with space; open a bin only if needed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q01c", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Sort the lengths descending using quick sort; show passes and pivots.", "answerGuide": "Identify each pivot, partition, and continue until descending order is complete.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q01d", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Pack the descending list using first-fit decreasing.", "answerGuide": "Run first fit on the sorted list and record the allocation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q02a", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Prim from A to list a minimum tree's arcs in selection order.", "answerGuide": "Grow the tree by the cheapest eligible arc joining a new building.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q02b", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the spanning tree's total cost.", "answerGuide": "Add its selected arc weights.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q02c", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Kruskal for a minimum tree that must include BC and EF.", "answerGuide": "Start with those links; examine increasing costs, adding links unless they form a cycle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q02d", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the revised cost when BC and EF are already free.", "answerGuide": "Sum the costs of the selected paid links.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q03a", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the shortest route from A to J by Dijkstra's algorithm.", "answerGuide": "Update tentative distances until J is permanent; state route and length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q03b", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the pipes to duplicate in a shortest closed inspection route from A.", "answerGuide": "Pair odd-degree vertices with minimum extra distance and duplicate the selected arcs.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q03c", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give a valid shortest closed inspection route and its length.", "answerGuide": "Trace a route using every original and duplicated arc, starting and ending at A.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q03d", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "After removing pipes incident to B, find optimal route endpoints and length.", "answerGuide": "Recheck odd vertices in the reduced network and minimize the extra distance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q04", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw the activity-on-arc network from the precedence table.", "answerGuide": "Join each activity after its predecessors; add only the dummy arcs needed to preserve precedence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q05a", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Write the budget constraint for the two toy types.", "answerGuide": "Multiply each count by its unit cost and constrain spending to the budget.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q05b", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Graph the constraints and shade the feasible region.", "answerGuide": "Draw each boundary and label the common allowed half-planes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q05c", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the objective for making as many toys as possible.", "answerGuide": "Maximize the sum of the two toy counts.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q05d", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(d)", "marks": 7, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find all feasible vertices and the production plan with most toys.", "answerGuide": "Solve boundary intersections, evaluate the objective at each vertex, and choose the largest.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q06a", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete early and late event times on the project network.", "answerGuide": "Use a forward pass for early times and backward pass for late times.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q06b", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the minimum project time.", "answerGuide": "Read the critical-path duration from the earliest finish time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q06c", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find a lower bound on workers for the minimum project time.", "answerGuide": "Divide total activity-hours by project duration and round up.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q06d", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw the project cascade chart.", "answerGuide": "Schedule each activity within its time window while observing precedence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q06e", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Show a three-worker schedule finishing one hour after the minimum time.", "answerGuide": "Avoid worker overlaps, respect precedence, and finish at the required time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q07a", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate the milkshake profit problem and simplified constraints.", "answerGuide": "Define sales variables, write the profit objective, translate and simplify each condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WDM11-01A-q07b", "paperId": "ial-2026-1-WDM11-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "With two sales quantities fixed, find minimum and maximum profits.", "answerGuide": "Find the allowed range for the remaining quantity and evaluate profit at both endpoints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1A_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q01a", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand the given function through the cubic term.", "answerGuide": "Use a binomial expansion and simplify coefficients through x cubed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q01b", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use part (a) to expand g through x cubed.", "answerGuide": "Substitute the first series into g and retain terms through x cubed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q01c", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use part (a) to expand h through x cubed.", "answerGuide": "Apply the stated transformation and collect terms through x cubed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q02", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": null, "marks": 7, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find both points on the implicit curve with the stated tangent gradient.", "answerGuide": "Differentiate implicitly, set the gradient, solve with the curve equation, and give both points.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q03a", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve the shrub-growth differential equation with the initial condition.", "answerGuide": "Separate and integrate, apply h(1)=1, and find the constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q03b", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the limiting height predicted by the model.", "answerGuide": "Take the height function's limit as time tends to infinity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q04", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": null, "marks": 7, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the rational definite integral exactly in the required logarithmic form.", "answerGuide": "Decompose or substitute, integrate, apply bounds, and simplify the logarithms.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q05", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find constants in two vector line equations from intersection and perpendicularity.", "answerGuide": "Use the intersection parameter to equate positions, then set direction-vector dot product to zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q06a", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Establish the exact logarithmic integral using u=2x−1.", "answerGuide": "Substitute, integrate the resulting terms, and evaluate the exact limits.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=16", "qpPages": [ 4, 4 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q06b", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use part (a) to find the exact volume of revolution.", "answerGuide": "Set up the disc integral πy² and apply the established result over the stated bounds.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=16", "qpPages": [ 4, 4 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q07a", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "differentiation", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the specified point on the parametric curve.", "answerGuide": "Substitute its parameter into both coordinate equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q07b", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "differentiation", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the curve's normal line at that point.", "answerGuide": "Form dy/dx from parameter derivatives, take its negative reciprocal, and use point-gradient form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q07c", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "coordinate-geometry", "differentiation", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Show the enclosed area equals the given parameter integral.", "answerGuide": "Combine line and curve contributions using dx/dθ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q07d", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "coordinate-geometry", "differentiation", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Evaluate the area integral exactly.", "answerGuide": "Integrate the trigonometric terms over the stated parameter interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=18", "qpPages": [ 5, 5 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q08a", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Factor the given difference of squares.", "answerGuide": "Use A²−B²=(A−B)(A+B).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=21", "qpPages": [ 6, 6 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q08b", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove by contradiction there are no positive integer solutions.", "answerGuide": "Assume a solution, use the factorization, and derive an impossible divisibility or parity condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=21", "qpPages": [ 6, 6 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q09a", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parallelogram's missing position vector.", "answerGuide": "Use a+c=b+d and rearrange.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=23", "qpPages": [ 6, 6 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q09b", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the angle between two adjacent sides.", "answerGuide": "Use the dot product and side-vector magnitudes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=23", "qpPages": [ 6, 6 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q09c", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parallelogram's area.", "answerGuide": "Take the magnitude of the cross product of adjacent sides.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=23", "qpPages": [ 6, 6 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WMA14-01A-q09d", "paperId": "ial-2026-1-WMA14-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01A", "variant": "Paper A · China regional", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the related trapezium area from the parallelogram area.", "answerGuide": "Use the midpoint condition to relate parallel sides and scale the area.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4A_MS.pdf#page=23", "qpPages": [ 6, 6 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q01a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find P's speed after the head-on collision.", "answerGuide": "Apply impulse equals change in momentum with a consistent direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q01b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Decide whether P reverses direction.", "answerGuide": "Compare the signed final velocity with the initial direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q01c", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find Q's post-collision speed.", "answerGuide": "Use momentum conservation and P's final velocity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q02", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": null, "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the strap tension as the suitcase moves steadily up a rough floor.", "answerGuide": "Resolve vertically for reaction, find friction, and balance horizontal forces.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q03a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the launch height from the first ball's motion and impact time.", "answerGuide": "Use vertical displacement under gravity over five seconds and set final height to zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q03b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the second ball's downward launch speed from the impact-speed relation.", "answerGuide": "Find the first impact speed, apply the stated ratio, then use constant-acceleration motion over the same drop.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q04a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Give the direction of motion as a bearing.", "answerGuide": "Use east/north velocity components and convert to a clockwise angle from north.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q04b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Write the position vector at time t.", "answerGuide": "Add t times velocity to the initial position vector.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q04c", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find when the particle is northwest of the origin.", "answerGuide": "Set coordinate magnitudes equal with opposite signs and solve for positive time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q05a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch both speed-time graphs from A to B.", "answerGuide": "Show rising acceleration segments followed by constant-speed segments.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q05b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the car's acceleration duration.", "answerGuide": "Use the area under its speed-time graph for the 816 m journey.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q05c", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the van's cruising speed.", "answerGuide": "Match its triangular and rectangular graph areas to 816 m.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q06a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the towbar thrust while braking.", "answerGuide": "Apply Newton's second law to the trailer with resistance and deceleration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q06b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the car's mass.", "answerGuide": "Apply Newton's second law to the car with brake, resistance, and towbar forces.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q06c", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the time until the car stops.", "answerGuide": "Use constant-deceleration equations with initial speed inferred from stopping distance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q07a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the rod's centre of mass has the stated distance from A.", "answerGuide": "Use vertical force balance and a moments equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=15", "qpPages": [ 8, 8 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q07b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the new reaction ratio after moving a support and adding a particle.", "answerGuide": "Use force balance and moments for the changed setup, then solve the ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=15", "qpPages": [ 8, 8 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q08a", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(a)", "marks": 8, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the string tension while both particles move.", "answerGuide": "Find friction, write each motion equation, and eliminate acceleration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=18", "qpPages": [ 9, 9 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q08b", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show Q's speed just before it reaches the plane.", "answerGuide": "Use constant-acceleration kinematics over the drop.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=18", "qpPages": [ 9, 9 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME01-01A-q08c", "paperId": "ial-2026-1-WME01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01A", "variant": "Paper A · China regional", "questionNumber": 8, "questionPart": "8(c)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Prove the string-length condition for P to stop before the pulley.", "answerGuide": "Find P's deceleration after Q lands, calculate further travel, and combine distances.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1A_MS.pdf#page=18", "qpPages": [ 9, 9 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q01a", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the magnitude of the impulse changing the particle's velocity.", "answerGuide": "Multiply velocity change by mass, then take the vector magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q01b", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the angle between the impulse and initial motion.", "answerGuide": "Use the vector dot product divided by the product of magnitudes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q02a", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find resistance as the cyclist climbs at steady speed and power.", "answerGuide": "Resolve along the slope and equate power to force times speed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q02b", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the cyclist's instantaneous downhill acceleration.", "answerGuide": "Use power/speed for driving force and apply Newton's second law along the slope.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q03a", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the lamina centre-of-mass distances from OD and OA.", "answerGuide": "Split the lamina into simple shapes, establish their areas and centroid coordinates, then use weighted distances for both axes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q03b", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the equilibrium inclination of CB when suspended at C.", "answerGuide": "The centre of mass lies below C; use its coordinates and a trigonometric ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q04a", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the velocity is parallel to the specified vector.", "answerGuide": "Differentiate position and equate component ratios to the direction ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q04b", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration magnitude at that time.", "answerGuide": "Differentiate velocity, substitute the time, then find the vector magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q05a", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the force expression for the hinged rod and attached particle.", "answerGuide": "Take moments about the hinge and substitute the stated tangent ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q05bi", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the horizontal hinge-force component.", "answerGuide": "Resolve the applied force horizontally and balance horizontal forces.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q05bii", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the vertical hinge-force component.", "answerGuide": "Balance vertical forces including both weights and the applied-force component.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q05c", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the hinge-force direction to find the mass parameter.", "answerGuide": "Set the hinge components equal in magnitude for the 45-degree direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q06ai", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show B's speed after the direct collision using momentum and restitution.", "answerGuide": "Write momentum conservation and restitution equations and solve simultaneously.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q06aii", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 1, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find A's speed after that collision.", "answerGuide": "Substitute B's speed into momentum conservation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q06b", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find where the particles collide again after B rebounds from the wall.", "answerGuide": "Apply wall restitution, write positions after the first impact, and solve for the second meeting.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q07a", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the projectile reaches maximum height.", "answerGuide": "Set vertical velocity to zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q07b", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the positive constant locating the stated point on the path.", "answerGuide": "Use both displacement equations at the same time and the equal-coordinate condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q07c", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the other position at the same height.", "answerGuide": "Use the symmetric vertical-motion time and substitute it into horizontal motion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME02-01A-q07d", "paperId": "ial-2026-1-WME02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when velocity is perpendicular to the stated vector.", "answerGuide": "Set the velocity-vector dot product to zero and solve for time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_M2A.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q01", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the force at a stated distance when speed depends on position.", "answerGuide": "Use a=v(dv/dx), differentiate the speed relation, and apply F=ma.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q02a", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show the inverse-square force constant from the surface weight.", "answerGuide": "Set K/R² equal to mg at the surface.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q02b", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 7, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the spacecraft's impact speed from its speed at three Earth radii.", "answerGuide": "Apply work-energy for the inverse-square force between 3R and R.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q03a", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the SHM period from a displacement, speed, and acceleration state.", "answerGuide": "Use a=−ω²x to find ω, then calculate 2π/ω.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q03b", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the SHM amplitude.", "answerGuide": "Use v²=ω²(A²−x²) at the supplied state.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q03c", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find time per cycle within half the amplitude of equilibrium.", "answerGuide": "Find the angular range for |x|≤A/2 and convert its period fraction to time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=12", "qpPages": [ 4, 4 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q04a", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the particle's weight from elastic-string equilibrium.", "answerGuide": "Use geometry for extension, Hooke's law for tension, and vertical resolution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q04b", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find speed after the specified fall.", "answerGuide": "Use energy conservation with gravity and elastic strain energy at both configurations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=14", "qpPages": [ 5, 5 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q05a", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "integration", "energy-momentum" ], "primaryTopicId": "integration", "summary": "Show the volume of the solid formed by rotating the sine-bounded region.", "answerGuide": "Integrate πsin²x across the interval and use a double-angle identity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q05b", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "integration", "energy-momentum" ], "primaryTopicId": "integration", "summary": "Find the solid's centre-of-mass x-coordinate in terms of π.", "answerGuide": "Calculate the first moment using xπsin²x and divide by volume.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=16", "qpPages": [ 6, 6 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q06a", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the motorcycle speed on a banked track without sideways friction.", "answerGuide": "Resolve the normal force vertically and radially to provide centripetal acceleration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q06b", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 8, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the limiting speed when sideways friction is present.", "answerGuide": "Set friction to μR and resolve vertically and toward the circle's centre.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=19", "qpPages": [ 7, 7 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q07a", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the minimum launch-speed condition for a complete vertical circle on a rigid rod.", "answerGuide": "Use energy between launch and the limiting position under the rod constraint.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=22", "qpPages": [ 8, 8 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01A-q07b", "paperId": "ial-2026-1-WME03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 11, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use least and greatest rod tensions to derive the launch-speed relation.", "answerGuide": "Write radial equations at top and bottom, relate speeds by energy, and apply the tension ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3A_MS.pdf#page=22", "qpPages": [ 8, 8 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q01a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Evaluate F(5).", "answerGuide": "Add probabilities for outcomes no greater than five.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q01b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected value.", "answerGuide": "Sum xP(X=x) over the table.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q01c", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance.", "answerGuide": "Calculate E(X²)−[E(X)]².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q02a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the median press-up count for each year group.", "answerGuide": "Order the data and take the central observation(s).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q02b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find Year 7's lower and upper quartiles.", "answerGuide": "Use the stated class data's quartile positions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q02c", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Compare skewness using medians and quartiles.", "answerGuide": "Compare lower and upper quartile gaps around each median.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q02d", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Give two reasons a normal model is unsuitable for Year 11 counts.", "answerGuide": "Refer to discreteness, non-negativity, bounds, or visible asymmetry.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q03a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate Svs and Sss from the coded sample totals.", "answerGuide": "Use Svs=Σvs−ΣvΣs/n and Sss=Σs²−(Σs)²/n.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q03b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate product-moment correlation.", "answerGuide": "Divide Svs by √(SvvSss).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q03c", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Judge whether correlation supports a linear model.", "answerGuide": "Use the coefficient's sign and strength as evidence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q03d", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the regression line of s on v.", "answerGuide": "Use gradient Svs/Svv and intercept s̄−b v̄.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q03e", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Convert the coded line to sales y against advertising x.", "answerGuide": "Substitute the code definitions and rearrange.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q03f", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(f)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Choose whether extra spending should promote textbooks or novels.", "answerGuide": "Compare marginal sales gains and note which model applies to each category.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q04a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(S=1), where selection stops only after both blue socks have appeared.", "answerGuide": "A single draw cannot contain both socks, so P(S=1)=0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q04b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability more than two socks are selected.", "answerGuide": "Enumerate first-two-draw outcomes without both blue socks.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q04c", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability stopping occurs on draw three.", "answerGuide": "Count sequences where the second blue sock first appears on draw three.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q04d", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Given draw two is blue, find the probability exactly three draws are needed.", "answerGuide": "Condition on blue at draw two and determine the colour of draw one.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q04e", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(e)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability all five socks are drawn before the second blue appears.", "answerGuide": "Use the position distribution of the later of the two blue socks.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q05a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the histogram width and height for the lightest class.", "answerGuide": "Use the reference-bar scale and frequency density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q05b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Explain why a histogram suits these data.", "answerGuide": "State that weights are continuous and grouped into intervals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q05c", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate grouped mean and standard deviation.", "answerGuide": "Use class midpoints for the first and second moments.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q05d", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median by interpolation.", "answerGuide": "Locate the median class and interpolate from cumulative frequency.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q05e", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(e)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the probability of a weight between 1.1 g and 1.6 g.", "answerGuide": "Use the included grouped frequencies, interpolating at a partial class.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q05f", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(f)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "State how a corrected bean weight changes the estimates in part (c).", "answerGuide": "Compare the correction with the original and centre to explain the direction of each change.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q06a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the jam-weight mean and standard deviation from two normal tail probabilities.", "answerGuide": "Standardize each boundary, use normal quantiles, and solve the pair of equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q06b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Amy's expected profit or loss per jar.", "answerGuide": "Weight each amount by the probability of its interval and sum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q07a", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(B) from the given conditional and union information.", "answerGuide": "Use P(A∩B)=P(A|B)P(B) with the complement identity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q07b", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(A′|B′).", "answerGuide": "Apply conditional probability to the complementary events.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q07c", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(B∩C).", "answerGuide": "Use independence to multiply P(B) and P(C).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01A-q07d", "paperId": "ial-2026-1-WST01-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete a three-event Venn diagram with region probabilities.", "answerGuide": "Apply mutual exclusion and independence, then fill each intersection and remaining region.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1A_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q01a", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the machine mean for a 5% chance that a rod is under 200 cm.", "answerGuide": "Standardize 200 using the lower-tail normal quantile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q01b", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the chance at least three of eight rods are short.", "answerGuide": "Use a binomial model with p=0.05 and evaluate its upper tail.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q01c", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the chance more than five of sixty rods are short.", "answerGuide": "Use a suitable approximation with mean np=3 and find six or more.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q02ai", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Write P(X=3) for the binomial variable in terms of p.", "answerGuide": "Use the binomial point-probability formula.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q02aii", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find p when P(X=3) is sixteen times P(X=7).", "answerGuide": "Form and simplify the probability ratio, then solve for valid p.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q02b", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find λ when P(Y=3) is five times P(Y=5).", "answerGuide": "Form the Poisson probability ratio and solve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q02c", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find n and the variance in the stated normal approximation.", "answerGuide": "Match np to the mean and use np(1−p) for variance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=2", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q03a", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find a with probability 0.4 above it under the given CDF.", "answerGuide": "Set F(a)=0.6 on the correct interval and solve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q03bi", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find E(X) from the CDF.", "answerGuide": "Differentiate to get the density, integrate xf(x), and evaluate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q03bii", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find Var(X).", "answerGuide": "Calculate E(X²) and subtract [E(X)]².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=11", "qpPages": [ 3, 3 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q04ai", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses for a two-sided test of the 35% claim.", "answerGuide": "Set H0:p=0.35 and H1:p≠0.35.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q04aii", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find a 5% critical region for the germination count.", "answerGuide": "Use B(15,0.35) and choose integer tails nearest 2.5% each.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q04b", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Decide whether eight germinations support the claim.", "answerGuide": "Check whether 8 lies in the critical region and state a contextual conclusion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q04c", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State the actual significance level.", "answerGuide": "Add probabilities in the selected tails.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=3", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=13", "qpPages": [ 3, 3 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q05a", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "State the distribution of the longer piece after a random cut.", "answerGuide": "Use symmetry to identify the longer-length range and density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q05b", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability the longer piece rounds to 28 cm.", "answerGuide": "Translate rounding to an interval for L.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q05c", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the chance the larger square area is below 64 cm².", "answerGuide": "Turn the area condition into a bound on L and use its distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q05d", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the chance the square areas differ by more than 81 cm².", "answerGuide": "Express the difference using L and 40−L, solve the inequality, and evaluate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=4", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=15", "qpPages": [ 4, 4 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q06a", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the piecewise density over its domain.", "answerGuide": "Plot each formula on its interval and show zero outside support.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q06b", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the mode.", "answerGuide": "Identify where density is greatest.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q06c", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show P(X>2)=0.8.", "answerGuide": "Integrate the upper portion of the density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q06d", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Define the CDF fully.", "answerGuide": "Integrate the density on each interval and give values outside support.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q06e", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(e)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find F(a) from the conditional probability.", "answerGuide": "Rewrite the conditional event using probabilities above 2 and below a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q06f", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(f)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a using the CDF.", "answerGuide": "Use the correct piece and check its interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=5", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=17", "qpPages": [ 5, 5 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q07a", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the least count threshold whose two-month Poisson upper tail is below 0.05.", "answerGuide": "Use the two-month Poisson mean and test integer tails.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q07b", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the least time with probability of no sightings below 0.05.", "answerGuide": "Solve exp(−m/2)<0.05.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q07c", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance of at least one sighting in each of three months.", "answerGuide": "Calculate the one-month probability and cube it by independence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q07d", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the chance the eight-month count equals its expectation.", "answerGuide": "Evaluate the Poisson point probability at its mean count.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01A-q07e", "paperId": "ial-2026-1-WST02-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(e)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Given four sightings in four months, find the chance more occurred in the last half.", "answerGuide": "Condition on the total; the equal-half split is binomial with p=1/2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2A_QP.pdf#page=6", "msUrl": "https://edemat.alevelclouds.com/26_01_MS_S2A.pdf#page=20", "qpPages": [ 6, 6 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q01a", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Calculate Spearman's rank correlation for distance and commission.", "answerGuide": "Rank each variable, find rank differences, and apply the coefficient formula.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q01b", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% for positive rank correlation.", "answerGuide": "State one-sided hypotheses, compare with the critical value, and conclude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q02a", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Show the sample mean is biased for α and state its bias.", "answerGuide": "Find the uniform mean from endpoints and compare E(X̄) with α.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q02b", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find k so kX̄ is unbiased for α.", "answerGuide": "Choose k so kE(X̄)=α.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q02c", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the upper endpoint using the sample.", "answerGuide": "Use the unbiased estimator and rearrange the uniform mean relation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q03a", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(a)", "marks": 10, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% for association between test result and test centre.", "answerGuide": "State hypotheses, find expected counts and the statistic, then compare with the correct critical value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q03b", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Recommend a centre for follow-up and give a reason.", "answerGuide": "Identify the largest departure from expectation and relate it to the observed results.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=8", "qpPages": [ 3, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q04a", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Construct a 95% confidence interval for mean mug mass.", "answerGuide": "Use the sample mean and known standard deviation with the normal critical value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q04b", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the chance more than three of twenty intervals miss the mean.", "answerGuide": "Use a binomial model with miss probability 0.05 and its upper tail.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q05a", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find missing observed and expected frequencies for five births.", "answerGuide": "Use the total weeks and the Poisson model with mean 2.8.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q05b", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at 5% whether a Poisson model fits the birth counts.", "answerGuide": "Combine cells if required, compute the chi-squared statistic and degrees of freedom, then compare.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06a", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Describe a proportionate stratified sample of 200 staff.", "answerGuide": "Allocate by the 6000:4000 group sizes and select randomly within each stratum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06b", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Give an advantage of stratified sampling.", "answerGuide": "Explain that both staff groups are guaranteed representation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06c", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(c)", "marks": 7, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test at 1% whether the two groups have different mean scores.", "answerGuide": "State two-sided hypotheses, calculate the standard error and statistic, and compare with the critical value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06d", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Explain the Central Limit Theorem's role.", "answerGuide": "Relate large independent samples to approximate normality of the sample-mean difference.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06e", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(e)", "marks": 1, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "State an assumption for the two-sample test.", "answerGuide": "Name a relevant condition such as independent random samples.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06f", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(f)", "marks": 2, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Interpret z=2.53 at the 1% level.", "answerGuide": "Compare its absolute value with the two-sided critical value and conclude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q06g", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 6, "questionPart": "6(g)", "marks": 1, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Interpret the pre- and post-training results together.", "answerGuide": "State what the tests show about differences without overclaiming causation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q07ia", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(i)(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the chance three independent task times total more than 90 minutes.", "answerGuide": "Add the means and variances to form the total-time distribution, then standardize.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q07ib", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(i)(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find P(A>B) for the two task times.", "answerGuide": "Form A−B with the difference of means and sum of variances, then evaluate above zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01A-q07ii", "paperId": "ial-2026-1-WST03-01A", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01A", "variant": "Paper A · China regional", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find k for the chance an observation exceeds its four-value sample mean by kσ.", "answerGuide": "Rewrite using X1−X̄, find its mean and variance, standardize, and use the 0.1 tail quantile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3A_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2025-w-11-q01", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1: Find the parameter values for which the stated quadratic equation has two distinct real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the discriminant greater than zero, simplify to a quadratic inequality in k, and solve to obtain k < 4 or k > 16." }, { "id": "cie-2025-w-11-q02a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "2(a): Use the second and fifth terms of the geometric progression to find the angle whose cosine is its common ratio.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Divide the fifth term by the second to get cos³θ = 1/64, so cos θ = 1/4 and θ ≈ 1.32 radians." }, { "id": "cie-2025-w-11-q02b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "2(b): Find the exact sum to infinity of the geometric progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use cos θ = 1/4 and the second term to find a = 32, then apply S∞ = a/(1 − r) to obtain 128/3." }, { "id": "cie-2025-w-11-q03", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "3: Compare the x⁴ terms in two binomial expansions to find the positive constant p from their stated coefficient.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Obtain the x⁴ contributions from each expansion, combine them according to the given difference, and solve the resulting quadratic in p²; retain the positive value p = 2." }, { "id": "cie-2025-w-11-q04a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "4(a): Rewrite the given quadratic in completed-square form a − (x + b)².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to obtain 10 − (x + 3)², so a = 10 and b = 3." }, { "id": "cie-2025-w-11-q04b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "4(b): Describe the reflection and translation that transform y = x² into the stated quadratic graph, and find m and n.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Reflect in the x-axis, then translate by (−3, 10); hence m = −3 and n = 10." }, { "id": "cie-2025-w-11-q05a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Show the stated identity by expressing the powers of tan θ using sine and cosine.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite tan⁴θ using sin⁴θ/cos⁴θ, use sin²θ = 1 − cos²θ, and simplify to the required expression." }, { "id": "cie-2025-w-11-q05b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Use the identity from part (a) to solve the trigonometric equation for 0° < θ < 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the identity to form an equation in cos²θ, solve for cos θ, and retain the two angles in the interval: approximately 70.5° and 109.5°." }, { "id": "cie-2025-w-11-q06a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "6(a): State the range of f for its specified domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The minimum occurs at the domain endpoint x = 0, giving f(x) ≥ −3; the range is [−3, ∞)." }, { "id": "cie-2025-w-11-q06b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "6(b): Find an expression for the inverse function f⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set y = (x + 3)² − 12, take the positive square root because x ≥ 0, and rearrange to f⁻¹(y) = √(y + 12) − 3." }, { "id": "cie-2025-w-11-q06c", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "6(c): Solve the equation involving the composite function gf(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute f(x) into g to get 2((x + 3)² − 12) − 5 = 69. Solve the quadratic and use x ≥ 0 to select x = 4." }, { "id": "cie-2025-w-11-q07a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "7(a): Show that the area of the circular segment is approximately 0.614r² cm².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the isosceles triangle area from the sector area using angle 2π/3, then simplify to approximately 0.614r²." }, { "id": "cie-2025-w-11-q07bi", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 3, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "7(b)(i): Find the rate of increase of the segment area when r = 20 and dr/dt = 0.4 cm s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate A = 0.61418…r², apply dA/dt = (dA/dr)(dr/dt), and evaluate at r = 20 to get 9.8 cm² s⁻¹." }, { "id": "cie-2025-w-11-q07bii", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 3, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "7(b)(ii): Find the rate of increase of the arc AB length when the radius grows at 0.4 cm s⁻¹.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use l = (2π/3)r and differentiate with respect to time to obtain (2π/3)(0.4) ≈ 0.84 cm s⁻¹." }, { "id": "cie-2025-w-11-q08a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(a): Find the shaded area bounded by the square-root curve and the given straight lines.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve over the stated x-interval and subtract that area from the bounding rectangle, giving 9/2 cm²." }, { "id": "cie-2025-w-11-q08b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "8(b): Find the exact volume formed by rotating the shaded region through 360° about the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express x² in terms of y and integrate πx² with respect to y over the stated limits; this gives 243π/10 cm³." }, { "id": "cie-2025-w-11-q09a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "9(a): Use the given partial sums of an arithmetic progression to find its common difference d.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write S₂ − 1, S₄ and S₉ in terms of d, equate them as consecutive terms of an arithmetic progression, and solve to get d = −1/5." }, { "id": "cie-2025-w-11-q09b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "9(b): Find the difference between the fifteenth terms of the two arithmetic progressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use d = −1/5 to find the second progression’s first term and difference, calculate both fifteenth terms, and report their difference, 59.6." }, { "id": "cie-2025-w-11-q10a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "10(a): Find the coordinates where the stated line intersects the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the line into the circle equation and solve the resulting quadratic to obtain (3, 2) and (5, −2)." }, { "id": "cie-2025-w-11-q10b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "10(b): Find the area of the triangle formed by the circle’s centre and the two intersection points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the centre (0, −2), the chord AB as the base, and the perpendicular height to obtain an area of 10 square units." }, { "id": "cie-2025-w-11-q11a", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(a): Find the equation of the normal to the curve at P(4, 3).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = 4 into dy/dx to get tangent gradient 1/10, use normal gradient −10, and form the line through (4, 3): y = −10x + 43." }, { "id": "cie-2025-w-11-q11b", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(b): Find the rate of change of the curve’s gradient at x = 4.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given dy/dx to find d²y/dx², then substitute x = 4 to obtain 7/100." }, { "id": "cie-2025-w-11-q11c", "paperId": "cie-2025-w-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(c): Use the point P and the curve’s derivative to find q when the curve passes through (−1, q).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_11.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_11.pdf", "qpPages": [ 17, 17 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate dy/dx, use (4, 3) to determine the constant of integration, then substitute x = −1 to obtain q = 11." }, { "id": "cie-2025-w-12-q01a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(a): Write the quadratic in the stated completed-square form and identify its constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to obtain 9(x − 2)² − 28, so p = 9, q = −2 and r = −28." }, { "id": "cie-2025-w-12-q01b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(b): Use the completed-square form to find when the quadratic equation has no real roots.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The minimum value is −28, so the equation has no real roots when k < −28." }, { "id": "cie-2025-w-12-q01c", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "1(c): Find the exact roots of the related quadratic equation with the stated right-hand side.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the completed-square form, isolate the squared term and solve to obtain x = 2 ± √13/3." }, { "id": "cie-2025-w-12-q02", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "2: Find the term independent of x in the given finite binomial expansion with positive and negative powers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the general binomial term and select the index that makes the total power of x zero; evaluate that term to obtain 4860." }, { "id": "cie-2025-w-12-q03a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "3(a): Describe the horizontal scaling and vertical translation that map y = f(x) to y = f(3x) + 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply a horizontal stretch with scale factor 1/3, then translate by (0, 2)." }, { "id": "cie-2025-w-12-q03b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "3(b): Write the equation after stretching y = g(x) vertically by factor 3 and reflecting in the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The vertical stretch multiplies outputs by 3 and reflection replaces x by −x, giving y = 3g(−x)." }, { "id": "cie-2025-w-12-q04a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "4(a): Use the curve’s gradient at S(2, 20) to determine the constant k.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the known x-coordinate and gradient into dy/dx to find k = 4." }, { "id": "cie-2025-w-12-q04b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "4(b): Integrate the curve’s derivative and use S to find the ordinate t at x = 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate dy/dx term by term with k = 4, use (2, 20) to find the constant of integration, then substitute x = 1 to obtain t = 4." }, { "id": "cie-2025-w-12-q05a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "5(a): Find the x-coordinate a of the maximum point on the stated radical curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and set dy/dx = 0 (or complete the square after substitution) to obtain a = 4." }, { "id": "cie-2025-w-12-q05b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "5(b): Use the positive x-intercept and maximum point to find the exact shaded area under the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the positive intercept b = 16, integrate between x = 4 and x = 16, and evaluate exactly to get 88/3 square units." }, { "id": "cie-2025-w-12-q06a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Sketch y = 3 sin x + 2 over one full cycle from 0 to 2π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw one sine cycle starting at y = 2 and increasing, with maximum 5 at π/2 and minimum −1 at 3π/2." }, { "id": "cie-2025-w-12-q06bi", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b)(i): Count the solutions of 3 sin x + 2 = x over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the sketch to count one intersection in the interval." }, { "id": "cie-2025-w-12-q06bii", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b)(ii): Count the solutions of 3 sin x + 2 = 5 − x over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the sketch to count three intersections in the interval." }, { "id": "cie-2025-w-12-q06c", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(c): Solve the trigonometric equation involving 3 sin x + 2 and 5 cos²x − 1 for 0 ≤ x ≤ 2π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Replace cos²x by 1 − sin²x, solve the resulting quadratic for sin x, and list the three solutions in range: approximately 0.41, 2.73 and 3π/2." }, { "id": "cie-2025-w-12-q07a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(a): Find the equation of the circle whose diameter has endpoints P(1, 1) and Q(7, 11).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the midpoint centre (4, 6) and radius √34, then write (x − 4)² + (y − 6)² = 34." }, { "id": "cie-2025-w-12-q07b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(b): Find the equation of the tangent to the circle at Q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The diameter slope is 5/3, so the tangent slope is −3/5; use Q(7, 11) to form the line." }, { "id": "cie-2025-w-12-q07c", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "7(c): Find where the tangents at Q and at the other point R with x-coordinate 7 intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find R = (7, 1), write the tangent there, and solve with the tangent from part (b) to obtain (46/3, 6)." }, { "id": "cie-2025-w-12-q08a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "8(a): Use the geometric and arithmetic progression conditions to prove the stated quadratic identity in a and c.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 12, 12 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use b² = ac for the geometric progression and 2b = a − 3c for the arithmetic progression; eliminate b and simplify to a² − 10ac + 9c² = 0." }, { "id": "cie-2025-w-12-q08bi", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 5, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "8(b)(i): With a = 9 and the smaller admissible c, find the exact sum to infinity of the geometric progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 12, 12 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve the identity to choose c = 1, then b = 3 and r = 1/3; apply S∞ = a/(1 − r) to obtain 27/2." }, { "id": "cie-2025-w-12-q08bii", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "8(b)(ii): Find the sum of the first 20 terms of the arithmetic progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 13, 13 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The arithmetic progression begins 9, 3, −3, so d = −6. Apply S₂₀ = 20/2[2(9) + 19(−6)] to obtain −960." }, { "id": "cie-2025-w-12-q09a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "9(a): Differentiate f and use the derivative to determine whether f is increasing or decreasing on its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 14, 14 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the quotient rule and show f′(x) < 0 for x > 2, so f is decreasing." }, { "id": "cie-2025-w-12-q09b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "9(b): State whether the inverse of f exists and justify your answer.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 14, 14 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Yes. The strictly decreasing function is one-to-one on its domain, so f⁻¹ exists." }, { "id": "cie-2025-w-12-q09c", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "9(c): State the range of g(x) = 4x − 3 for x > a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 15, 15 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Since x > a, the range is (4a − 3, ∞)." }, { "id": "cie-2025-w-12-q09d", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "9(d): Find the values of a for which the composite function fg exists.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 15, 15 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Require every value in the range of g to lie in the domain x > 2 of f; this gives 4a − 3 ≥ 2, hence a ≥ 5/4." }, { "id": "cie-2025-w-12-q10a", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(a): Use the given circle and chord geometry to show that s = √3r.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 16, 16 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the equal chord lengths and the cosine rule (or the right-triangle relation) to show s² = 3r², hence s = √3r." }, { "id": "cie-2025-w-12-q10b", "paperId": "cie-2025-w-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 7, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "10(b): Find an exact expression for the shaded area in terms of r, in the requested two-constant form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_12.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_12.pdf", "qpPages": [ 17, 17 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the central angles to calculate the relevant sectors, subtract the corresponding triangle areas, and substitute s = √3r to express the shaded region exactly as a multiple of r²." }, { "id": "cie-2025-w-13-q01a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "1(a): Expand the stated binomial in ascending powers of x through the cubic term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial expansion and retain the constant, x, x² and x³ terms in ascending order." }, { "id": "cie-2025-w-13-q01b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "1(b): Use the expansion from part (a) to find the specified x² coefficient in the product.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Collect the contributions to x² from the relevant terms of the two factors and add them." }, { "id": "cie-2025-w-13-q02a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(a): Solve the equation involving the inverse tangent of a linear expression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply tangent to both sides, then solve the resulting linear equation for x." }, { "id": "cie-2025-w-13-q02b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2(b): Solve the stated sine-cosine equation for θ over the given interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Replace cos²θ with 1 − sin²θ, solve the quadratic in sin θ, and list every angle in the interval." }, { "id": "cie-2025-w-13-q03a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "3(a): Use the curve equation to find the missing ordinate k at the specified nearby x-coordinate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = 8.001 into f(x) = x³(x − 2)² and round the result to six decimal places." }, { "id": "cie-2025-w-13-q03b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "3(b): Calculate the gradient of chord AD from the given point coordinates.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the difference quotient between A and D and give the result to four decimal places." }, { "id": "cie-2025-w-13-q03c", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "3(c): Infer the value of f′(8) suggested by the listed nearby chord gradients.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "As the chord endpoint approaches x = 8, the gradients approach 30; state f′(8) = 30." }, { "id": "cie-2025-w-13-q04a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(a): Assuming the first three terms form an arithmetic progression, find its thirtieth term.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Equate the consecutive differences to find the common difference, then use the nth-term formula with n = 30." }, { "id": "cie-2025-w-13-q04b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "4(b): Instead assume the first three terms form a geometric progression and find its sum to infinity.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use equality of consecutive ratios to determine the common ratio, check |r| < 1, and apply S∞ = a/(1 − r)." }, { "id": "cie-2025-w-13-q05a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Find the exact central angle subtended by the given chord of a circle of radius 4 cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the chord formula 4√3 = 2(4)sin(θ/2), giving θ = 2π/3 radians." }, { "id": "cie-2025-w-13-q05b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Find the exact area of the major circular segment cut off by the chord.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the minor segment (sector area minus triangle area) from the full circle area, using θ = 2π/3." }, { "id": "cie-2025-w-13-q06a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "6(a): Describe, in order, a sequence of transformations mapping the displayed cubic parent graph to y = f(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Read the horizontal and vertical changes from the graph, then state the stretch/reflection and translation in the required order." }, { "id": "cie-2025-w-13-q06b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "6(b): Use the displayed transformed graph to identify the constants in the given cubic form for f(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Match the graph’s horizontal shift, vertical shift and scale to the constants a, b and c in the stated function form." }, { "id": "cie-2025-w-13-q07", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7", "marks": 4, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "7: Find the inverse of the rational function g and verify the stated self-inverse property when a = 6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set y = g(x), interchange x and y, and rearrange. Substitute a = 6 and show the resulting inverse equals g." }, { "id": "cie-2025-w-13-q08a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(a): Find the equation of the circle whose diameter endpoints are (6, 1) and (2, 7).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 9, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the midpoint centre (4, 4) and radius √13, then use the standard circle equation." }, { "id": "cie-2025-w-13-q08b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "8(b): Find the exact x-coordinates of the points where tangents with the stated gradient touch the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 9, 9 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the circle equation and the condition that a radius is perpendicular to the tangent; solve for both contact points’ x-coordinates." }, { "id": "cie-2025-w-13-q09", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9", "marks": 8, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "9: Find the exact volume formed by revolving the shaded region between the curve and y = 4.5 about the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 10, 11 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find both curve-line intersections for the integration limits, then integrate π times the difference of the squared radii and evaluate exactly." }, { "id": "cie-2025-w-13-q10ai", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)(i)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "10(a)(i): Complete the square for the quadratic when p = 2 and q = 10.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 12, 12 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Collect the x-terms and write f(x) in the form a(x + b)² + c by completing the square." }, { "id": "cie-2025-w-13-q10aii", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)(ii)", "marks": 1, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "10(a)(ii): State the range of the completed-square quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 12, 12 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the vertex value as the minimum or maximum, depending on the sign of the squared-term coefficient." }, { "id": "cie-2025-w-13-q10b", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "10(b): Use the two specified roots to find the constants p and m when q = 5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 13, 13 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the sum and product of roots to the quadratic, equate them to the coefficient relations, and solve for p and m." }, { "id": "cie-2025-w-13-q11a", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(a): Find where the tangent to the curve at (3, 5) intersects the line y = 8x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 14, 14 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate to find the tangent gradient at x = 3, form the tangent through (3, 5), and solve with y = 8x." }, { "id": "cie-2025-w-13-q11bi", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)(i)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(b)(i): Find the x-coordinates of all stationary points of the curve.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 15, 15 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set dy/dx = 0 and solve for each stationary x-coordinate." }, { "id": "cie-2025-w-13-q11bii", "paperId": "cie-2025-w-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)(ii)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "11(b)(ii): Find the second derivative and use it to classify the stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_13.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_13.pdf", "qpPages": [ 15, 15 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate again to obtain d²y/dx², evaluate it at each stationary x-coordinate, and classify each point from the sign." }, { "id": "cie-2025-w-15-q01a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "1(a): State the equation of a circle with the given centre and radius.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the centre (2, 6) and radius 10 into (x − h)² + (y − k)² = r²." }, { "id": "cie-2025-w-15-q01b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "1(b): Find both possible values of k for the point (8, k) on the circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = 8 into the circle equation and solve the resulting equation for both y-values." }, { "id": "cie-2025-w-15-q02a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2(a): Find the common ratio of the geometric progression with the stated surd first and second terms.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Divide the second term by the first, rationalize and simplify to the required form 2 + p." }, { "id": "cie-2025-w-15-q02b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "2(b): Find the sum to infinity of the geometric progression.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Check that the common ratio has modulus less than one, then use S∞ = a/(1 − r)." }, { "id": "cie-2025-w-15-q03a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "3(a): Rewrite the quadratic in completed-square form with rational constants.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Complete the square to obtain 4(x + 5/4)² − 1/4." }, { "id": "cie-2025-w-15-q03b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "3(b): State the value of k for which the horizontal line meets the quadratic at exactly one point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "A single intersection occurs at the vertex, so k equals the minimum value −1/4." }, { "id": "cie-2025-w-15-q04a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(a): Find the perimeter of the logo formed from the circular sector and rhombus.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Combine the exposed rhombus edges with the major arc length, using the stated angle and radius." }, { "id": "cie-2025-w-15-q04b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4(b): Use the 200 cm perimeter to find the logo’s area to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve the perimeter equation for r, then add the major-sector area and rhombus area and round to three significant figures." }, { "id": "cie-2025-w-15-q05", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5: Solve the reciprocal cubic equation by converting it to a quadratic in a substituted variable.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Multiply through by x³, set u = x³, solve the quadratic in u, and take the real cube roots." }, { "id": "cie-2025-w-15-q06a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(a): Show that the stated trigonometric equation reduces to the given quadratic in cos θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write tan θ as sin θ/cos θ, clear denominators, and use sin²θ + cos²θ = 1 to obtain 6cos²θ − cos θ − 2 = 0." }, { "id": "cie-2025-w-15-q06b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "6(b): Use the quadratic from part (a) to solve the original trigonometric equation over the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 9, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic in cos θ and list every angle in the specified interval that has an admissible cosine value." }, { "id": "cie-2025-w-15-q07a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(a): Show that an open cylinder with fixed outside area has the stated volume formula in terms of r.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 10, 10 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the base and curved-surface areas to express h in terms of r, then substitute into V = πr²h." }, { "id": "cie-2025-w-15-q07b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(b): Find the exact radius that maximizes the cylinder’s volume.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the volume expression with respect to r, set the derivative to zero, and select the positive maximizing radius." }, { "id": "cie-2025-w-15-q07c", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7(c): Find the maximum volume of the cylinder.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 11, 11 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the maximizing radius from part (b) into the volume formula." }, { "id": "cie-2025-w-15-q08a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "8(a): Use the volume-of-revolution condition to establish the stated quadratic inequality in a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate πy² from x = a to x = 2a, compare with 46π, and rearrange to 9a² + 5a − 46 ≥ 0." }, { "id": "cie-2025-w-15-q08b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "8(b): Find the possible positive values of a that satisfy the volume condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic inequality and retain the positive interval, using a > 0." }, { "id": "cie-2025-w-15-q09a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "9(a): Use equal coefficients of x and x² in the binomial expansion to find the simplest ratio p:q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 13, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write both coefficients using the binomial formula, equate them and simplify the positive ratio p:q." }, { "id": "cie-2025-w-15-q09b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "9(b): Use the specified x² coefficient to find the positive constants p and q.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 13, 13 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the ratio from part (a) into the x² coefficient, set it equal to 486, and solve with p,q > 0." }, { "id": "cie-2025-w-15-q10a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "10(a): Use f(a) = 4 to determine the input a for the stated restricted-domain function.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the function expression into f(a) = 4 and solve, checking that the result satisfies a > 2." }, { "id": "cie-2025-w-15-q10b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "10(b): Find f⁻¹ and state its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 14, 14 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set y = f(x), interchange x and y, and rearrange for y; use the range of f on x > 2 as the inverse’s domain." }, { "id": "cie-2025-w-15-q10c", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "10(c): Show that the composite fg(x) is kx and determine k for the stated function g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 15, 15 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute g(x) into f, simplify the rational expression and identify the constant multiplier k." }, { "id": "cie-2025-w-15-q11a", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(a)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(a): Find the equation of the tangent at P, where P has x-coordinate 3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 16, 16 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, evaluate its gradient at x = 3, and use P to write the tangent in y = mx + c form." }, { "id": "cie-2025-w-15-q11b", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(b): Find the exact area bounded by the curve, the x-axis and its tangent at P.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 17, 17 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find the relevant intersections, integrate the difference between the tangent and curve over the bounded interval, and evaluate exactly." }, { "id": "cie-2025-w-15-q11ci", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(c)(i)", "marks": 1, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(c)(i): Find the equation of the curve after the stated horizontal stretch.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 17, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Replace x by x divided by the horizontal scale factor in the original equation and simplify to the requested cubic form." }, { "id": "cie-2025-w-15-q11cii", "paperId": "cie-2025-w-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2025, "session": "October/November", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(c)(ii)", "marks": 2, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "11(c)(ii): State the transformed point Q and the area of the transformed shaded region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_15.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_15.pdf", "qpPages": [ 17, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Scale the x-coordinate of P by 1/3 and scale the area by the same horizontal factor." }, { "id": "cie-2025-w-21-q01", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "1: Evaluate the indefinite integral involving sin²x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use sin²x = (1 − cos 2x)/2, then integrate term by term to obtain 3x − (3/2)sin 2x + C." }, { "id": "cie-2025-w-21-q02", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "2: Solve the equation that is quadratic in e^(2x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set u = e^(2x) > 0, factor u² − 8u − 48 = 0, reject the negative root, and take logarithms to get x = ½ ln 12." }, { "id": "cie-2025-w-21-q03a", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "3(a): Solve the equation equating the absolute values of two linear expressions in x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Split into the two sign cases and solve to obtain x = −5/3 and x = 1/7." }, { "id": "cie-2025-w-21-q03b", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "3(b): Use part (a) to solve the related secant equation over π < θ < 2π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the admissible secant value cos θ = −3/5, select the solution in the specified interval, and give θ ≈ 4.07 radians." }, { "id": "cie-2025-w-21-q04", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Solve the cotangent–tangent equation for an angle between 0° and 90°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set t = tan θ and apply the addition formula for tan(θ + 45°), giving 7t² − 6t + 1 = 0. Retain the two angles in range, approximately 12.8° and 32.2°." }, { "id": "cie-2025-w-21-q05a", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "5(a): Show that the x-coordinate of B on the exponential curve is 6 ln 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the curve equal to zero and use logarithm rules to solve for the intercept x = 6 ln 2." }, { "id": "cie-2025-w-21-q05b", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "5(b): Find the exact shaded area between the exponential curve and its intercept chord.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve to obtain the area under it, subtract from the chord triangle area, and simplify to 27 ln 2 − 14." }, { "id": "cie-2025-w-21-q06a", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "6(a): For the parametric curve, derive dy/dx as a function of θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to θ, divide dy/dθ by dx/dθ, and simplify to 6cos⁵θ − 5cos³θ." }, { "id": "cie-2025-w-21-q06b", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "6(b): Find the exact normal at the point where the parametric curve crosses the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set y = 0 to obtain θ = π/4, find tangent gradient −1/√2, then use the reciprocal negative gradient and the point to get y = √2x − √2." }, { "id": "cie-2025-w-21-q07a", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra", "functions", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(a): Use the factor theorem to find k, given that x + 2 is a factor of p(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = −2 into p(x) and solve the resulting equation to obtain k = 4." }, { "id": "cie-2025-w-21-q07b", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra", "functions", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(b): Identify the integer root α and show that the other root β satisfies the stated fixed-point equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use α = −2, divide p(x) by x + 2, and rearrange the remaining cubic to the required equation for β." }, { "id": "cie-2025-w-21-q07c", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "algebra", "functions", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(c): Show by calculation that β lies between −1.4 and −1.0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the cubic or fixed-point difference at −1.4 and −1.0; the sign change brackets the root in that interval." }, { "id": "cie-2025-w-21-q07d", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "algebra", "functions", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(d): Use an iterative formula to find β to three significant figures, showing iterates to five significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate the given fixed-point formula with sufficient working precision to obtain β = −1.26 to 3 significant figures." }, { "id": "cie-2025-w-21-q08", "paperId": "cie-2025-w-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "8: Find the exact coordinates of the stationary point of the stated exponential curve with a square-root factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_21.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_21.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product rule, set the derivative to zero to get x = 7/12, then substitute to find y = 2√3e^(−1/6)." }, { "id": "cie-2025-w-22-q01", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "1: Solve the logarithmic equation involving the difference of two natural logarithms, giving an exact answer.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Combine the logarithms, exponentiate and solve the resulting rational equation to obtain x = (2e⁴ + 5)/(e⁴ − 3); check the domain x > 2." }, { "id": "cie-2025-w-22-q02", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "2: Solve the equation involving tan²θ and sec θ over −180° < θ < 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Replace tan²θ with sec²θ − 1, solve the quadratic in sec θ and retain all four angles: approximately ±66.4° and ±104.5°." }, { "id": "cie-2025-w-22-q03a", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "3(a): Solve the absolute-value inequality in x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Split the modulus inequality at the sign changes and solve each case to get −1/5 ≤ x ≤ 9." }, { "id": "cie-2025-w-22-q03b", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "3(b): Find the largest integer N satisfying the related inequality with powers of 70.01.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the positive upper bound from part (a), take logarithms to find the threshold, and check adjacent integers; N = 112." }, { "id": "cie-2025-w-22-q04a", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(a): Divide the polynomial by x² + 3 and verify that the remainder is 11.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Carry out polynomial division to obtain quotient x² − 10x + 17 and remainder −11." }, { "id": "cie-2025-w-22-q04b", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "4(b): Use the division result to find the exact real roots of p(x) + 11 = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor p(x) + 11 as (x² + 3)(x² − 10x + 17); only the second factor gives real roots, x = 5 ± 2√2." }, { "id": "cie-2025-w-22-q05a", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "5(a): Find the coordinates of both maximum points of the trigonometric curve on 0 ≤ x ≤ π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate and solve for stationary points, then evaluate the curve at each maximum to obtain A = (π/6, 6) and B = (5π/6, 6)." }, { "id": "cie-2025-w-22-q05b", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "5(b): Find the exact area enclosed by the curve and the horizontal segment joining the two maxima.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve between the two x-coordinates, then subtract that area from the rectangle of height 6 to get 4π − 6√3." }, { "id": "cie-2025-w-22-q06a", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "integration", "exponentials", "numerical-methods" ], "primaryTopicId": "integration", "summary": "6(a): Use the definite exponential integral condition to derive the stated equation for the positive parameter a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each exponential term, apply the limits and rearrange to the given logarithmic fixed-point equation." }, { "id": "cie-2025-w-22-q06b", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "integration", "exponentials", "numerical-methods" ], "primaryTopicId": "integration", "summary": "6(b): Show by calculation that a lies between 1.0 and 1.2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the equation’s residual at 1.0 and 1.2 and show the signs differ, so the root lies between them." }, { "id": "cie-2025-w-22-q06c", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "integration", "exponentials", "numerical-methods" ], "primaryTopicId": "integration", "summary": "6(c): Apply the given iteration to find a correct to four significant figures, showing iterates to six significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate to convergence and justify the rounded result a = 1.159." }, { "id": "cie-2025-w-22-q07a", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "7(a): Implicitly differentiate the curve, then find its gradient at the point where y = 0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each term implicitly to obtain dy/dx = (7 − 10xy)/(5x² + 8e^(2y)); use y = 0 and the curve equation to get x = 2 and gradient 1/4." }, { "id": "cie-2025-w-22-q07b", "paperId": "cie-2025-w-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "7(b): Show that no point on the curve has a tangent parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_22.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_22.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The denominator of dy/dx is 5x² + 8e^(2y), which is always positive, so the gradient cannot be undefined." }, { "id": "cie-2025-w-23-q01", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "integration", "summary": "1: Evaluate the indefinite integral involving sin²x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use sin²x = (1 − cos 2x)/2, then integrate term by term to obtain 3x − (3/2)sin 2x + C." }, { "id": "cie-2025-w-23-q02", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "2: Solve the equation that is quadratic in e^(2x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set u = e^(2x) > 0, factor u² − 8u − 48 = 0, reject the negative root, and take logarithms to get x = ½ ln 12." }, { "id": "cie-2025-w-23-q03a", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "3(a): Solve the equation equating the absolute values of two linear expressions in x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Split into the two sign cases and solve to obtain x = −5/3 and x = 1/7." }, { "id": "cie-2025-w-23-q03b", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "3(b): Use part (a) to solve the related secant equation over π < θ < 2π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the admissible secant value cos θ = −3/5, select the solution in the specified interval, and give θ ≈ 4.07 radians." }, { "id": "cie-2025-w-23-q04", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Solve the cotangent–tangent equation for an angle between 0° and 90°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set t = tan θ and apply the addition formula for tan(θ + 45°), giving 7t² − 6t + 1 = 0. Retain the two angles in range, approximately 12.8° and 32.2°." }, { "id": "cie-2025-w-23-q05a", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "integration", "summary": "5(a): Show that the x-coordinate of B on the exponential curve is 6 ln 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the curve equal to zero and use logarithm rules to solve for the intercept x = 6 ln 2." }, { "id": "cie-2025-w-23-q05b", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "integration", "summary": "5(b): Find the exact shaded area between the exponential curve and its intercept chord.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve to obtain the area under it, subtract from the chord triangle area, and simplify to 27 ln 2 − 14." }, { "id": "cie-2025-w-23-q06a", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): For the parametric curve, derive dy/dx as a function of θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to θ, divide dy/dθ by dx/dθ, and simplify to 6cos⁵θ − 5cos³θ." }, { "id": "cie-2025-w-23-q06b", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Find the exact normal at the point where the parametric curve crosses the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set y = 0 to obtain θ = π/4, find tangent gradient −1/√2, then use the reciprocal negative gradient and the point to get y = √2x − √2." }, { "id": "cie-2025-w-23-q07a", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(a): Use the factor theorem to find k, given that x + 2 is a factor of p(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = −2 into p(x) and solve the resulting equation to obtain k = 4." }, { "id": "cie-2025-w-23-q07b", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(b): Identify the integer root α and show that the other root β satisfies the stated fixed-point equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use α = −2, divide p(x) by x + 2, and rearrange the remaining cubic to the required equation for β." }, { "id": "cie-2025-w-23-q07c", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(c): Show by calculation that β lies between −1.4 and −1.0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the cubic or fixed-point difference at −1.4 and −1.0; the sign change brackets the root in that interval." }, { "id": "cie-2025-w-23-q07d", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(d): Use an iterative formula to find β to three significant figures, showing iterates to five significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate the given fixed-point formula with sufficient working precision to obtain β = −1.26 to 3 significant figures." }, { "id": "cie-2025-w-23-q08", "paperId": "cie-2025-w-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "8: Find the exact coordinates of the stationary point of the stated exponential curve with a square-root factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_23.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_23.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product rule, set the derivative to zero to get x = 7/12, then substitute to find y = 2√3e^(−1/6)." }, { "id": "cie-2025-w-25-q01", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "integration", "summary": "1: Evaluate the indefinite integral involving sin²x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use sin²x = (1 − cos 2x)/2, then integrate term by term to obtain 3x − (3/2)sin 2x + C." }, { "id": "cie-2025-w-25-q02", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "2: Solve the equation that is quadratic in e^(2x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set u = e^(2x) > 0, factor u² − 8u − 48 = 0, reject the negative root, and take logarithms to get x = ½ ln 12." }, { "id": "cie-2025-w-25-q03a", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "3(a): Solve the equation equating the absolute values of two linear expressions in x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Split into the two sign cases and solve to obtain x = −5/3 and x = 1/7." }, { "id": "cie-2025-w-25-q03b", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "3(b): Use part (a) to solve the related secant equation over π < θ < 2π.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the admissible secant value cos θ = −3/5, select the solution in the specified interval, and give θ ≈ 4.07 radians." }, { "id": "cie-2025-w-25-q04", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "4: Solve the cotangent–tangent equation for an angle between 0° and 90°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set t = tan θ and apply the addition formula for tan(θ + 45°), giving 7t² − 6t + 1 = 0. Retain the two angles in range, approximately 12.8° and 32.2°." }, { "id": "cie-2025-w-25-q05a", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "integration", "summary": "5(a): Show that the x-coordinate of B on the exponential curve is 6 ln 2.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the curve equal to zero and use logarithm rules to solve for the intercept x = 6 ln 2." }, { "id": "cie-2025-w-25-q05b", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "exponentials", "integration" ], "primaryTopicId": "integration", "summary": "5(b): Find the exact shaded area between the exponential curve and its intercept chord.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve to obtain the area under it, subtract from the chord triangle area, and simplify to 27 ln 2 − 14." }, { "id": "cie-2025-w-25-q06a", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(a): For the parametric curve, derive dy/dx as a function of θ.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to θ, divide dy/dθ by dx/dθ, and simplify to 6cos⁵θ − 5cos³θ." }, { "id": "cie-2025-w-25-q06b", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "differentiation", "summary": "6(b): Find the exact normal at the point where the parametric curve crosses the x-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set y = 0 to obtain θ = π/4, find tangent gradient −1/√2, then use the reciprocal negative gradient and the point to get y = √2x − √2." }, { "id": "cie-2025-w-25-q07a", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(a): Use the factor theorem to find k, given that x + 2 is a factor of p(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute x = −2 into p(x) and solve the resulting equation to obtain k = 4." }, { "id": "cie-2025-w-25-q07b", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(b): Identify the integer root α and show that the other root β satisfies the stated fixed-point equation.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use α = −2, divide p(x) by x + 2, and rearrange the remaining cubic to the required equation for β." }, { "id": "cie-2025-w-25-q07c", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(c): Show by calculation that β lies between −1.4 and −1.0.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 11, 11 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the cubic or fixed-point difference at −1.4 and −1.0; the sign change brackets the root in that interval." }, { "id": "cie-2025-w-25-q07d", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "7(d): Use an iterative formula to find β to three significant figures, showing iterates to five significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate the given fixed-point formula with sufficient working precision to obtain β = −1.26 to 3 significant figures." }, { "id": "cie-2025-w-25-q08", "paperId": "cie-2025-w-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2025, "session": "October/November", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8", "marks": 6, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "8: Find the exact coordinates of the stationary point of the stated exponential curve with a square-root factor.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_25.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_25.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate using the product rule, set the derivative to zero to get x = 7/12, then substitute to find y = 2√3e^(−1/6)." }, { "id": "cie-2025-w-31-q01", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "1: Evaluate the definite integral of ln(3x) from 1 to 2 in the stated exact logarithmic form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use integration by parts to find x ln(3x) − x, substitute 1 and 2, and simplify to ln 12 − 1." }, { "id": "cie-2025-w-31-q02a", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "2(a): Show that the logarithmic equation can be rewritten as a quadratic equation in x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Combine the base-4 logarithms with log laws, convert to an equation without logarithms and expand to the required quadratic." }, { "id": "cie-2025-w-31-q02b", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "2(b): Solve the original logarithmic equation, respecting its domain.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic form and discard any root outside the logarithm domains; the valid root is (19 + 4√31)/9." }, { "id": "cie-2025-w-31-q03a", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(a): Express the sine-plus-cosine expression as R cos(x − α), with the stated restrictions on R and α.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Expand R cos(x − α), match sine and cosine coefficients, and obtain R = 5 and α ≈ 36.9°." }, { "id": "cie-2025-w-31-q03b", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "3(b): Use part (a) to solve the related trigonometric equation over 0° < θ < 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the single-cosine form and the triple-angle relation, then select the two solutions in range: 60.0° and approximately 84.6°." }, { "id": "cie-2025-w-31-q04", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "differentiation", "trigonometry", "exponentials" ], "primaryTopicId": "differentiation", "summary": "4: Find the x-coordinate of the maximum point of the exponential–trigonometric curve on the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate with the product and chain rules, set the derivative to zero, reduce to a quadratic in sin 2x and select the stationary point in range, x ≈ 0.113." }, { "id": "cie-2025-w-31-q05a", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(a): Write two inequalities describing the shaded Argand region bounded by a circle and a vertical line.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express the vertical boundary as Re(z) ≥ 2 and the circular boundary as |z − (1 − 2i)| ≤ 2." }, { "id": "cie-2025-w-31-q05b", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "5(b): Find the least argument of a point in the shaded Argand region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 9, 9 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the lower intersection of the circle with Re(z) = 2; its argument is approximately −61.8° (or 298.2° on a 0°–360° scale)." }, { "id": "cie-2025-w-31-q06", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "6: Solve the quadratic equation with complex coefficients and express both roots as x + iy.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quadratic formula, use i² = −1 and simplify the roots to −3/5 + 4i/5 and −1." }, { "id": "cie-2025-w-31-q07", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "7: For the parametrically defined curve, simplify dy/dx in terms of t.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x and y with respect to t, divide dy/dt by dx/dt and factor to obtain 1/[2(2t + 1)(2t − 1)(t + 1)]." }, { "id": "cie-2025-w-31-q08", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": "8", "marks": 8, "topicIds": [ "integration", "exponentials", "differentiation" ], "primaryTopicId": "integration", "summary": "8: Solve the separable differential equation using the two point conditions, then evaluate y at x = √3.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables and integrate, use y = 0 at x = 0 and the second point condition to determine the constants, then substitute x = √3 to obtain y = −½ ln(2e − 1)." }, { "id": "cie-2025-w-31-q09a", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "9(a): Use a suitable pair of graphs to show the equation has exactly one root in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the two functions and show they have a single intersection in the interval." }, { "id": "cie-2025-w-31-q09b", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "9(b): Show by calculation that the root lies between 0.9 and 1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the difference of the two sides at 0.9 and 1 and show the signs differ." }, { "id": "cie-2025-w-31-q09c", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "9(c): Show that any convergent sequence from the given iteration converges to the root from part (a).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "At a limit, set consecutive iterates equal and rearrange to recover the equation in part (a)." }, { "id": "cie-2025-w-31-q09d", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(d)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "9(d): Use the iteration to find the root to three decimal places, showing iterates to five decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate with the stated formula until the values agree to the requested precision; the root is approximately 0.978." }, { "id": "cie-2025-w-31-q10a", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "algebra", "binomial" ], "primaryTopicId": "algebra", "summary": "10(a): Express the rational function in partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 16, 16 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve it as 1 − 4/(x + 3) + (x − 3)/(x² + 2), or an equivalent decomposition." }, { "id": "cie-2025-w-31-q10b", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "algebra", "binomial" ], "primaryTopicId": "algebra", "summary": "10(b): Use the partial fractions to expand f(x) in ascending powers through x².", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 17, 17 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Expand each partial-fraction term about x = 0, combine coefficients, and obtain −11/6 + (17/18)x + (65/108)x²." }, { "id": "cie-2025-w-31-q11a", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Find a vector equation for the line through A and B.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 18, 18 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the position vector of A and direction vector B − A to write the line in vector form." }, { "id": "cie-2025-w-31-q11b", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b): Find the position vector where line m through A and B intersects line CD.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 18, 18 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write both lines parametrically, equate corresponding components and solve to obtain (−3, 1, −5)." }, { "id": "cie-2025-w-31-q11c", "paperId": "cie-2025-w-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "11(c): Find the position vector of the perpendicular foot from C to line m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_31.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_31.pdf", "qpPages": [ 19, 19 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Let the foot lie on m and impose that its displacement from C has zero dot product with the line direction; the foot is (−1, 3, −1)." }, { "id": "cie-2025-w-32-q01a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "1(a): Sketch y = |x + 3a| for positive a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw the V-shaped graph with vertex at (−3a, 0) and gradient −1 to the left and +1 to the right." }, { "id": "cie-2025-w-32-q01b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "1(b): Use the graph or algebra to solve the inequality |x + 3a| > a − 2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Compare the V-shaped graph with the line a − 2x and solve the boundary cases, retaining x > −2a/3." }, { "id": "cie-2025-w-32-q02", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "2: Solve the equation involving powers of 2 and 3, giving the answer to three significant figures.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Express both sides with compatible exponential bases, take logarithms and solve for x; round the result to 3 significant figures." }, { "id": "cie-2025-w-32-q03a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "3(a): Find the exact complex number represented by P, an intersection of the stated line and circle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use Re(z) = 2 in the circle equation |z − (3 + i)| = 2 and solve for the imaginary part to get P = 2 + (1 − √3)i." }, { "id": "cie-2025-w-32-q03b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "3(b): Find the greatest possible argument of a point in the shaded Argand region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Identify the boundary point with the greatest angle from the positive real axis and calculate its argument, approximately 1.01 radians." }, { "id": "cie-2025-w-32-q04", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "4: Evaluate exactly the definite integral involving x tan⁻¹x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use integration by parts and simplify the remaining rational integral to obtain π/4 − 1/2." }, { "id": "cie-2025-w-32-q05a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "5(a): Show that (x − a) is a factor of the derivative when f(x) = (x − a)²g(x).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate by the product rule and factor the result to expose a factor (x − a)." }, { "id": "cie-2025-w-32-q05b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "5(b): Find p and q when (x − 3)² is a factor of the stated quartic polynomial.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 6, 6 ], "msPages": [ 15, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "A double root at 3 requires f(3) = 0 and f′(3) = 0; solve these equations to obtain p = −30 and q = 72." }, { "id": "cie-2025-w-32-q06a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "6(a): Use a suitable graph sketch to show the trigonometric equation has exactly one root in the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Sketch cot 2x and 2 sin 2x − 1 over the interval and show they intersect once." }, { "id": "cie-2025-w-32-q06b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "6(b): Show by calculation that the root lies between 0.4 and 0.6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the difference of the two sides at 0.4 and 0.6 and verify a sign change." }, { "id": "cie-2025-w-32-q06c", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "6(c): Use the given iterative formula to find the root to two decimal places, showing iterates to four decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 7, 7 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Iterate with the specified formula until stable to the required precision; the root is approximately 0.49." }, { "id": "cie-2025-w-32-q07a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "7(a): Implicitly differentiate the curve and obtain the stated expression for dy/dx.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 8, 8 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides with respect to x and collect dy/dx terms to get dy/dx = (x² + 2xy)/(2y² − x²)." }, { "id": "cie-2025-w-32-q07b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "7(b): Find the coordinates where the normal to the curve is parallel to the y-axis.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 9, 9 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "A vertical normal corresponds to a horizontal tangent, so set the numerator of dy/dx to zero, substitute into the curve and obtain (0, 2) and (4, −2)." }, { "id": "cie-2025-w-32-q08a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "8(a): Prove the stated identity for sin 4x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 10, 10 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use double-angle identities to expand sin 4x and factor the result into the required expression." }, { "id": "cie-2025-w-32-q08b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "8(b): Use the identity from part (a) to evaluate the related trigonometric integral exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 11, 11 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the identity into the integrand, use a cosine substitution for the resulting powers and apply the given limits to obtain (12 + √2)/35." }, { "id": "cie-2025-w-32-q09a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "9(a): Express the parameter-dependent rational function in partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 12, 12 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write f(x) as 1 + A/(x + 2a) + B/(x + 3a), compare coefficients and obtain A = 2a and B = −3a." }, { "id": "cie-2025-w-32-q09b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "algebra", "integration" ], "primaryTopicId": "algebra", "summary": "9(b): Integrate the function between the stated symmetric limits and give the exact result in the requested form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 13, 13 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the partial fractions as logarithms over the limits −a to a and simplify to a(2 + ln(9/8))." }, { "id": "cie-2025-w-32-q10a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "10(a): Use the tank’s inflow, outflow and dimensions to derive the stated differential equation for water depth h.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 14, 14 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Relate the volume change to the difference between inflow and h²-proportional outflow, then use the rate at h = 20 to determine the constant and obtain dh/dt = (500 − h²)/250." }, { "id": "cie-2025-w-32-q10b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "10(b): Find the time required for the water depth to increase from 0 to 20 cm.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 15, 15 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables in the differential equation, integrate from h = 0 at t = 0 to h = 20, and evaluate the time, approximately 16.1 seconds." }, { "id": "cie-2025-w-32-q11a", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "11(a): Find vectors MB and MC from the given position vectors and midpoint M of AB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 27, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Calculate M = (A + B)/2, then subtract M from B and C to obtain MB = (−1, 5, 2) and MC = (−5, −3, 4)." }, { "id": "cie-2025-w-32-q11b", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "11(b): Find the exact cosine of angle CMB.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 16, 16 ], "msPages": [ 28, 28 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use cos θ = (MB · MC)/(|MB||MC|) and simplify to −√15/75." }, { "id": "cie-2025-w-32-q11c", "paperId": "cie-2025-w-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 11, "questionPart": "11(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "11(c): Find the exact area of triangle ABC.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_32.pdf#page=17", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_32.pdf", "qpPages": [ 17, 17 ], "msPages": [ 29, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the two vectors from M and the sine of their included angle to calculate the triangle area, obtaining 2√374." }, { "id": "cie-2025-w-33-q01", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "1: Solve a strict inequality comparing two absolute-value expressions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Split at the zeroes of the expressions or solve the two boundary equations, then identify the intervals where the inequality holds: x<1/9 or x>5/3." }, { "id": "cie-2025-w-33-q02", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "2: Find the quotient and remainder when a quartic polynomial is divided by x+1.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Polynomial division gives quotient 3x³−3x²+x−1 and remainder 1; the remainder can also be found by evaluating the dividend at x=−1." }, { "id": "cie-2025-w-33-q03", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "3: Solve an exponential equation and give the result as an exact logarithmic value.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply logarithm laws to collect x; the exact answer is ln(48)/ln(40)." }, { "id": "cie-2025-w-33-q04", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "4: Sketch the Argand-plane region satisfying a circle condition and a distance comparison.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw the circle centred at (0,−2) with radius 3 and the perpendicular bisector between (0,−2) and (2,−4), then shade the region satisfying both conditions." }, { "id": "cie-2025-w-33-q05a", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(a): Prove the stated trigonometric identity by rewriting cos 4x using double-angle formulae.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use cos 4x=1−2sin²2x and sin 2x=2sin x cos x, then replace cos²x with 1−sin²x to obtain 8sin⁴x−6sin²x." }, { "id": "cie-2025-w-33-q05b", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "5(b): Use the identity from part (a) to solve the trigonometric equation on the inclusive interval −180° to 180°.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor as sin²x(8sin²x−6)=0. Include x=0°, ±180° and the four solutions with sin²x=3/4: ±60°, ±120°." }, { "id": "cie-2025-w-33-q06", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6", "marks": 6, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "6: Evaluate exactly a definite integral involving x² sin 2x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate by parts twice and apply the limits 0 and π/6 to obtain −π²/144+√3π/24−1/8." }, { "id": "cie-2025-w-33-q07", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7", "marks": 6, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "7: Solve a complex-number equation involving z and its conjugate.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Write z=x+iy and z*=x−iy, clear the denominator and equate real and imaginary parts; the roots are 2−5i and −16/5−12i/5." }, { "id": "cie-2025-w-33-q08a", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "differentiation", "exponentials", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "8(a): Differentiate the given exponential-logarithmic function and derive the equation satisfied by its stationary point.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the product rule and set the derivative to zero to obtain ln(5p)=1/(5p)." }, { "id": "cie-2025-w-33-q08b", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "differentiation", "exponentials", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "8(b): Show graphically that the equation for p has only one root.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 12, 12 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Sketch y=ln(5p) and y=1/(5p); the first is increasing and the second decreasing for p>0, so they meet once." }, { "id": "cie-2025-w-33-q08c", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "differentiation", "exponentials", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "8(c): Verify by calculation that the root p lies between 0.2 and 0.6.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 13, 13 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute 0.2 and 0.6 into the two sides of ln(5p)=1/(5p) and show the difference changes sign." }, { "id": "cie-2025-w-33-q08d", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "differentiation", "exponentials", "numerical-methods" ], "primaryTopicId": "differentiation", "summary": "8(d): Use the supplied rearrangement to iterate and find p correct to two decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 13, 13 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the given fixed-point formula to successive values until stable; p rounds to 0.35." }, { "id": "cie-2025-w-33-q09a", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(a): Find the direction vector of the second line using its perpendicularity and coordinate restrictions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use its zero x-component and the condition that it is perpendicular to (3,−2,1) to determine a suitable direction vector." }, { "id": "cie-2025-w-33-q09b", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(b): Find the acute angle between the two line directions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=14", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the scalar-product formula with the direction vectors; the acute angle is approximately 28.6°." }, { "id": "cie-2025-w-33-q09c", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "9(c): Find the position vector where the two three-dimensional lines intersect.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=15", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 15, 15 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Equate corresponding coordinates, solve for the line parameters (−2 and −4), and substitute to obtain i−j−14k." }, { "id": "cie-2025-w-33-q10a", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "algebra", "integration", "trigonometry" ], "primaryTopicId": "algebra", "summary": "10(a): Express the rational function in partial fractions.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 16, 16 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Factor 1−9y²=(1−3y)(1+3y) and solve for the two coefficients; both are 1." }, { "id": "cie-2025-w-33-q10b", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 6, "topicIds": [ "algebra", "integration", "trigonometry" ], "primaryTopicId": "algebra", "summary": "10(b): Solve the separable differential equation using its trigonometric coefficient and initial condition.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=16", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 16, 16 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables and integrate to obtain (1/3)ln((1+3y)/(1−3y))=(1/3)tan(3x)+C; use y=0 at x=π/12 to determine C, then rearrange for y." }, { "id": "cie-2025-w-33-q11a", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 6, "topicIds": [ "differentiation", "integration", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "11(a): Find the x-coordinate of the minimum of sec²x√(3+2tan x) on the stated interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=18", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 18, 18 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set t=tan x and differentiate; the stationary equation factors to (t+1)(5t+1)=0, giving the interior minimum at tan x=−1/5 and x≈−0.197 radians." }, { "id": "cie-2025-w-33-q11b", "paperId": "cie-2025-w-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "differentiation", "integration", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "11(b): Find the exact area under the curve using u=3+2tan x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_33.pdf#page=19", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_33.pdf", "qpPages": [ 19, 19 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "With du=2sec²x dx and limits u=1 to 5, the area is (1/2)∫₁⁵√u du=(5√5−1)/3." }, { "id": "cie-2025-w-35-q01a", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "1(a): Sketch y=|3x−6|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a V-shaped graph with vertex (2,0) and y-intercept 6." }, { "id": "cie-2025-w-35-q01b", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "1(b): Solve the strict inequality 5x−3<|3x−6|.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Split at x=2 and solve each linear case; the combined solution is x<9/8." }, { "id": "cie-2025-w-35-q02", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "2: Shade the Argand-plane region inside a radius-2 circle centred at 1+3i and outside a radius-4 circle centred at the origin.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw |z−(1+3i)|=2 and |z|=4, then shade the intersection of the smaller circle’s interior and the larger circle’s exterior." }, { "id": "cie-2025-w-35-q03a", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "3(a): Show that Ay=b^x gives a straight-line relationship between ln y and x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Take logarithms to obtain ln y=ln A+x ln b, which is linear in x." }, { "id": "cie-2025-w-35-q03b", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "3(b): Use two observations of x and ln y to estimate A and b.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Solve the two simultaneous logarithmic equations; to 2 significant figures, b=2.1 and A=5.2 (5.3 also accepted)." }, { "id": "cie-2025-w-35-q04", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4", "marks": 5, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "4: Implicitly differentiate the logarithmic curve and find its exact gradient at (2,3).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate x²ln(2y)−yln(2+x²)=ln6 implicitly, substitute x=2,y=3, and solve for dy/dx; the gradient is (6−12ln6)/(4−3ln6)." }, { "id": "cie-2025-w-35-q05", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "5: Solve the complex equation involving z times its conjugate, giving answers in x+iy form.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute z=x+iy and z*=x−iy, then equate real and imaginary parts. This gives x=2 and y=2 or 3, so z=2+2i or 2+3i." }, { "id": "cie-2025-w-35-q06a", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry", "functions" ], "primaryTopicId": "differentiation", "summary": "6(a): Differentiate tan x/(5+2sin x) and simplify the result using sin x alone.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quotient rule and simplify to dy/dx=(5+2sin³x)/[(1−sin²x)(5+2sin x)²]." }, { "id": "cie-2025-w-35-q06b", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "differentiation", "trigonometry", "functions" ], "primaryTopicId": "differentiation", "summary": "6(b): Show that the curve has no stationary points.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The derivative numerator satisfies 3≤5+2sin³x≤7, so it never vanishes; hence there are no stationary points." }, { "id": "cie-2025-w-35-q07a", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "7(a): Use u=x²−3 to transform the definite integral and find the transformed limits.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Since du/dx=2x, x=√7 maps to u=4 and x=√12 maps to u=9; substituting gives ∫₄⁹ 2(u+3)/√u du." }, { "id": "cie-2025-w-35-q07b", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "7(b): Evaluate the transformed definite integral exactly.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate 2(u+3)/√u between 4 and 9 to obtain 112/3." }, { "id": "cie-2025-w-35-q08a", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(a): Express the trigonometric combination as R sin(θ+α), finding exact R and α to 3 decimal places.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_35.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_35.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Expand the shifted sine to get (5/2)sinθ+(3√3/2)cosθ; R=√13 and α≈0.805 radians." }, { "id": "cie-2025-w-35-q08b", "paperId": "cie-2025-w-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2025, "session": "October/November", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "8(b): Use the result from part (a) to solve the trigonometric equation on 04.5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_43.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_43.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Conserve momentum with directions signed; Q’s velocity is (2m/3−3)u, and requiring its stated direction gives m>4.5." }, { "id": "cie-2025-w-43-q06b", "paperId": "cie-2025-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6(b): Find the largest integer m consistent with the wall rebound and no further collision.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_43.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_43.pdf", "qpPages": [ 11, 11 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quarter-speed rebound and the no-recollision condition; the allowed range gives the largest integer m=6." }, { "id": "cie-2025-w-43-q07a", "paperId": "cie-2025-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(a): Show the stated formula for the coefficient of friction in limiting equilibrium.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_43.pdf", "qpPages": [ 12, 12 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the horizontal force and weight parallel and perpendicular to the plane, set friction equal to μR, and eliminate R to obtain μ=(5tanθ−1)/(5+tanθ)." }, { "id": "cie-2025-w-43-q07b", "paperId": "cie-2025-w-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/43", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "7(b): Find the possible acute values of the plane angle.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_43.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_43.pdf", "qpPages": [ 12, 12 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Require positive friction in the given formula with 0<θ<90°; the range is arctan(1/5)<θ<90°, i.e. 11.3°<θ<90°." }, { "id": "cie-2025-w-45-q01", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "1: Find two unknown force magnitudes from the equilibrium of four coplanar forces.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally and vertically using the diagram’s directions, solve the simultaneous equations, and obtain P≈27.6 N and Q≈37.2 N." }, { "id": "cie-2025-w-45-q02a", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(a): Sketch the athlete’s velocity-time graph.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Draw a trapezium profile: linear acceleration from rest, a constant-speed section at V, then linear deceleration to rest." }, { "id": "cie-2025-w-45-q02b", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(b): Express the constant-speed duration T in terms of V and derive the stated quadratic.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the total time to write T=246−V/0.5−V/0.2, then equate the trapezium area to 1350 m; this gives 7V²−492V+2700=0." }, { "id": "cie-2025-w-45-q02c", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "2(c): Find the feasible cruise speed and duration.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 3, 3 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Reject the larger quadratic root because it makes T negative; V=6 m/s and T=204 s." }, { "id": "cie-2025-w-45-q03a", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "3(a): Find the resistance and engine power from the car’s two constant-speed journeys.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 4, 4 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use P=Rv on the level road and P=(R+mg sinθ)v uphill with sinθ=0.15; solve to obtain R=1800 N and P=36 kW." }, { "id": "cie-2025-w-45-q03b", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "3(b): Find the acceleration uphill at 10 m/s when engine power is 32 kW.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 5, 5 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The engine force is P/v; subtract resistance and the component of weight down the slope, then divide by 800 kg to obtain 0.25 m/s²." }, { "id": "cie-2025-w-45-q04a", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(a): Show that the particle never has zero velocity only if k>4.5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 6, 6 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate a=kt−3 using v(0)=1 to get v=½kt²−3t+1; require the quadratic discriminant to be negative, giving k>4.5." }, { "id": "cie-2025-w-45-q04b", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "4(b): Find the total distance travelled from t=1 to t=3 when k=2.5.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 7, 7 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Substitute k=2.5 into velocity, find its zero at t=2, and integrate |v| on [1,2] and [2,3]; the total distance is 2 m." }, { "id": "cie-2025-w-45-q05a", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "5(a): Find both string tensions and the acceleration of the three-particle system.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 8, 8 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to Q, R and P with the supplied tension ratio; the tensions are 40g/9 N and 20g/9 N, and the acceleration is g/9." }, { "id": "cie-2025-w-45-q05b", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "5(b): Find the coefficient of friction for P on the rough plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 9, 9 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces on P along the plane, using the acceleration and both tensions; the resulting coefficient is μ≈0.877." }, { "id": "cie-2025-w-45-q05c", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "5(c): Find the total length of the plane.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 9, 9 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "P starts at the midpoint and reaches the lower pulley in 1.5 s from rest with acceleration g/9; use s=½at² to find the half-length and double it, giving 2.5 m." }, { "id": "cie-2025-w-45-q06", "paperId": "cie-2025-w-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2025, "session": "October/November", "paperCode": "9709/45", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6", "marks": 8, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "6: Find B’s speed after a collision with a specified kinetic-energy loss.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_45.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_45.pdf", "qpPages": [ 10, 10 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use momentum to relate the post-collision speed of B to k and u. Set the final kinetic energy to 65% of the initial total, solve for k, discard the nonphysical root, and obtain vB=2.5u." }, { "id": "cie-2025-w-51-q01a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "1(a): Complete the probability distribution when each value has probability proportional to its square.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use total probability to find k=1/18, giving probabilities 4/18, 1/18, 4/18 and 9/18 for the stated values." }, { "id": "cie-2025-w-51-q01b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "1(b): Find the expectation of X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use E(X)=ΣxP(X=x) to obtain 14/9." }, { "id": "cie-2025-w-51-q01c", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "1(c): Find a conditional probability among the positive values of X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Restrict to X>0 and renormalise the relevant probabilities; the result is 5/7." }, { "id": "cie-2025-w-51-q02a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "2(a): Find the probability that the first six appears on the eighth die throw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric probability (5/6)^7(1/6)." }, { "id": "cie-2025-w-51-q02b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "2(b): Find the probability that the third six appears on the seventh throw.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The first six throws contain exactly two sixes and the seventh is a six: C(6,2)(1/6)^3(5/6)^4." }, { "id": "cie-2025-w-51-q03a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "3(a): Read the median and interquartile range from the back-to-back stem-and-leaf display.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "For Browns, the median is $32,400 and the interquartile range is $2,200." }, { "id": "cie-2025-w-51-q03b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "3(b): Find the mean and standard deviation for the combined 54 salaries.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Pool the supplied sums and squared sums for both groups; the mean is about $32,867 and the standard deviation about $1,360." }, { "id": "cie-2025-w-51-q04a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Complete the probability tree for the three marble selections and transfers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Update each bag’s red and blue counts after each draw and label every branch with its conditional probability." }, { "id": "cie-2025-w-51-q04b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Find the probability that all three selected marbles have the same colour.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Add the all-red and all-blue path probabilities to obtain approximately 0.432." }, { "id": "cie-2025-w-51-q05a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "5(a): Find the chance Cooper wears a blue jumper on at least five of seven days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the upper tail of Bin(7,0.6), giving approximately 0.420." }, { "id": "cie-2025-w-51-q05b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "5(b): Approximate the probability Cooper wears no jumper on fewer than 22 of 150 days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 7, 8 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The no-jumper count is Bin(150,0.1), approximated by N(15,13.5); use a continuity correction at 21.5 to obtain about 0.962." }, { "id": "cie-2025-w-51-q06a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "6(a): Estimate the number of bars with weights between 148 g and 160 g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardise both endpoints under N(155,6²); the interval probability is about 0.6759, giving about 236 of 350 bars." }, { "id": "cie-2025-w-51-q06b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "6(b): Find the mean and standard deviation for the second factory’s normal weight model.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Convert the 8% upper tail and 20% lower tail to standard normal scores, then solve the two linear equations; μ=105 g and σ≈6.45 g." }, { "id": "cie-2025-w-51-q07a", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "7(a): Count unrestricted distinct arrangements of SEYCHELLES.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Divide 10! by the factorials for the repeated letters to obtain 151,200." }, { "id": "cie-2025-w-51-q07b", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "7(b): Count arrangements with exactly two letters between the Ss, one of them C.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Fix the S positions and place C in one of the two intervening positions, then arrange the remaining repeated letters; the count is 5,880." }, { "id": "cie-2025-w-51-q07c", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "7(c): Count arrangements with Ss at both ends and the Es not all consecutive.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Arrange the remaining letters with S fixed at both ends and subtract cases where the three Es form one block; the count is 3,000." }, { "id": "cie-2025-w-51-q07d", "paperId": "cie-2025-w-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/51", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "7(d): Find the probability a random five-letter selection contains all three Es.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_51.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_51.pdf", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Choose the other two from the seven non-E letters: C(7,2)/C(10,5)=1/12." }, { "id": "cie-2025-w-52-q01a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(a): Find the probability the first head occurs on throw five.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the geometric model: 0.6⁴×0.4=0.05184." }, { "id": "cie-2025-w-52-q01b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "1(b): Find the probability the first head occurs after throw six.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The first six throws must all be tails, giving 0.6⁶≈0.0467." }, { "id": "cie-2025-w-52-q02a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "2(a): Give the probability distribution for the draw on which the first green marble appears.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The first green can occur on draws 1–5; the probabilities are 5/15, 4/15, 3/15, 2/15 and 1/15." }, { "id": "cie-2025-w-52-q02b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "2(b): Find the variance of the number of draws until the first green marble.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the distribution from part (a) to calculate E(X) and E(X²), giving Var(X)=14/9." }, { "id": "cie-2025-w-52-q03a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Estimate how many senior members have heights within 5 cm of the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardise 182.4 cm and 192.4 cm under N(187.4,6.4²), then multiply the interval probability by 124; the result is about 70." }, { "id": "cie-2025-w-52-q03b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Find the standard deviation of the junior members’ heights from a lower-tail percentage.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use Φ((170.3−172.7)/σ)=0.23 and solve for σ≈3.25 cm." }, { "id": "cie-2025-w-52-q04a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(a): Use the matching-probability condition to find Ivy’s pack size x.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set (6/18)((x−12)/x) equal to twice (10/18)(9/x); solve to obtain x=42." }, { "id": "cie-2025-w-52-q04b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "4(b): Find the probability that the two selected cards show the same vehicle type.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Add the bus, car and train matching probabilities to obtain 23/63≈0.365." }, { "id": "cie-2025-w-52-q05a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(a): Draw a histogram for the competitors’ grouped times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use frequency divided by class width for the bar heights; the frequency densities are 1.2, 3.8, 13.6, 15.2 and 2.3." }, { "id": "cie-2025-w-52-q05b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "5(b): Estimate the mean time from the grouped data.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 9, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use each class midpoint weighted by its frequency and divide by 240; the estimate is 25.875 minutes." }, { "id": "cie-2025-w-52-q06a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Find the probability that none of ten people has a weekend birthday.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use (5/7)^10≈0.0346." }, { "id": "cie-2025-w-52-q06b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Find the probability fewer than three of ten people have a Wednesday birthday.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the cumulative probability through 2 for Bin(10,1/7), approximately 0.838." }, { "id": "cie-2025-w-52-q06c", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "6(c): Approximate the probability more than 65 of 392 people have Friday birthdays.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Approximate Bin(392,1/7) by N(56,48), apply continuity correction at 65.5 and obtain about 0.0851." }, { "id": "cie-2025-w-52-q07a", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "7(a): Count arrangements of ZOOLOGICAL with the three Os together and the two Ls separated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 12, 12 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Treat OOO as a block and subtract arrangements where Ls are adjacent; the count is 15,120." }, { "id": "cie-2025-w-52-q07b", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "7(b): Count arrangements with exactly five letters between the two Ls.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 12, 12 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Fix the L positions five places apart and arrange the remaining multiset; the count is 26,880." }, { "id": "cie-2025-w-52-q07c", "paperId": "cie-2025-w-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/52", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "7(c): Find the probability that two randomly selected letters are different.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_52.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_52.pdf", "qpPages": [ 13, 13 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the probability of drawing a matching pair from 1 to obtain 41/45." }, { "id": "cie-2025-w-53-q01a", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "1(a): Find the probability that all three selected students avoid travelling by car.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The probability is (1−0.4)^3=0.216." }, { "id": "cie-2025-w-53-q01b", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "1(b): Find the probability fewer than nine of eleven students travel by car or bus.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use X~Bin(11,0.75) and sum P(X≤8), giving approximately 0.545." }, { "id": "cie-2025-w-53-q02", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 4, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "probability", "summary": "2: Count six-person committees with at least two swimmers and at least two divers.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Sum the cases (4,2), (3,3) and (2,4): C(16,4)C(10,2)+C(16,3)C(10,3)+C(16,2)C(10,4)=174,300." }, { "id": "cie-2025-w-53-q03a", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "3(a): Count DAFFODILS arrangements with D at each end and the two Fs separated.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Fix D at both ends and subtract the cases where F are adjacent; the count is 1,800." }, { "id": "cie-2025-w-53-q03b", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "probability" ], "primaryTopicId": "algebra", "summary": "3(b): Find the probability a random arrangement has exactly four letters between the Ds.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 5, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Count 10,080 favourable arrangements out of 90,720 distinct arrangements, giving 1/9." }, { "id": "cie-2025-w-53-q04a", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(a): Construct the probability distribution for the score from the two spinners.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the 16 equally likely outcomes, assigning zero on a tie; probabilities for X=0,1,2,3,4 are 3/16, 1/16, 3/16, 5/16 and 4/16." }, { "id": "cie-2025-w-53-q04b", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(b): Find Var(X) for the spinner score.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the distribution from part (a) to find E(X) and E(X²); Var(X)=127/64≈1.98." }, { "id": "cie-2025-w-53-q04c", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "4(c): Find the chance the red spinner first lands on 4 before spin eight.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the complement that the first seven spins are not 4: 1−(3/4)^7≈0.867." }, { "id": "cie-2025-w-53-q05a", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(a): Draw the fully labelled probability tree for the counter and two marble draws.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "First branch by white/yellow counter, then show the two without-replacement marble outcomes using each selected box’s conditional probabilities." }, { "id": "cie-2025-w-53-q05b", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(b): Find the probability of obtaining one red and one blue marble.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 9, 9 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Combine the red-blue and blue-red paths under both counter outcomes; the probability is 39/70." }, { "id": "cie-2025-w-53-q05c", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "5(c): Find the probability the counter was white given that one red and one blue marble were obtained.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 9, 9 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use Bayes’ rule with the white and yellow path probabilities; the conditional probability is 8/13." }, { "id": "cie-2025-w-53-q06a", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(a): Draw a back-to-back stem-and-leaf diagram for the two cycling teams.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 10, 10 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Place the Linnets on the left and Puffins on the right, preserving all 11 times and the given key." }, { "id": "cie-2025-w-53-q06b", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(b): Find the interquartile range of the Linnets’ times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 10, 10 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "From the ordered data, Q1=54 and Q3=65, so the IQR is 11 minutes." }, { "id": "cie-2025-w-53-q06c", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(c): Draw box-and-whisker plots for both teams on the same scale.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 11, 11 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use each team’s minimum, lower quartile, median, upper quartile and maximum from the data." }, { "id": "cie-2025-w-53-q06d", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "6(d): Make one comparison between the two teams’ times.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 11, 11 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "For example, the Puffins’ times are more spread out." }, { "id": "cie-2025-w-53-q07a", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(a): Find the probability Obi takes more than 15.6 minutes to walk to work.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 12, 12 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Standardise under N(14.8,1.5²); the upper-tail probability is about 0.297." }, { "id": "cie-2025-w-53-q07b", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(b): Find the time exceeded on 90% of days.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=12", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 12, 12 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the 10th percentile of the normal distribution to obtain t≈12.9 minutes." }, { "id": "cie-2025-w-53-q07c", "paperId": "cie-2025-w-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2025, "session": "October/November", "paperCode": "9709/53", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "7(c): Find the expected annual number of days the time is within one minute of the mean.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_53.pdf#page=13", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_53.pdf", "qpPages": [ 13, 13 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find P(13.82) for independent X~Po(3), Y~Po(2).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use X+Y~Po(5) and find the upper tail above 2, approximately 0.875." }, { "id": "cie-2025-w-61-q01c", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "1(c): Use a normal approximation for the total of 100 X-values and 150 Y-values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The total has mean and variance 600; approximate by N(600,600), use a continuity correction at 559.5 and find P(T<560)≈0.0492." }, { "id": "cie-2025-w-61-q02", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2", "marks": 8, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "2: Test at 5% whether the population mean packet mass differs from 500 g.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use x̄=499.5 g and sample variance≈4.83 to form the two-sided test statistic z≈−1.762. Since |z|<1.96, do not reject H₀." }, { "id": "cie-2025-w-61-q03a", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "3(a): Approximate the number of incorrect characters and find P(X<4).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use Poisson λ=np=1.45 and sum probabilities for 0 to 3 errors." }, { "id": "cie-2025-w-61-q03b", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "3(b): Find the revised error rate that doubles the probability of zero errors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set e^(−14500p)=2e^(−1.45) and solve; p≈0.0000522." }, { "id": "cie-2025-w-61-q04a", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "4(a): Find α from the ratio of confidence-interval widths.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal critical values and the width ratio 1.414 to obtain α≈98%." }, { "id": "cie-2025-w-61-q04b", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "4(b): Find the probability the 90% interval covers μ, given the α% interval does.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The conditional coverage probability is 90/98=45/49≈0.918." }, { "id": "cie-2025-w-61-q05a", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "5(a): Find the actual Type I error probability for the one-sided binomial test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Under Bin(40,0.2), the rejection region at 2.5% is X≤2 and P(X≤2)≈0.00794." }, { "id": "cie-2025-w-61-q05b", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "5(b): State the rejection region for the test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The critical region is X≤2." }, { "id": "cie-2025-w-61-q05c", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "5(c): Explain whether the observed result can be a Type II error.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The observed X=2 is in the rejection region, so H₀ is rejected and a Type II error has not occurred for this result." }, { "id": "cie-2025-w-61-q06a", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "6(a): Find the chance the total mass of one large and one small potato bag exceeds 3.55 kg.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The sum is N(3.3,0.07); standardise 3.55 to obtain P≈0.172." }, { "id": "cie-2025-w-61-q06b", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "6(b): Find the probability a large bag is lighter than three times a small bag.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "For D=L−3S, use mean 0.1 and variance 0.23; standardise zero to obtain P(D<0)≈0.417." }, { "id": "cie-2025-w-61-q07a", "paperId": "cie-2025-w-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/61", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "7(a): Find the probability a student takes more than 4.5 minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_61.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_61.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density over 4.50.5; the largest integer is r=13." }, { "id": "cie-2025-w-62-q04a", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "4(a): Use a binomial test at 5% to assess the claim that the defect rate is below 18%.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Assume independent cups with common defect probability. Under p=0.18, P(X≤3)≈0.0542>0.05, so there is insufficient evidence for a lower rate." }, { "id": "cie-2025-w-62-q04b", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "4(b): Explain why the Poisson approximation is unsuitable for the test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Here np=7.2 is not small, so the usual rare-event approximation criterion is not met." }, { "id": "cie-2025-w-62-q05a", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "5(a): Find the normalising constant k for the probability density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate k(2x²−x³) over 0≤x≤2 and set the total area to one; k=3/4." }, { "id": "cie-2025-w-62-q05bi", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 1, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "5(b)(i): State P(X≤m) for the median m.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "By definition of the median, P(X≤m)=1/2." }, { "id": "cie-2025-w-62-q05bii", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "5(b)(ii): Find P(E(X)≤X≤m).", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Find E(X)=6/5, use the median condition to locate m, then integrate the density between E(X) and m; the probability is about 0.0248." }, { "id": "cie-2025-w-62-q06a", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "6(a): Test whether the mean weekly profit has fallen after the management change.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use H₀:μ=736 against H₁:μ<736 with σ=26 and n=35. The statistic is z≈−2.503, beyond the 2% lower-tail critical value, so reject H₀." }, { "id": "cie-2025-w-62-q06b", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "6(b): State the Type I error probability.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "At a 2% significance level, the Type I error probability is 0.02." }, { "id": "cie-2025-w-62-q06c", "paperId": "cie-2025-w-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/62", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "6(c): Find the Type II error probability if the true mean is 718.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_62.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_62.pdf", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the test’s critical boundary with sampling distribution N(718,26²/35); the probability of not rejecting H₀ is about 0.0205." }, { "id": "cie-2025-w-63-q01a", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(a): Find P(22) for independent Poisson variables.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=2", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Since X+Y~Po(5), evaluate the upper tail above 2 to get approximately 0.875." }, { "id": "cie-2025-w-63-q01c", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "1(c): Use a normal approximation for the total of 100 X-values and 150 Y-values.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=3", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The aggregate has mean and variance 600; use N(600,600) with continuity correction at 559.5 to obtain P(T<560)≈0.0492." }, { "id": "cie-2025-w-63-q02", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2", "marks": 8, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "2: Test whether the mean tea-packet mass differs from 500 g at the 5% level.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=4", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Calculate x̄=499.5 and the unbiased variance≈4.83; the two-sided test statistic is z≈−1.762, so do not reject H₀ because |z|<1.96." }, { "id": "cie-2025-w-63-q03a", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(a): Use a Poisson approximation to find P(X<4) incorrect characters.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "With λ=np=1.45, sum the Poisson probabilities for 0,1,2,3 errors to obtain about 0.940." }, { "id": "cie-2025-w-63-q03b", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "3(b): Find a new error rate that doubles the probability of no errors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set e^(−14500p)=2e^(−1.45) and solve to get p≈0.0000522." }, { "id": "cie-2025-w-63-q04a", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(a): Find α from the stated ratio of confidence-interval widths.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the normal critical values; the resulting confidence level is approximately 98%." }, { "id": "cie-2025-w-63-q04b", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "4(b): Find the conditional coverage probability for the 90% interval.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The probability is 0.90/0.98=45/49≈0.918." }, { "id": "cie-2025-w-63-q05a", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "5(a): Find the actual Type I error probability for the one-sided household test.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Under Bin(40,0.2), the rejection region is X≤2; the actual Type I probability is P(X≤2)≈0.00794." }, { "id": "cie-2025-w-63-q05b", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "5(b): State the rejection region.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Reject H₀ when X≤2." }, { "id": "cie-2025-w-63-q05c", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "5(c): State whether the observed result can be a Type II error.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Since the observed X=2 is in the rejection region, H₀ is rejected; this result is not a Type II error." }, { "id": "cie-2025-w-63-q06a", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(a): Find the probability the total mass of a large and small potato bag exceeds 3.55 kg.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The total is N(3.3,0.07); standardising gives a probability of about 0.172." }, { "id": "cie-2025-w-63-q06b", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "6(b): Find the probability a large bag weighs less than three times a small bag.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "For D=L−3S, use mean 0.1 and variance 0.23; P(D<0)≈0.417." }, { "id": "cie-2025-w-63-q07a", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "7(a): Find the probability a student takes longer than 4.5 minutes.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density from 4.5 to 5 to obtain 5/32." }, { "id": "cie-2025-w-63-q07b", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "7(b): State the median of X.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_63.pdf#page=11", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_63.pdf", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "By symmetry, the median is 4 minutes." }, { "id": "cie-2025-w-63-q07c", "paperId": "cie-2025-w-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/63", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "7(c): Find P(3.50.025, so do not reject H₀." }, { "id": "cie-2025-w-65-q03b", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "3(b): Explain which testing errors may have occurred.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=5", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Because H₀ was not rejected, a Type II error may have occurred; a Type I error did not occur for this decision." }, { "id": "cie-2025-w-65-q04ai", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a)(i): Approximate the probability of at least four incorrect characters in a magazine.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use Poisson λ=2430/540=4.5; calculate P(X≥4)≈0.658." }, { "id": "cie-2025-w-65-q04aii", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(a)(ii): Justify the Poisson approximation for the character errors.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=6", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The number of characters is large and the error probability per character is small." }, { "id": "cie-2025-w-65-q04bi", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b)(i): State a condition for monthly sales to follow a Poisson distribution.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "One valid condition is independent sales occurring at a constant average rate." }, { "id": "cie-2025-w-65-q04bii", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "4(b)(ii): Approximate the chance that more than 130 magazines sell in a month.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=7", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated Poisson mean 123.4 for the monthly count and approximate with a normal distribution, applying continuity correction at 130.5; the upper-tail probability is about 0.261." }, { "id": "cie-2025-w-65-q05a", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(a): Find P(L>S+0.5) for the independent tomato-bag masses.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=8", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The difference L−S is N(0.59,0.21); standardise 0.5 to obtain approximately 0.578." }, { "id": "cie-2025-w-65-q05b", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "5(b): Find the probability the combined tomato price is below $16.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=9", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "The total price is 4.30(L+S), normal with mean about 15.523 and variance 3.8829; standardise $16 to get about 0.596." }, { "id": "cie-2025-w-65-q06a", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "6(a): Find b in terms of a so the triangular graph is a probability density.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Set the triangle area to one: ab/2=1, so b=2/a." }, { "id": "cie-2025-w-65-q06b", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2025, "session": "October/November", "paperCode": "9709/65", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "6(b): Use the expected value to find a.", "qpUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_qp_65.pdf#page=10", "msUrl": "https://papers.xtremepape.rs/CAIE/AS%20and%20A%20Level/Mathematics%20(9709)/9709_w25_ms_65.pdf", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "sourceEvidencePages": null, "tagConfidence": "reviewed", "answerGuide": "Integrate x f(x) over the support and set E(X)=4/9; with b=2/a this gives a=3/2 and b=4/3." }, { "id": "cie-2025-w-65-q06c", "paperId": "cie-2025-w-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", 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3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Determine the coefficient of the term in x 7 of the binomial expansion of \u001b \u001a \u0019 x giving your answer in simplest form.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-que-20240117.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wma12-01-rms-20240307.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Apply the relevant sequence, expansion or index rule, show the intermediate algebra, and state the exact form requested.", "sourceFiles": { "questionPaper": "wma12-01-que-20240117.pdf", "markScheme": "wma12-01-rms-20240307.pdf", "examinerReport": "wma12-01-pef-20240307.pdf" 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"kinematics", "summary": "Determine the speed of Q immediately after receiving the impulse.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME02-01-q3a", "paperId": "ial-2024-1-WME02-01", 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momentum equation, define the quantities before and after the event, and state the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME02-01-q3b", "paperId": "ial-2024-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Apply the model and the work‑energy principle to find the speed of the block at the instant it reaches A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME02-01-q4a", "paperId": "ial-2024-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that d = a.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME02-01-q4b", "paperId": "ial-2024-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine F in terms of W.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=10", "qpPages": [ 10, 10 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=11", "qpPages": [ 14, 14 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", 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"hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The report calls this part more challenging; errors often came from using the wrong resistance force or signs in the trailer equation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-pef-20240307.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the acceleration from the system equation, then apply Newton’s second law to the trailer with gravity and its resistance acting downhill.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" }, 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[ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the magnitude of the resultant force acting on the beam at A.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=12", "qpPages": [ 18, 18 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=14", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME02-01-q7b", "paperId": "ial-2024-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Determine the total kinetic energy lost in the collision between P and Q.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=14", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Set up the relevant energy 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range of possible values of f.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=14", "qpPages": [ 22, 22 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": 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"hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME02-01-q8b", "paperId": "ial-2024-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the value of T.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-que-20240112.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=15", "qpPages": [ 26, 26 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme02-01-rms-20240307.pdf#page=15", "qpPages": [ 26, 26 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme02-01-que-20240112.pdf", "markScheme": "wme02-01-rms-20240307.pdf", "examinerReport": "wme02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "640F92C2F75BB69C061A972A7B9E4722544847359BCD5E9B6BFB8DAB73553AA8", "markScheme": "5998017823A62BB02CB3490601306A35DD61AD9077A152752B5DFA14A4697FE7", "examinerReport": "3EA1C35541A714639F13D6EB669AD9A3162D082420C3F21644C36C70E6E06A4A" } }, { "id": "ial-2024-1-WME03-01-q1a", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that v gR x C 2 where C is a constant.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q1b", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine, in terms of g and R, the speed of S as it hits the surface of the planet.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q2a", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Apply the principle of conservation of mechanical energy to show that 100 91 mg.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Set up the relevant energy or momentum equation, define the quantities before and after the event, and state the requested result with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q2b", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the acceleration of P when it is released from rest at B.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q3a", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Apply the model and algebraic integration to show that the x coordinate of the centre of mass of S is 31 24.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q3b", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the value of α.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q4", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4", "marks": 9, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Demonstrate that the maximum possible angular speed is 16 g r.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q5a", "paperId": "ial-2024-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2024, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Apply algebraic integration to show that the centre of mass of a uniform semicircular disc of radius r and centre O is at a distance 4 r π from the diameter through O [You may assume, without proof, that the area of a.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-que-20240120.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wme03-01-rms-20240307.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "sourceFiles": { "questionPaper": "wme03-01-que-20240120.pdf", "markScheme": "wme03-01-rms-20240307.pdf", "examinerReport": "wme03-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "5CFFF8D620E30A8F3891F4AFAFA581C27718A285F60BE691F3EC0814EE129D96", "markScheme": "DE0B55ED44A20B60D04C50F396E807DC7C9C68DBD5E49A89EDDB9B0739B18644", "examinerReport": "0DE3EEBA72CC88F9B6321AA1A18212836D8004C6DECD6907A6B7162C83779B5C" } }, { "id": "ial-2024-1-WME03-01-q6b", "paperId": "ial-2024-1-WME03-01", 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Give a reason for your answer.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": "wst01-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "B2DE83F047F963A7E58B3E51AFE4525505D3BF0511D5DF5CE9A898B15A48658F", "markScheme": "F8B2108528ED11C54BF3052D7A7EA070A9AB448A25EFDA32469E74F339275994", "examinerReport": "31809717AFA63248606B60BCF2500D0BEA47D792586181B71090941EC6AA1E3E" } }, { "id": "ial-2024-1-WST01-01-q1d", "paperId": "ial-2024-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Estimate the probability that both of their heights are between 8cm and 14cm.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": "wst01-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "B2DE83F047F963A7E58B3E51AFE4525505D3BF0511D5DF5CE9A898B15A48658F", "markScheme": "F8B2108528ED11C54BF3052D7A7EA070A9AB448A25EFDA32469E74F339275994", "examinerReport": "31809717AFA63248606B60BCF2500D0BEA47D792586181B71090941EC6AA1E3E" } }, { "id": "ial-2024-1-WST01-01-q2a", "paperId": "ial-2024-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Work out the mean of the 12 values of the average minimum monthly temperature. (ii) Show that the standard deviation of the 12 values of the average minimum monthly temperature is 8.57°F to 3 significant figures.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": "wst01-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "B2DE83F047F963A7E58B3E51AFE4525505D3BF0511D5DF5CE9A898B15A48658F", "markScheme": "F8B2108528ED11C54BF3052D7A7EA070A9AB448A25EFDA32469E74F339275994", "examinerReport": "31809717AFA63248606B60BCF2500D0BEA47D792586181B71090941EC6AA1E3E" } }, { "id": "ial-2024-1-WST01-01-q2b", "paperId": "ial-2024-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Work out the product moment correlation coefficient between x and y.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": "wst01-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "B2DE83F047F963A7E58B3E51AFE4525505D3BF0511D5DF5CE9A898B15A48658F", "markScheme": "F8B2108528ED11C54BF3052D7A7EA070A9AB448A25EFDA32469E74F339275994", "examinerReport": "31809717AFA63248606B60BCF2500D0BEA47D792586181B71090941EC6AA1E3E" } }, { "id": "ial-2024-1-WST01-01-q2c", "paperId": "ial-2024-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Apply this formula and the values from part (a) to calculate, in °C, the mean and the standard deviation of the 12 values of the average minimum monthly temperature in Kolkata. Give your answers to 3 significant figures.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=7", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": "wst01-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "B2DE83F047F963A7E58B3E51AFE4525505D3BF0511D5DF5CE9A898B15A48658F", "markScheme": 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"Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the tree diagram, in terms of p where necessary.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Express the requested event using the given probabilities or 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"https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": "wst01-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "B2DE83F047F963A7E58B3E51AFE4525505D3BF0511D5DF5CE9A898B15A48658F", "markScheme": "F8B2108528ED11C54BF3052D7A7EA070A9AB448A25EFDA32469E74F339275994", 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"Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine the probability that the student is in Year 12.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-que-20240111.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst01-01-rms-20240307.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "sourceFiles": { "questionPaper": "wst01-01-que-20240111.pdf", "markScheme": "wst01-01-rms-20240307.pdf", "examinerReport": 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many omitted a tail probability or confused a tail probability with the critical region.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-pef-20240307.pdf", "tagConfidence": "reviewed", "answerGuide": "Under the null model, compare the lower and upper cumulative binomial probabilities, choose integer tail cutoffs near 0.025, and state both tail probabilities and the critical region.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" }, "sourceEvidencePages": { "examinerReport": [ 4 ] } }, { "id": "ial-2024-1-WST02-01-q3c", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Determine the actual level of significance of the test based on your critical region from part (b).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q3d", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Comment on Rowan’s belief. Give a reason for your answer.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q3e", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 5, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test, at the 5% level of significance, whether or not there is evidence that the proportion of type B vacuum tubes that shatter when exposed to alternating high and low temperatures is lower than 35% You should state.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=8", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q4a", "paperId": "ial-2024-1-WST02-01", "board": 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state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q4b", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine P1 26 GG 2 |.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=9", "qpPages": [ 12, 12 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q4c", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine E2 33 2 HG Show your working clearly. (Solutions relying on calculator technology are not acceptable.).", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=9", "qpPages": [ 12, 12 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q5a", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Demonstrate that a = 12.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q5bi", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(i)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the value of b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": "EE6E0BF55FC8BAEA4337A10FDEFE12E51B0246D7743DA7424E7E8F1EDE46618F", "examinerReport": "0ECF9657D8D5015D5691CFABA42D95EB914F5CEDBA3DA64CDEAF606EB3253AA5" } }, { "id": "ial-2024-1-WST02-01-q5bii", "paperId": "ial-2024-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine P \u001b \u001a \u0019 12 << W b.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-que-20240113.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International%20Advanced%20Level/Mathematics/2018/Exam-materials/wst02-01-rms-20240307.pdf#page=11", "qpPages": [ 16, 16 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "source-derived", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "sourceFiles": { "questionPaper": "wst02-01-que-20240113.pdf", "markScheme": "wst02-01-rms-20240307.pdf", "examinerReport": "wst02-01-pef-20240307.pdf" }, "sourceSha256": { "questionPaper": "4BE8AE3A24F649D6715B786F80C9C1590AE808E036D9E697BBE0ECA021DBF11F", "markScheme": 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"https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q02ai", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)(i)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand the first binomial expression through its x² term.", "answerGuide": "Apply the binomial coefficients through the second power and simplify each coefficient.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q02aii", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)(ii)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Expand the second binomial expression through its x² term.", "answerGuide": "Use the binomial theorem and retain terms up to x².", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q02b", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Combine the two expansions to obtain the product’s x² coefficient.", "answerGuide": "Multiply only term pairs whose powers add to two, then add their coefficients.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q03", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Write the equation after a horizontal stretch and a translation.", "answerGuide": "Replace the original x-input using the inverse horizontal scale and shift, then apply the vertical translation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q04", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over the stated interval.", "answerGuide": "Rewrite one squared trigonometric term using the Pythagorean identity, solve the resulting equation, and retain every solution in range.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q05", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 5, "questionPart": null, "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find an integer-form normal at a specified point on a curve.", "answerGuide": "Differentiate for the tangent gradient, take the negative reciprocal for the normal, and use the point to form its line equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q06", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 6, "questionPart": null, "marks": 6, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the difference between two arithmetic progressions’ twentieth terms from linked tenth-to-twelfth terms.", "answerGuide": "Use the constant-difference condition to determine the parameter and each progression’s common difference; extrapolate both to term 20 and subtract.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q07a", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact perimeter of a shaded sector-based shape.", "answerGuide": "Use the given midpoint and chord length with the cosine rule to recover the central angle, then combine the straight edges and arc length.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q07b", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact shaded area of the sector-based shape.", "answerGuide": "Subtract the relevant triangle area from the sector area, using the exact angle and radius relationships from part (a).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q08", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 8, "questionPart": null, "marks": 8, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact volume when the region between a reciprocal curve and a line is rotated about the x-axis.", "answerGuide": "Find the intersection limits, integrate π times the difference of the squared radii, and evaluate exactly.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q09a", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the inverse of a function defined on all real numbers.", "answerGuide": "Reverse the input-output relation and isolate the original input, preserving the appropriate root.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q09bi", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)(i)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Use a composite-function condition to determine a positive parameter.", "answerGuide": "Evaluate the required composition carefully, substitute the supplied value, and solve the resulting equation for the positive parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q09bii", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find parameter values for which an equation involving two function compositions has no real roots.", "answerGuide": "Expand the relevant compositions, equate them to form a quadratic, then use its discriminant condition and solve the parameter inequality.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=17", "qpPages": [ 13, 13 ], "msPages": [ 17, 17 ], "hardestPart": "9(b)(ii)", "studyPriority": 1, "difficultyEvidence": "The June 2026 report says fully correct responses were rare and notes errors in expansion, signs and the final inequality for this exact part.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q10a", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 1, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Classify a stationary point using the second derivative.", "answerGuide": "Evaluate the second derivative at the given point; its sign identifies a local maximum or minimum.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q10b", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Recover a curve equation from its second derivative and a known stationary point.", "answerGuide": "Integrate twice, use zero gradient at the stationary point, then use the point’s coordinates to determine the constants.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=18", "qpPages": [ 14, 14 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q10ci", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)(i)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find exact x-coordinates of the other stationary points.", "answerGuide": "Set the first derivative to zero, use the supplied factorisation hint, and solve each resulting factor.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=18", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": "10(c)(i)", "studyPriority": 3, "difficultyEvidence": "The report says few candidates used the supplied hint and few found the other stationary x-values.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q10cii", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(c)(ii)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Classify the additional stationary points.", "answerGuide": "Evaluate the second derivative at each x-coordinate and use its sign to classify the point.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=18", "qpPages": [ 15, 15 ], "msPages": [ 18, 18 ], "hardestPart": "10(c)(ii)", "studyPriority": 2, "difficultyEvidence": "The report says few candidates correctly used the second derivative and many omitted the classification part.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-11-q11", "paperId": "cie-2026-s-11", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/11", "variant": "Variant 1", "questionNumber": 11, "questionPart": null, "marks": 9, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of a quadrilateral formed by two tangents to a circle and the coordinate axes.", "answerGuide": "Use radius-tangent perpendicularity to find the tangent equations and their intersection, then calculate the quadrilateral area from coordinates or integrals.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2011.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2011.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "The report notes that many candidates did not attempt the quadrilateral area and that only a few methods handled the limits correctly.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q01", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an equation that becomes quadratic after substituting for x².", "answerGuide": "Clear the denominator, set u=x², solve the quadratic, reject any negative u, and recover the real x-values.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q02a", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Describe the transformations that map one piecewise graph to another.", "answerGuide": "Match corresponding vertices and line segments to identify the ordered vertical and horizontal shifts and scale changes.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=10", "qpPages": [ 2, 2 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q02b", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Express the transformed graph using the original function and constants.", "answerGuide": "Translate the graph correspondences into the form a f(bx+c)+d, checking the input transformation is reversed inside f.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=11", "qpPages": [ 2, 2 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q03", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find where a line meets the perpendicular bisector of a segment.", "answerGuide": "Find the segment midpoint and slope, use the negative reciprocal for the bisector slope, then solve the two line equations together.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=12", "qpPages": [ 2, 2 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q04ai", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(i)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first three terms of a binomial expansion.", "answerGuide": "Use the first three binomial terms and simplify them in ascending powers of x.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=13", "qpPages": [ 2, 2 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q04aii", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(a)(ii)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first three terms of a second binomial expansion involving a parameter.", "answerGuide": "Expand through the x² term while retaining the parameter in the coefficients.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=13", "qpPages": [ 2, 2 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q04b", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use a product coefficient condition to solve for a parameter.", "answerGuide": "Form the x² coefficient from the valid products of terms in the two expansions, equate it to the given value, and solve.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=13", "qpPages": [ 2, 2 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q05", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 5, "questionPart": null, "marks": 7, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the other stationary point when a curve’s second derivative and two minima are given.", "answerGuide": "Integrate the second derivative, apply zero gradient at the two minima to determine constants, then solve the first derivative for stationary x-values and use the curve conditions to identify the remaining point.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=14", "qpPages": [ 3, 3 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q06", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 6, "questionPart": null, "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over a full-turn interval.", "answerGuide": "Convert the equation to one trigonometric function with standard identities and list all solutions in the specified interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=15", "qpPages": [ 3, 3 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q07a", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum of a finite arithmetic progression from its second, fourth and last terms.", "answerGuide": "Use consecutive-term differences to find the common difference and number of terms, then apply the finite-sum formula.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=16", "qpPages": [ 3, 3 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q07bi", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(i)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the parameter in a geometric progression from three consecutive terms.", "answerGuide": "Apply the geometric mean relation to the three terms and solve, using the stated negative sign to select the valid parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=16", "qpPages": [ 3, 3 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q07bii", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)(ii)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum to infinity of the geometric progression.", "answerGuide": "Use the parameter to recover the first term and common ratio, verify convergence, then use the sum-to-infinity formula.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=17", "qpPages": [ 3, 3 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q08a", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of a circle from the endpoints of a diameter.", "answerGuide": "Use the midpoint as the centre and half the diameter length as the radius in the circle equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=18", "qpPages": [ 3, 3 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q08b", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 7, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact chord length cut from a circle by a line.", "answerGuide": "Substitute the line into the circle, solve for the intersection points, then use the distance formula and simplify the surd.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=19", "qpPages": [ 3, 3 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q09a", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the area under the curve from the y-axis to the x-coordinate of R.", "answerGuide": "Integrate the curve above the x-axis from x=0 to x=4.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=20", "qpPages": [ 4, 4 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q09b", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 7, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Choose a parameter so that a coordinate-based triangle has the same area as the region.", "answerGuide": "Find the normal equation and its intercepts, express the triangle area in terms of the parameter, and equate it to part (a).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=21", "qpPages": [ 4, 4 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q10a", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find parameter values that give a repeated root in a quadratic.", "answerGuide": "Set the discriminant equal to zero and solve for the parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=22", "qpPages": [ 4, 4 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q10bi", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(i)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Show the least domain cutoff that makes a restricted quadratic invertible.", "answerGuide": "Use the vertex and monotonic branch of the quadratic to establish the smallest admissible domain boundary.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=22", "qpPages": [ 4, 4 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-12-q10bii", "paperId": "cie-2026-s-12", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/12", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse of a restricted quadratic for a specified parameter.", "answerGuide": "Complete the square or use the quadratic formula, interchange x and y, and select the root consistent with the restricted domain.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2012.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2012.pdf#page=23", "qpPages": [ 4, 4 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q01a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection points of a curve and a straight line.", "answerGuide": "Equate the equations, reduce to a quadratic, solve for x, and substitute back to find both y-coordinates.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q01b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the intersection points to solve a related inequality.", "answerGuide": "Use the roots from part (a) to identify where the relevant curve lies above or below the line.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q02a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rewrite a radical expression as a rational power.", "answerGuide": "Express the root as a fractional index and combine powers using index laws.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q02b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an equation involving fractional powers of x.", "answerGuide": "Use the fractional-power substitution from part (a), solve the resulting quadratic, then retain only admissible real x-values.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q03a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity.", "answerGuide": "Rewrite tangent in terms of sine and cosine and use the Pythagorean identity to simplify one side to the other.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q03b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the proved identity to solve a trigonometric equation on the stated interval.", "answerGuide": "Substitute the identity, collect terms and factor; solve both trigonometric factors and include every in-range angle.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q04a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find a curve parameter from the gradient of a normal at a point.", "answerGuide": "Differentiate, convert the given parallel normal direction to the corresponding curve gradient, and solve for the parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q04b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find another point where the curve normal has the same direction.", "answerGuide": "Set the derivative to the same gradient, solve for the second x-coordinate, and substitute into the curve.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q05a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first four terms of a binomial expansion.", "answerGuide": "Use the binomial theorem through x³ and simplify each coefficient.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q05b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Multiply the expansion by a linear factor and retain terms through x³.", "answerGuide": "Multiply each term of the expansion by both terms of the factor, then collect like powers up to x³.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q05c", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "binomial", "numerical-methods" ], "primaryTopicId": "binomial", "summary": "Use the truncated expansion to approximate a numerical value.", "answerGuide": "Choose x so the expression matches the target, substitute into the series from part (b), and round to five decimal places.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q06a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find all stationary points of a rational-plus-linear function.", "answerGuide": "Differentiate, solve for where the derivative is zero, and substitute the x-values into the function.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q06b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Use a reflection condition on a vertically shifted function to find the shift.", "answerGuide": "Substitute g(x)=f(x)+a into g(−x)=−g(x), compare the constant terms, and solve for a.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q07a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of a garden made from two circular arcs and an equilateral triangle.", "answerGuide": "Add the straight side and the two equal arc lengths, using the 60-degree central angles.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q07b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact area of the garden for a specified side length.", "answerGuide": "Combine the triangle area with the two circular-sector contributions, simplifying the exact surds and π terms.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q08", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 8, "questionPart": null, "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the upper x-limit when the area under a curve from x=1 is specified.", "answerGuide": "Integrate the curve, evaluate the definite integral from 1 to t, equate it to the given area, and solve for the admissible t.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q09a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for a curve defined by a relation between x and y.", "answerGuide": "Differentiate implicitly with respect to x and isolate dy/dx.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=17", "qpPages": [ 11, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q09b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the rate of change of y when x changes at a stated rate.", "answerGuide": "Find x at the given y, evaluate dy/dx there, then use dy/dt=(dy/dx)(dx/dt).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=18", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q10a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Show a quadratic relation for the common ratio of linked geometric and arithmetic progressions.", "answerGuide": "Write the stated terms in terms of first term, common ratio and common difference, equate corresponding terms, then eliminate the other variables.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=19", "qpPages": [ 12, 12 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q10b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the arithmetic progression’s common difference from the geometric series sum.", "answerGuide": "Use the admissible common ratio from part (a), apply the infinite geometric sum to find the first term, and substitute into the linked AP terms.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=20", "qpPages": [ 13, 13 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q11a", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the circle equation from the endpoints of a diameter.", "answerGuide": "Use the diameter’s midpoint and radius to write and expand the circle equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=21", "qpPages": [ 14, 14 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q11b", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter values for which a line is tangent to the circle.", "answerGuide": "Substitute the parameterized line into the circle and set the resulting quadratic discriminant to zero.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=22", "qpPages": [ 15, 15 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-13-q11c", "paperId": "cie-2026-s-13", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/13", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of a quadrilateral bounded by two tangents and the axes.", "answerGuide": "Use the two tangent parameter values to find the intercepts and vertices, then split the quadrilateral into simple coordinate-based areas.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2013.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2013.pdf#page=23", "qpPages": [ 15, 15 ], "msPages": [ 23, 23 ], "hardestPart": "11(c)", "studyPriority": 1, "difficultyEvidence": "The June report says some candidates found the intersection vertices but did not complete a method for the full area in this exact question.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q01", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a pair of simultaneous equations, one nonlinear and one linear.", "answerGuide": "Use the linear equation to substitute for one variable, solve the resulting quadratic, and recover both coordinate pairs.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q02", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 2, "questionPart": null, "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the exact rate of change of water height from a volume-height relation.", "answerGuide": "Differentiate volume with respect to height and use the chain rule with the supplied filling rate at the stated height.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q03a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Relate two coefficients in binomial expansions to express p in terms of q.", "answerGuide": "Identify the terms that generate x⁴ and x², equate their coefficients and rearrange for p.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q03b", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find two possible values of a coefficient after using a specified parameter relation.", "answerGuide": "Use part (a) to determine q, then form the requested coefficient in each of the two resulting cases.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q04a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the stationary x-coordinate of a quadratic curve.", "answerGuide": "Differentiate and set the gradient equal to zero.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q04b", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "Find the transformed stationary x-coordinate after a horizontal stretch.", "answerGuide": "Apply the horizontal scale factor to the original x-coordinate.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q04c", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Write the transformed quadratic equation after vertical and horizontal stretches.", "answerGuide": "Apply the vertical scale to y and replace x using the inverse horizontal scale, then collect coefficients.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q05a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Complete the square for a quadratic expression.", "answerGuide": "Factor out the leading coefficient and add/subtract the square of half the linear coefficient.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q05b", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a quadratic equation using the completed-square form.", "answerGuide": "Set the completed square equal to the remaining constant and take both square-root branches.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q06", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 6, "questionPart": null, "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter range for a line to meet a quadratic curve.", "answerGuide": "Substitute the line into the curve and require the resulting quadratic to have a non-negative discriminant.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q07a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove a trigonometric identity.", "answerGuide": "Rewrite tangent using sine and cosine, use the Pythagorean identity, and show the two sides reduce to the same expression.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q07b", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the identity to solve a trigonometric equation in degrees.", "answerGuide": "Substitute the result from part (a), factor or reduce to a standard trig equation, and list solutions in the stated range.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q08a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find a hypotenuse length in a right-angled triangle with a split side.", "answerGuide": "Use the angle and given side lengths to resolve the relevant right triangle with an exact trigonometric ratio.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q08bi", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)(i)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Show an exact sine value for an angle in the divided triangle.", "answerGuide": "Use the exact length from part (a) and the sine ratio in the smaller right triangle, then simplify to the requested inverse-sine form.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": "8(b)(i)", "studyPriority": 1, "difficultyEvidence": "The June report says only a few candidates transformed the correct sine expression into the requested exact form, and working often did not link the two.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q08bii", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)(ii)", "marks": 1, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find an exact angle using the sine result from the previous part.", "answerGuide": "Use the exact inverse-sine value and the angle sum in the right triangle to obtain the requested angle.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=15", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q09ai", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(a)(i)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate a restricted-domain function.", "answerGuide": "Apply the appropriate power rule to each term and simplify.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q09aii", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(a)(ii)", "marks": 1, "topicIds": [ "functions", "differentiation" ], "primaryTopicId": "functions", "summary": "Show that the function is increasing on its domain.", "answerGuide": "Show that the derivative is positive throughout the stated domain.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q09bi", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(b)(i)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find an expression for the inverse function.", "answerGuide": "Set y=f(x), rearrange for x, interchange x and y, and keep the branch consistent with the original domain.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q09bii", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the domain and range of the inverse function.", "answerGuide": "Swap the original function’s domain and range, respecting the restricted interval and endpoint values.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=16", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q10", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 10, "questionPart": null, "marks": 7, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the area enclosed by a quadratic curve and a line.", "answerGuide": "Find intersection limits, integrate the difference of the two functions across the bounded interval, and simplify the area.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=17", "qpPages": [ 11, 11 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q11a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find where two tangent lines meet a circle.", "answerGuide": "Substitute each tangent equation into the circle, solve for the point of tangency and report both coordinate pairs.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=18", "qpPages": [ 12, 12 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q11b", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the shaded circular segment area cut off by the chord.", "answerGuide": "Use the centre and radius with the chord endpoints to find the central angle, then subtract the triangle area from the sector area.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=19", "qpPages": [ 13, 13 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q12a", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 12, "questionPart": "12(a)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the first negative term of an arithmetic progression.", "answerGuide": "Use the nth-term expression and common difference to locate the first term below zero, then calculate that term.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=20", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q12bi", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 12, "questionPart": "12(b)(i)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find when a trigonometric geometric progression converges.", "answerGuide": "Form the common ratio from the first two terms and solve the strict condition that its magnitude is less than one in the given interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=20", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-15-q12bii", "paperId": "cie-2026-s-15", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 1", "year": 2026, "session": "May/June", "paperCode": "9709/15", "variant": "Variant 5", "questionNumber": 12, "questionPart": "12(b)(ii)", "marks": 3, "topicIds": [ "sequences", "trigonometry" ], "primaryTopicId": "sequences", "summary": "Find an exact sum to infinity at a specified angle.", "answerGuide": "Evaluate the first term and common ratio at the angle, verify convergence, and substitute into the infinite geometric sum.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2015.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2015.pdf#page=21", "qpPages": [ 15, 15 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q01", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Linearise a relation between x and y using natural logarithms.", "answerGuide": "Take logarithms of both sides, apply log laws with brackets, and rearrange to identify the exact gradient and intercept.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "The matching June report describes frequent bracket omissions and many answers that did not use logarithm laws to reach the required form.", "difficultySourceUrl": "https://www.studocu.com/en-nz/document/acg-strathallan/mathematics/9709-mathematics-june-2026-principal-examiner-report-for-teachers/173431681", "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q02a", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an equation involving two exponential terms.", "answerGuide": "Rearrange the two powers into a linear equation in x, taking care to preserve signs.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q02b", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the exponential-equation result to solve a cotangent equation.", "answerGuide": "Set the cotangent expression equal to the x-solutions from part (a), convert to tangent, and give each angle to the requested precision.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q03a", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Approximate a definite integral with the trapezium rule using three intervals.", "answerGuide": "Calculate the four ordinates, apply the trapezium-rule weights, and round the estimate to three significant figures.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q03b", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Decide whether the trapezium-rule estimate is high or low.", "answerGuide": "Use the curve’s concavity over the interval to determine whether the trapezia lie above or below it.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q04a", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write a sine-cosine combination as a single shifted cosine.", "answerGuide": "Set R to the amplitude from the coefficient squares, then match coefficients to find the phase angle in the required range.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=11", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q04b", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation using the single-cosine form.", "answerGuide": "Use the phase form from part (a), find the principal angles, and include all solutions within the given degree interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q05a", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide a polynomial and verify the stated remainder.", "answerGuide": "Use polynomial or synthetic division and show the quotient together with the remainder.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q05b", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Identify one root and show that the other root satisfies a given equation.", "answerGuide": "Use the remainder/factor information to determine the known intercept, then factor the remaining polynomial to obtain the equation for the second root.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q05c", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Bracket a positive root between two decimal values.", "answerGuide": "Evaluate the defining function at both endpoints and show the signs differ.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q05d", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use an iterative formula to find the positive root to four significant figures.", "answerGuide": "Start from a suitable initial estimate, iterate with the prescribed rearrangement, and show successive values to six significant figures.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=13", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q06a", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for a curve given parametrically.", "answerGuide": "Differentiate x and y with respect to the parameter, then divide dy/dt by dx/dt and simplify.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q06b", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the coordinates of the maximum point on a parametric curve.", "answerGuide": "Set dy/dx to zero, solve for the parameter in the stated interval, and substitute into both coordinate expressions.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q06c", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the curve gradient at a specified x-coordinate.", "answerGuide": "Solve the x-parametrisation for the corresponding parameter and substitute into dy/dx.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-21-q07", "paperId": "cie-2026-s-21", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/21", "variant": "Variant 1", "questionNumber": 7, "questionPart": null, "marks": 9, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area under a trigonometric curve between two maximum points.", "answerGuide": "Differentiate to locate the maxima, then integrate the curve between their x-coordinates and simplify the exact result.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2021.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2021.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q01", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate a definite integral containing exponential terms and a positive parameter.", "answerGuide": "Expand the integrand into separate exponentials, integrate term by term, and apply both limits carefully.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q02", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosecant-cotangent equation in degrees.", "answerGuide": "Use reciprocal and Pythagorean identities to obtain an equation in one trigonometric function, then list every solution in the interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q03", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Determine positive constants from a straight-line graph of ln y against x.", "answerGuide": "Take logarithms of the model, use the two plotted points to find the gradient and intercept, then recover the constants.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q04", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find a normal to an exponential curve at its y-intercept.", "answerGuide": "Find the intercept, differentiate using the quotient or product rule, evaluate the tangent gradient there, and form the perpendicular line.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q05a", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an inequality involving exponential expressions.", "answerGuide": "Find the equality boundaries by solving the corresponding linear equation after separating the exponential terms, then test the correct intervals.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q05b", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the smallest and largest integers satisfying a logarithmic version of the inequality.", "answerGuide": "Use monotonicity of the logarithm to convert to bounds on N and identify the integer endpoints.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q06a", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find a parameter using a known linear factor of a quartic.", "answerGuide": "Apply the factor theorem at the stated root and solve for the parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q06b", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "Show that the positive root satisfies a fixed-point equation.", "answerGuide": "Divide out the known factor, rearrange the remaining factor equation, and isolate the positive-root term.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q06c", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration to find the positive root to four significant figures.", "answerGuide": "Iterate using the specified fixed-point rearrangement and show values to six significant figures until they stabilize.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q07", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 7, "questionPart": null, "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of a shaded region under a trigonometric curve.", "answerGuide": "Use the curve’s zeros or stated bounds as the limits, integrate with the appropriate trigonometric identity, and evaluate the exact area.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q08a", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of a parametric curve at a specified parameter value.", "answerGuide": "Differentiate both coordinate expressions with respect to the parameter and evaluate dy/dx at the given value.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-22-q08b", "paperId": "cie-2026-s-22", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/22", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact y-coordinate of a maximum point on a parametric curve.", "answerGuide": "Set dy/dx to zero, solve the resulting trigonometric equation in the given interval, then substitute the valid parameter into y.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2022.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2022.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q01", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate a definite integral containing exponential terms and a positive parameter.", "answerGuide": "Expand the integrand into separate exponentials, integrate term by term, and apply both limits carefully.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q02", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosecant-cotangent equation in degrees.", "answerGuide": "Use reciprocal and Pythagorean identities to obtain an equation in one trigonometric function, then list every solution in the interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q03", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Determine positive constants from a straight-line graph of ln y against x.", "answerGuide": "Take logarithms of the model, use the two plotted points to find the gradient and intercept, then recover the constants.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q04", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find a normal to an exponential curve at its y-intercept.", "answerGuide": "Find the intercept, differentiate using the quotient or product rule, evaluate the tangent gradient there, and form the perpendicular line.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q05a", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an inequality involving exponential expressions.", "answerGuide": "Find the equality boundaries by solving the corresponding linear equation after separating the exponential terms, then test the correct intervals.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q05b", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the smallest and largest integers satisfying a logarithmic version of the inequality.", "answerGuide": "Use monotonicity of the logarithm to convert to bounds on N and identify the integer endpoints.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q06a", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find a parameter using a known linear factor of a quartic.", "answerGuide": "Apply the factor theorem at the stated root and solve for the parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q06b", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "Show that the positive root satisfies a fixed-point equation.", "answerGuide": "Divide out the known factor, rearrange the remaining factor equation, and isolate the positive-root term.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q06c", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration to find the positive root to four significant figures.", "answerGuide": "Iterate using the specified fixed-point rearrangement and show values to six significant figures until they stabilize.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q07", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of a shaded region under a trigonometric curve.", "answerGuide": "Use the curve’s zeros or stated bounds as the limits, integrate with the appropriate trigonometric identity, and evaluate the exact area.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q08a", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of a parametric curve at a specified parameter value.", "answerGuide": "Differentiate both coordinate expressions with respect to the parameter and evaluate dy/dx at the given value.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-23-q08b", "paperId": "cie-2026-s-23", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/23", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact y-coordinate of a maximum point on a parametric curve.", "answerGuide": "Set dy/dx to zero, solve the resulting trigonometric equation in the given interval, then substitute the valid parameter into y.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2023.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2023.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q01", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate a definite integral containing exponential terms and a positive parameter.", "answerGuide": "Expand the integrand into separate exponentials, integrate term by term, and apply both limits carefully.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=9", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q02", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a cosecant-cotangent equation in degrees.", "answerGuide": "Use reciprocal and Pythagorean identities to obtain an equation in one trigonometric function, then list every solution in the interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q03", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Determine positive constants from a straight-line graph of ln y against x.", "answerGuide": "Take logarithms of the model, use the two plotted points to find the gradient and intercept, then recover the constants.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=10", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q04", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find a normal to an exponential curve at its y-intercept.", "answerGuide": "Find the intercept, differentiate using the quotient or product rule, evaluate the tangent gradient there, and form the perpendicular line.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q05a", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an inequality involving exponential expressions.", "answerGuide": "Find the equality boundaries by solving the corresponding linear equation after separating the exponential terms, then test the correct intervals.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q05b", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the smallest and largest integers satisfying a logarithmic version of the inequality.", "answerGuide": "Use monotonicity of the logarithm to convert to bounds on N and identify the integer endpoints.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=12", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q06a", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find a parameter using a known linear factor of a quartic.", "answerGuide": "Apply the factor theorem at the stated root and solve for the parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q06b", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra", "numerical-methods" ], "primaryTopicId": "algebra", "summary": "Show that the positive root satisfies a fixed-point equation.", "answerGuide": "Divide out the known factor, rearrange the remaining factor equation, and isolate the positive-root term.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=13", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q06c", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use iteration to find the positive root to four significant figures.", "answerGuide": "Iterate using the specified fixed-point rearrangement and show values to six significant figures until they stabilize.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=13", "qpPages": [ 9, 9 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q07", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 7, "questionPart": null, "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Find the exact area of a shaded region under a trigonometric curve.", "answerGuide": "Use the curve’s zeros or stated bounds as the limits, integrate with the appropriate trigonometric identity, and evaluate the exact area.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=14", "qpPages": [ 10, 10 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q08a", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the gradient of a parametric curve at a specified parameter value.", "answerGuide": "Differentiate both coordinate expressions with respect to the parameter and evaluate dy/dx at the given value.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-25-q08b", "paperId": "cie-2026-s-25", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 2", "year": 2026, "session": "May/June", "paperCode": "9709/25", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact y-coordinate of a maximum point on a parametric curve.", "answerGuide": "Set dy/dx to zero, solve the resulting trigonometric equation in the given interval, then substitute the valid parameter into y.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2025.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2025.pdf#page=15", "qpPages": [ 13, 13 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "cie-2026-s-31-q01", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve a strict inequality involving two absolute-value expressions.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split at the expressions’ zeroes, solve the resulting linear cases, and combine the valid intervals: x < -13 or x > -11/5." }, { "id": "cie-2026-s-31-q02", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use two points on a straight-line plot of ln y against x to find the constants in an exponential model.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the model as a linear relation in ln y and x, then use the two plotted points to solve for the gradient and intercept and recover A and k." }, { "id": "cie-2026-s-31-q03", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal to a rational curve at x = 1 and give its equation with integer coefficients.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, find the point and tangent gradient at x = 1, take the negative reciprocal, and use point-gradient form." }, { "id": "cie-2026-s-31-q04", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find both exact Cartesian square roots of a complex number by first forming a quartic equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Let a root be x + iy, equate real and imaginary parts after squaring, eliminate one variable to get a quartic, and state both opposite roots." }, { "id": "cie-2026-s-31-q05a", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Prove the stated identity involving a product of sine, cosine, cosecant and cotangent terms.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the reciprocal-trigonometric factors and use a double-angle identity to reduce the left side to the required expression." }, { "id": "cie-2026-s-31-q05b", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Use the identity from part (a) to solve the related trigonometric equation over the stated interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the simplified expression from part (a) equal to 3, solve for the angle, and retain all solutions in [0, π]." }, { "id": "cie-2026-s-31-q06a", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Show that the x-coordinates of the curve’s stationary points satisfy tan x = x + 2.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=15", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the given exponential-trigonometric curve, set dy/dx to zero, and rearrange to the required equation." }, { "id": "cie-2026-s-31-q06b", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use a sketch of two graphs to show the equation from part (a) has exactly one root in the stated interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=16", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch y = tan x and y = x + 2 over the specified interval, using the tangent branch behavior to justify a single intersection." }, { "id": "cie-2026-s-31-q06c", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Verify by calculation that the root lies between 1.2 and 1.3.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=16", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate tan x - x - 2 at both endpoints in radians and show that the signs differ." }, { "id": "cie-2026-s-31-q06d", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Apply the stated iteration to find the root to 3 decimal places, recording iterations to 5 decimal places.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=16", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate in radians using the supplied recurrence, retain 5 d.p. at each step, and check successive values support the rounded root." }, { "id": "cie-2026-s-31-q07a", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Express the shaded Argand region as two inequalities in z.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the vertical boundary as Re z ≥ -2 and the circular boundary as |z + 3 + i| ≤ 2; both boundaries are included." }, { "id": "cie-2026-s-31-q07b", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the greatest modulus attained by a point in the shaded Argand region.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the distance from the origin along the feasible boundary and identify the farthest point; the maximum modulus is about 3.39." }, { "id": "cie-2026-s-31-q08", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 8, "questionPart": null, "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate exactly a definite integral of x²e^(3x) from 0 to 1.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=18", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate by parts twice, substitute both limits, and simplify the exact result." }, { "id": "cie-2026-s-31-q09a", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector equation for the line through two given position-vector points.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=19", "qpPages": [ 12, 12 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the position vectors to obtain a direction vector and use either point in the line equation." }, { "id": "cie-2026-s-31-q09b", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Classify the relationship between the vector line from part (a) and a second 3D line.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=20", "qpPages": [ 12, 12 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare direction vectors to rule out parallel lines, then solve the coordinate equations and show the lines do not intersect; conclude they are skew." }, { "id": "cie-2026-s-31-q10a", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine two polynomial coefficients from a known factor and a derivative-remainder condition.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=22", "qpPages": [ 14, 14 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use p(2) = 0 and p′(2) = 24 to form two simultaneous equations, then solve for a and b." }, { "id": "cie-2026-s-31-q10bi", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factor the resulting polynomial completely.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=22", "qpPages": [ 15, 15 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the coefficients from part (a), divide by the known linear factor, and factor the remaining quadratic." }, { "id": "cie-2026-s-31-q10bii", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 10, "questionPart": "10(b)(ii)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the strict inequality p(x) > 0 using the factorisation from part (b)(i).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=22", "qpPages": [ 15, 15 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the real roots and a sign chart to select the intervals where the polynomial is positive." }, { "id": "cie-2026-s-31-q11a", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(a)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate a sine-cosine expression to establish the stated derivative identity under cot y = cot(x/4).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=23", "qpPages": [ 16, 16 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate by the product rule, then substitute the cotangent relation and simplify with a Pythagorean identity." }, { "id": "cie-2026-s-31-q11b", "paperId": "cie-2026-s-31", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/31", "variant": "Variant 1", "questionNumber": 11, "questionPart": "11(b)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Solve the given separable differential equation with an initial condition, then choose y at x = 1 from the required range.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2031.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2031.pdf#page=23", "qpPages": [ 16, 16 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate the x and y terms, integrate using the identity from part (a), apply the initial value, and select the solution in the stated interval." }, { "id": "cie-2026-s-32-q01", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Divide a quartic polynomial by a quadratic and state the quotient and remainder.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use polynomial long division or compare coefficients in dividend = divisor × quotient + remainder, then verify by expansion." }, { "id": "cie-2026-s-32-q02", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation and express the exact result as a ratio of natural logarithms.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=10", "qpPages": [ 2, 2 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Isolate the exponential term, take natural logarithms, collect the unknown, and write the result in the requested ln ratio form." }, { "id": "cie-2026-s-32-q03a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Multiply two complex numbers in exponential form and express the product in the required argument range.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=11", "qpPages": [ 2, 2 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the moduli and add the arguments, reducing the result to the specified interval." }, { "id": "cie-2026-s-32-q03b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Describe the scale change and rotation represented by multiplying one complex number by another.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=11", "qpPages": [ 2, 2 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the modulus of the multiplier for the scale factor and its argument for the anticlockwise rotation." }, { "id": "cie-2026-s-32-q04", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find exact Cartesian square roots of a complex number by forming a quartic equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=12", "qpPages": [ 2, 2 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the root as x + iy, equate real and imaginary parts after squaring, eliminate one variable, and give both opposite roots." }, { "id": "cie-2026-s-32-q05a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric equations to express dy/dx in terms of tan(2θ).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=13", "qpPages": [ 2, 2 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form dy/dθ divided by dx/dθ, simplify the quotient, and express it using tan(2θ)." }, { "id": "cie-2026-s-32-q05b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact coordinates where the parametric curve has tangent gradient -1.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=14", "qpPages": [ 2, 2 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative from part (a) equal to -1 in the allowed parameter interval, then substitute the parameter into both coordinate functions." }, { "id": "cie-2026-s-32-q06a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradient of the trigonometric curve at the specified x-value.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=15", "qpPages": [ 2, 2 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve and substitute the given angle using exact trigonometric values." }, { "id": "cie-2026-s-32-q06b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate exactly a definite integral involving the trigonometric curve from part (a).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=16", "qpPages": [ 2, 2 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the curve using a suitable trigonometric identity, integrate over the stated limits, and substitute exact endpoint values." }, { "id": "cie-2026-s-32-q07a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Use the given gradient condition to show that the parameter a satisfies the stated trigonometric equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=17", "qpPages": [ 3, 3 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the curve, set its gradient at x = a to the given value, and simplify to the required equation." }, { "id": "cie-2026-s-32-q07b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the equation from part (a) for every a in the stated interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=17", "qpPages": [ 3, 3 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the trigonometric equation over the full interval and list every admissible value." }, { "id": "cie-2026-s-32-q08a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Use a graph sketch to show the trigonometric equation has exactly one root in the interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=18", "qpPages": [ 3, 3 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the relevant curves over the specified interval and explain why they intersect exactly once." }, { "id": "cie-2026-s-32-q08b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Show by calculation that the root lies between 1 and 1.2.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=19", "qpPages": [ 3, 3 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate a suitable rearranged expression at 1 and 1.2 in radians and show a sign change." }, { "id": "cie-2026-s-32-q08c", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Show that any convergent value of the supplied iteration must equal the equation’s root.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=20", "qpPages": [ 3, 3 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set consecutive terms equal at a fixed point and rearrange to the original trigonometric equation." }, { "id": "cie-2026-s-32-q08d", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration to calculate the root to 3 decimal places, giving iterations to 5 decimal places.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=21", "qpPages": [ 3, 3 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Iterate in radians using the supplied formula, retain 5 d.p. at each step, and round the converged value to 3 d.p." }, { "id": "cie-2026-s-32-q09a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector equation for the line through two given points in 3D.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=22", "qpPages": [ 3, 3 ], "msPages": [ 22, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the difference between the position vectors as the direction vector and take either point as the base point." }, { "id": "cie-2026-s-32-q09b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Decide whether the two given 3D lines are parallel, intersecting or skew.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=23", "qpPages": [ 3, 3 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare their direction vectors, then equate coordinates of general points to test for an intersection." }, { "id": "cie-2026-s-32-q09c", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 9, "questionPart": "9(c)", "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the position vector of a point on one line that makes a 60° angle with the specified point-to-point vector.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=24", "qpPages": [ 3, 3 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the point on the line using a parameter, form the relevant vectors, apply the dot-product angle formula, and solve for the admissible parameter." }, { "id": "cie-2026-s-32-q10a", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the given rational function as partial fractions.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=25", "qpPages": [ 4, 4 ], "msPages": [ 25, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the expression equal to the required partial-fraction form, clear denominators, and compare coefficients." }, { "id": "cie-2026-s-32-q10b", "paperId": "cie-2026-s-32", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/32", "variant": "Variant 2", "questionNumber": 10, "questionPart": "10(b)", "marks": 7, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the first-order differential equation using its initial condition and state a relation between x and y.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2032.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2032.pdf#page=26", "qpPages": [ 4, 4 ], "msPages": [ 26, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate the variables, integrate each side, use the condition y = 1 at x = 4 to fix the constant, and rearrange." }, { "id": "cie-2026-s-33-q01", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 1, "questionPart": null, "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve a logarithmic equation and give the simplified exact solution.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms using valid log rules, solve the resulting equation, and reject any value outside the logarithm domains." }, { "id": "cie-2026-s-33-q02", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve an absolute-value inequality and give the interval endpoints to 3 significant figures.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split at the absolute-value expressions’ critical points, solve each case, combine the intervals, and round the endpoints." }, { "id": "cie-2026-s-33-q03", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Shade the intersection of two regions defined by complex-modulus inequalities on an Argand diagram.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate each modulus into its circle or perpendicular-bisector boundary, mark included boundaries, and shade the common region." }, { "id": "cie-2026-s-33-q04", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 4, "questionPart": null, "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation in degrees over the stated interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the sine and cosine terms using a compound-angle identity, solve the resulting equation, and retain every solution in the interval." }, { "id": "cie-2026-s-33-q05a", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the parametric equations and write dy/dx in the requested form.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide dy/dt by dx/dt, simplify, and compare with the stated trigonometric form to identify its constant." }, { "id": "cie-2026-s-33-q05b", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find an exact tangent equation at the specified parameter value.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=15", "qpPages": [ 7, 7 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use part (a) for the gradient, substitute the parameter into both coordinates, and apply point-gradient form." }, { "id": "cie-2026-s-33-q06", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 6, "questionPart": null, "marks": 6, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate exactly a trigonometric definite integral using the stated substitution.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=16", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set u equal to the given expression, convert the differential and limits, simplify the algebraic integrand, and evaluate at the new endpoints." }, { "id": "cie-2026-s-33-q07", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 7, "questionPart": null, "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find exact Cartesian square roots of a complex number using a quartic equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=17", "qpPages": [ 9, 9 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Let the root be x + iy, equate real and imaginary parts after squaring, eliminate one variable to form a quartic, and state both roots." }, { "id": "cie-2026-s-33-q08a", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Use graphs to establish that the stated equation has exactly one real root.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the two sides and justify that their graphs intersect exactly once." }, { "id": "cie-2026-s-33-q08b", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Show that a convergent sequence defined by the iteration converges to the root from part (a).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=18", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assume the sequence converges, equate its limiting value to its successor in the recurrence, and rearrange to the original equation." }, { "id": "cie-2026-s-33-q08c", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the root lies between 1 and 2.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=19", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the equation’s left-minus-right side at 1 and 2 and show that the values have opposite signs." }, { "id": "cie-2026-s-33-q08d", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Iterate to calculate the root to 2 decimal places, recording values to 5 decimal places.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=20", "qpPages": [ 11, 11 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the recurrence with the supplied starting value, preserve 5 d.p. at each step, and round the limiting root to 2 d.p." }, { "id": "cie-2026-s-33-q09a", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(a)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Express the rational function as partial fractions.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=21", "qpPages": [ 12, 12 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor the denominator, write the correct partial-fraction form, clear denominators, and compare coefficients." }, { "id": "cie-2026-s-33-q09b", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the partial fractions to expand the function in ascending powers through x³.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=21", "qpPages": [ 13, 13 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand each partial-fraction term as a geometric series and collect the coefficients through x³." }, { "id": "cie-2026-s-33-q09c", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 9, "questionPart": "9(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "State the interval of x for which the power-series expansion is valid.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=21", "qpPages": [ 13, 13 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the convergence condition for each geometric series and take the intersection of the resulting bounds." }, { "id": "cie-2026-s-33-q10", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 10, "questionPart": null, "marks": 8, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve a first-order differential equation explicitly for y using the given initial condition.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=22", "qpPages": [ 14, 14 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate, apply the specified point to determine the constant, and isolate y." }, { "id": "cie-2026-s-33-q11a", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a vector equation for the line through two position-vector points.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=23", "qpPages": [ 16, 16 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Subtract the position vectors for a direction vector and use either point as the base point." }, { "id": "cie-2026-s-33-q11b", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the acute angle between the two 3D lines.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=24", "qpPages": [ 16, 16 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the dot product of direction vectors, divide by the product of their magnitudes, and take the inverse cosine." }, { "id": "cie-2026-s-33-q11c", "paperId": "cie-2026-s-33", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/33", "variant": "Variant 3", "questionNumber": 11, "questionPart": "11(c)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the perpendicular distance from a point to the specified 3D line.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2033.pdf#page=17", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2033.pdf#page=25", "qpPages": [ 17, 17 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Project the point-to-line vector onto the line direction or use the cross-product distance formula, then simplify." }, { "id": "cie-2026-s-35-q01a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the coefficient of x² in the stated binomial expansion.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the expression in generalized-binomial form and select the term whose power of x is 2." }, { "id": "cie-2026-s-35-q01b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "State the range of x for which the binomial expansion is valid.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=9", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the generalized-binomial convergence condition |u| < 1 to the expression inside the power." }, { "id": "cie-2026-s-35-q02", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Implicitly differentiate the curve and find its gradient at the point where y = 1.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=10", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate every term with respect to x, collect dy/dx, solve for the corresponding x when y = 1, and substitute; the gradient is 2/9." }, { "id": "cie-2026-s-35-q03a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Show that the given power-law model becomes linear when both variables are logged.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take natural logarithms and rearrange to a straight-line relation between ln y and ln x." }, { "id": "cie-2026-s-35-q03b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Estimate the two model constants from the two supplied data points, to 2 significant figures.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute each observation into the power-law model, solve the two equations, and round both constants; they are 2.5 and 40." }, { "id": "cie-2026-s-35-q04a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the stated absolute-value inequality.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Split the expression at its critical values, solve each linear case, and combine the valid intervals." }, { "id": "cie-2026-s-35-q04b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "algebra", "exponentials" ], "primaryTopicId": "algebra", "summary": "Use the bound from part (a) to find the largest integer satisfying the logarithmic inequality.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the sequence expression into the upper bound, apply logarithms with the correct inequality direction, and choose the greatest admissible integer." }, { "id": "cie-2026-s-35-q05a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Differentiate the trigonometric curve and simplify dy/dx to a fraction involving cos x.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quotient rule, then use double-angle and Pythagorean identities to express the result in cos x only." }, { "id": "cie-2026-s-35-q05b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find an exact equation for the normal at x = π/2.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the point and tangent gradient there, take the negative reciprocal, and use point-gradient form." }, { "id": "cie-2026-s-35-q06", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 6, "questionPart": null, "marks": 7, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Form and solve a differential equation when the curve gradient is proportional to y, using two given points.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=14", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write dy/dx = ky, separate and integrate, then use both points to determine the constants and express y in terms of x." }, { "id": "cie-2026-s-35-q07a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use a complex-number identity to prove that the real parameter a satisfies a quadratic equation.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=15", "qpPages": [ 8, 8 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply by the conjugate or rationalise the denominator, use i² = -1, compare real and imaginary parts, and eliminate the other real parameter." }, { "id": "cie-2026-s-35-q07b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find all possible values of a and the corresponding values of the second real parameter.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=16", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic from part (a), then substitute each root into either real or imaginary component equation." }, { "id": "cie-2026-s-35-q08a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Sketch graphs to show that the stated trigonometric equation has exactly one root in the interval.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sketch the secant and exponential expressions on the required interval and justify their single intersection." }, { "id": "cie-2026-s-35-q08b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the root lies between the two stated decimal bounds.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=17", "qpPages": [ 10, 10 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate a suitable difference at each endpoint in radians and show a sign change." }, { "id": "cie-2026-s-35-q08c", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Show that any convergent sequence from the given iteration converges to the root.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=18", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set consecutive terms equal at the fixed point and rearrange to the original trigonometric equation." }, { "id": "cie-2026-s-35-q08d", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Iterate from the supplied initial value to calculate the root to 3 decimal places.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=18", "qpPages": [ 11, 11 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the recurrence in radians, retain the requested precision at each step, and round the converged value to 3 d.p." }, { "id": "cie-2026-s-35-q09a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite a sine-cosine combination in the form R cos(θ + α), giving exact R and the requested form for α.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=19", "qpPages": [ 12, 12 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match coefficients to R cos α and R sin α, then express the phase using the required inverse-tangent notation." }, { "id": "cie-2026-s-35-q09b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "integration", "summary": "Use the result from part (a) to evaluate a definite trigonometric integral exactly.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=19", "qpPages": [ 13, 13 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the integrand using R cos(θ + α), integrate, substitute the exact limits, and use the phase from part (a)." }, { "id": "cie-2026-s-35-q10a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Use u = sin(x/2) to transform the given definite integral and identify its exact transformed limits and constants.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=14", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=20", "qpPages": [ 14, 14 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the substitution, convert all trigonometric factors and limits to u, and simplify to the stated algebraic integral." }, { "id": "cie-2026-s-35-q10b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate exactly the definite integral obtained in part (a).", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=15", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=20", "qpPages": [ 15, 15 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the transformed algebraic expression and substitute its exact endpoints." }, { "id": "cie-2026-s-35-q11a", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Use position vectors to show that the quadrilateral is a trapezium but not a parallelogram.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=21", "qpPages": [ 16, 16 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find opposite-side direction vectors, show one pair is parallel, and establish that the other pair is not parallel." }, { "id": "cie-2026-s-35-q11b", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(b)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact perpendicular distance from A to the line through C and D.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=16", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=22", "qpPages": [ 16, 16 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take a general point on CD and minimise its distance from A, or use an equivalent projection or cross-product method." }, { "id": "cie-2026-s-35-q11c", "paperId": "cie-2026-s-35", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Pure Mathematics 3", "year": 2026, "session": "May/June", "paperCode": "9709/35", "variant": "Variant 5", "questionNumber": 11, "questionPart": "11(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of the trapezium in simplified form.", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2035.pdf#page=17", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2035.pdf#page=24", "qpPages": [ 17, 17 ], "msPages": [ 24, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use half the sum of the parallel side lengths multiplied by the perpendicular height found in part (b)." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=9", "id": "cie-2026-s-41-q01", "questionNumber": 1, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Combine four coplanar forces shown in a diagram and give the resultant magnitude and direction.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Resolve horizontally and vertically, combine the components, then use the correct quadrant for the direction. The resultant is about 39.5 N at 12.7° above the positive horizontal." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=10", "id": "cie-2026-s-41-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find a car's acceleration on a level road from its engine power, speed, mass and resistance.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Use P=Fv to find the driving force, then apply Newton's second law after subtracting the resistance. The acceleration is 0.75 m s⁻²." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=10", "id": "cie-2026-s-41-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the incline of a hill when a car climbs it at steady speed with specified power and resistance.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Convert engine power to driving force with P=Fv and balance it against resistance and the component of weight down the hill. The incline is about 5.66°." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=11", "id": "cie-2026-s-41-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Differentiate a particle's displacement function and find when its velocity is zero.", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "answerGuide": "Differentiate displacement, set the resulting quadratic velocity to zero and solve; the stopping times are 2.5 s and 4 s." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=11", "id": "cie-2026-s-41-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled over a time interval that includes a change of direction.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Evaluate displacement at both interval endpoints and at the stationary time inside the interval, then add the absolute changes. The total is 40.5 m." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=11", "id": "cie-2026-s-41-q03c", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the magnitude of a particle's acceleration at a specified time from its displacement model.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Differentiate velocity to obtain acceleration, substitute t=3 and take the magnitude; the result is 6 m s⁻²." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=12", "id": "cie-2026-s-41-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use the tension in a three-particle pulley system to determine the mass of a particle attached to one hanging mass.", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "answerGuide": "Choose directions consistently and apply Newton's second law to the two sides of the string while all three masses move together. Eliminate the acceleration using the given tension; m=3.5 kg." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=13", "id": "cie-2026-s-41-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "After one attached mass separates, find the total time until the other hanging mass first stops.", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "answerGuide": "Use the first-stage acceleration to find the speed after the 0.8 m rise. Recalculate the acceleration for the remaining two masses and use v=u+at for the stopping phase; total time is 1.4 s." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=14", "id": "cie-2026-s-41-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the friction coefficient for a particle held in limiting equilibrium on an incline by a horizontal force.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "answerGuide": "Resolve the horizontal force and weight parallel and perpendicular to the plane, then use limiting friction F=μR. This gives μ=√3/5." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=15", "id": "cie-2026-s-41-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "After the horizontal force is removed, find the particle's speed after it slides down the plane.", "qpPages": [ 9, 9 ], "msPages": [ 15, 16 ], "answerGuide": "Use the coefficient from part (a) to find the downslope acceleration, then apply v²=u²+2as over 1.5 m. The speed is √6, about 2.45 m s⁻¹." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=16", "id": "cie-2026-s-41-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Use the stated velocities before and after a one-dimensional collision to find an unknown sphere mass.", "qpPages": [ 10, 10 ], "msPages": [ 16, 16 ], "answerGuide": "Write conservation of momentum for spheres A and B, substitute the given initial and final velocities, and solve the resulting quadratic subject to the stated mass condition. m=3.5 kg." }, { "paperId": "cie-2026-s-41", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/41", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2041.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2041.pdf#page=17", "id": "cie-2026-s-41-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "After sphere B collides with a stationary sphere C, decide whether B will collide again with sphere A.", "qpPages": [ 11, 11 ], "msPages": [ 17, 17 ], "answerGuide": "Use momentum and the specified kinetic-energy loss to determine B's speed after its collision with C, then compare it with A's speed and direction. Since A is faster behind B, a further collision occurs." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=9", "id": "cie-2026-s-42-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the engine power of a car moving steadily against a constant resistance.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "At constant speed the driving force equals the resistance, so use P=Fv. The power is 15.5 kW." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=9", "id": "cie-2026-s-42-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the car's instantaneous acceleration immediately after its power is increased.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Add the power increase to the original engine power, convert the new power to driving force at the unchanged instant speed, then apply Newton's second law. The acceleration is 0.5 m s⁻²." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=10", "id": "cie-2026-s-42-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the speed after two particles coalesce and the combined particle rebounds from a wall with a reduced speed.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Conserve momentum for the particle collision to get the speed before the wall, then reduce it by 25%. The rebound speed is 0.9375 m s⁻¹." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=10", "id": "cie-2026-s-42-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the elapsed time from the first particle collision until the rebounding combined particle is again 9 m from the wall.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Add the travel time for A to reach B to the post-rebound travel time from the wall. The total is 16.8 s." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=11", "id": "cie-2026-s-42-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension from the measured displacement and time of a particle moving down a smooth incline.", "qpPages": [ 4, 4 ], "msPages": [ 11, 12 ], "answerGuide": "Find the common acceleration from s=½at², then resolve forces on the inclined particle and apply Newton's second law. The tension is about 3.55 N." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=12", "id": "cie-2026-s-42-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction coefficient for the particle on the rough horizontal plane.", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "answerGuide": "Use the acceleration and tension from the earlier parts in Newton's second law for the horizontal particle, then divide the friction force by its normal reaction. μ≈0.336." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=13", "id": "cie-2026-s-42-q03c", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "After the string breaks, find the work done by friction as the horizontal particle stops.", "qpPages": [ 5, 5 ], "msPages": [ 13, 14 ], "answerGuide": "Find the speed at the break, use the frictional deceleration to obtain stopping distance, and apply work=force×distance (equivalently use the loss of kinetic energy). Friction removes about 0.563 J." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=15", "id": "cie-2026-s-42-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use work and energy to show the distance a particle travels uphill before stopping on a rough plane.", "qpPages": [ 6, 6 ], "msPages": [ 15, 16 ], "answerGuide": "Resolve weight and friction along the plane, then equate the initial kinetic energy to the work against both forces. The distance is 0.8 m." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=17", "id": "cie-2026-s-42-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the full time for the particle to travel up the rough plane, stop, and return to its starting point.", "qpPages": [ 7, 7 ], "msPages": [ 17, 18 ], "answerGuide": "Find the uphill time from constant deceleration and the downhill time from the friction-adjusted acceleration, then add them. The total is about 1.32 s." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=19", "id": "cie-2026-s-42-q05", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the common height of two particles released or projected downwards at different times when they reach the ground together.", "qpPages": [ 8, 8 ], "msPages": [ 19, 21 ], "answerGuide": "Write a constant-acceleration height equation for each particle, relate their flight times by the one-second delay, and solve the resulting equation. h=13.2 m." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=22", "id": "cie-2026-s-42-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the weight of a crate pulled across rough horizontal ground by an angled rope while accelerating.", "qpPages": [ 10, 10 ], "msPages": [ 22, 22 ], "answerGuide": "Resolve the rope and weight vertically to find the normal reaction, use friction=μR, and apply Newton's second law horizontally. W=66.25 N." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=23", "id": "cie-2026-s-42-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find how far the crate slides after the rope is removed and it decelerates to rest.", "qpPages": [ 11, 11 ], "msPages": [ 23, 23 ], "answerGuide": "After removing the rope, friction alone provides a=μg. Use v²=u²+2as with initial speed 8 m s⁻¹ and final speed zero; AB=12.8 m." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=24", "id": "cie-2026-s-42-q07a", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Use the given velocity function and deceleration at a specified time to find one of its constants.", "qpPages": [ 12, 12 ], "msPages": [ 24, 24 ], "answerGuide": "Differentiate the velocity expression, substitute t=1.25 and acceleration −0.5 m s⁻², then solve. k₁=6." }, { "paperId": "cie-2026-s-42", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/42", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2042.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2042.pdf#page=25", "id": "cie-2026-s-42-q07b", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use a later instantaneous-rest condition to find the distance travelled over the interval from t=0 to t=1.", "qpPages": [ 13, 13 ], "msPages": [ 25, 27 ], "answerGuide": "Set v=0 at t=3.5 to find the remaining constant, integrate the given velocity and evaluate between 0 and 1. The distance is about 6.53 m." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=9", "id": "cie-2026-s-43-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use equal heights at two specified times for an upward vertical projection to find its launch speed.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Equate the two height expressions using s=ut−½gt² and solve; u=50 m s⁻¹." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=9", "id": "cie-2026-s-43-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the maximum height reached by the vertically projected particle.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "At maximum height set v=0 and use the constant-acceleration equations. The greatest height is 125 m." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=9", "id": "cie-2026-s-43-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find when the acceleration is zero from a particle's quadratic velocity function.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Differentiate velocity with respect to time and set the acceleration to zero; t=1.5 s." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=10", "id": "cie-2026-s-43-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled over a time interval when the velocity remains negative throughout.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Integrate the velocity over 0≤t≤3 and take the magnitude of the displacement, since the particle does not stop in the interval. The distance is 18 m." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=11", "id": "cie-2026-s-43-q03", "questionNumber": 3, "questionPart": null, "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the mass of a block pulled up a rough incline by a known force when its acceleration and friction coefficient are given.", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "answerGuide": "Resolve weight and friction down the plane and apply Newton's second law in the direction of motion. The mass is about 1.91 kg." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=12", "id": "cie-2026-s-43-q04", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find a train's constant acceleration and a time parameter from distances and its speed at an intermediate point.", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "answerGuide": "Use the constant-acceleration displacement and velocity equations on AB and BC, form two equations in a and t, and solve; a=0.8 m s⁻² and t=5 s." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=13", "id": "cie-2026-s-43-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the tow-bar tension in a powered lorry-trailer system from the instantaneous power, speed and resistances.", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "answerGuide": "Use P=Fv for the driving force, then apply Newton's second law to the trailer (or the whole system and lorry). The tow-bar tension is 462.5 N." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=14", "id": "cie-2026-s-43-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use work and energy for a lorry-trailer set descending a hill to find the distance over which its speed increases.", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "answerGuide": "Equate the change in kinetic energy of the combined mass to gravitational work plus driving work minus resistance work. The distance is about 120 m." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=15", "id": "cie-2026-s-43-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude and direction of the resultant of three diagrammed forces for a specified force magnitude.", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "answerGuide": "Resolve each force into x- and y-components using the diagram's angles, add the components, then find the resultant magnitude and direction." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=16", "id": "cie-2026-s-43-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Choose a force magnitude that makes the resultant's vertical component zero, then find the remaining resultant.", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "answerGuide": "Set the sum of vertical components to zero to find F; the resultant is then horizontal, with magnitude equal to the absolute horizontal component." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=17", "id": "cie-2026-s-43-q07a", "questionNumber": 7, "questionPart": "7(a)", "marks": 8, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "For two particles moving towards each other on a rough incline, verify one acceleration and determine the other's launch speed from their collision time.", "qpPages": [ 10, 10 ], "msPages": [ 17, 18 ], "answerGuide": "Resolve each particle's weight and friction along the plane to establish the signed accelerations, then use their displacements in 0.25 s and the 3 m initial gap. The stated acceleration is 0.88 m s⁻² and u≈4.48 m s⁻¹." }, { "paperId": "cie-2026-s-43", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/43", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2043.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2043.pdf#page=19", "id": "cie-2026-s-43-q07b", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "After the collision, find how far the rebounding particle travels up the incline before stopping.", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "answerGuide": "Use momentum to find P's velocity immediately before impact, then apply conservation of momentum to the collision. Find P's uphill deceleration from gravity and friction and use v²=u²+2as; the distance is about 0.174 m." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=9", "id": "cie-2026-s-45-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Use the signed area under a piecewise velocity-time graph to find the athlete's displacement at 90 s.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Add the signed trapezium and triangle areas, subtracting the parts below the time axis. The displacement OP is 140 m." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=9", "id": "cie-2026-s-45-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the athlete's total distance over the 90-second motion from the velocity-time graph.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Add the absolute areas above and below the time axis; the total distance is 260 m." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=10", "id": "cie-2026-s-45-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "For two connected particles on a smooth table and over a pulley, find the string tension and acceleration.", "qpPages": [ 3, 3 ], "msPages": [ 10, 11 ], "answerGuide": "Apply Newton's second law to each mass with a shared acceleration and eliminate the acceleration. T=2.4 N and the acceleration is 4 m s⁻²." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=11", "id": "cie-2026-s-45-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time for the hanging particle to fall 0.5 m from rest.", "qpPages": [ 3, 3 ], "msPages": [ 11, 11 ], "answerGuide": "Use s=½at² with the acceleration from part (a), giving t=0.5 s." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=12", "id": "cie-2026-s-45-q03", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Resolve four forces in equilibrium to determine an unknown force and its angle.", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "answerGuide": "Set horizontal and vertical component sums to zero, solve for the force components, then use their ratio and magnitude. F≈30.5 N and θ≈11.9°." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=13", "id": "cie-2026-s-45-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the constant road resistance from the car's steady speed, engine power and travel distance and time.", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "answerGuide": "Find the steady speed from distance/time and use P=Rv. R=750 N." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=14", "id": "cie-2026-s-45-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the time for a car to cover a known distance after its engine power increases, given its final speed and constant resistance.", "qpPages": [ 5, 5 ], "msPages": [ 14, 14 ], "answerGuide": "Using the QP values, set engine work equal to the kinetic-energy gain plus work against resistance: 25 000t = ½(1200)(40²−20²) + 750(500), so t=43.8 s. The paired MS conflicts: it uses 42 kW, 460 m and 32 m s⁻¹, giving 17.1 s; that result is not for the QP values." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=15", "id": "cie-2026-s-45-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Integrate a time-dependent velocity and use the particle's position after 25 s to find its initial distance from a fixed point.", "qpPages": [ 6, 6 ], "msPages": [ 15, 15 ], "answerGuide": "Integrate v=6t−t^(3/2), evaluate the displacement from 0 to 25 s (625 m), then use the given 700 m position. OA=75 m." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=16", "id": "cie-2026-s-45-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 6, "topicIds": [ "kinematics", "differentiation", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the distance between the time acceleration becomes zero and the later time velocity becomes zero.", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "answerGuide": "Solve dv/dt=0 and v=0 for the positive times (16 s and 36 s), then integrate velocity between them. The distance is about 419.2 m." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=17", "id": "cie-2026-s-45-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show that two particles at rest on a rough incline remain stationary when released.", "qpPages": [ 8, 8 ], "msPages": [ 17, 18 ], "answerGuide": "For each particle compare the downslope component of its weight with the maximum available friction; the friction is sufficient for both, so neither moves." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=19", "id": "cie-2026-s-45-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the speed of one particle just before it reaches and collides with the other on the rough incline.", "qpPages": [ 8, 8 ], "msPages": [ 19, 20 ], "answerGuide": "While P moves down the plane, use gravity and opposing friction to find its acceleration, then apply v²=u²+2as over 2.75 m. The speed is 5 m s⁻¹." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=21", "id": "cie-2026-s-45-q06c", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "After the collision, find the separation of both particles once they have each come to rest.", "qpPages": [ 9, 9 ], "msPages": [ 21, 22 ], "answerGuide": "Use momentum to find Q's post-collision speed, then find each particle's stopping distance under its own frictional acceleration. Add the distances as they move in opposite directions; separation≈2.32 m." }, { "paperId": "cie-2026-s-45", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Mechanics", "year": 2026, "session": "May/June", "paperCode": "9709/45", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2045.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2045.pdf#page=23", "id": "cie-2026-s-45-q07", "questionNumber": 7, "questionPart": null, "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Use three equal-duration constant-acceleration stages with two known distances to find acceleration and the total speed increase.", "qpPages": [ 10, 10 ], "msPages": [ 23, 24 ], "answerGuide": "Write displacement or average-speed equations for AB, BC and CD and eliminate the unknown starting speed. This gives a=0.25 m s⁻²; over 30 s the speed increase is 7.5 m s⁻¹." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=9", "id": "cie-2026-s-51-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Build the probability distribution for the sum of the results from two specified four-sided spinners.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "List the 16 equally likely pairs, combine equal sums and divide each frequency by 16." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=10", "id": "cie-2026-s-51-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of the spinner-sum random variable.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Use the distribution from part (a) to calculate E(X) and E(X²), then subtract E(X)²." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=10", "id": "cie-2026-s-51-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a randomly selected student studies Art or Business.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Add the mutually exclusive Art and Business counts and divide by 150." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=11", "id": "cie-2026-s-51-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Test whether studying Business and being male are independent events.", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "answerGuide": "Compare P(M∩B) with P(M)P(B), or compare P(B|M) with P(B), then state the conclusion." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=11", "id": "cie-2026-s-51-q02c", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability a student is male, given that the student studies Art.", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "answerGuide": "Use the male Art count over the total Art count." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=12", "id": "cie-2026-s-51-q02d", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that a student is neither male nor studying Business.", "qpPages": [ 4, 4 ], "msPages": [ 12, 12 ], "answerGuide": "Use the female non-Business count over 150." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=13", "id": "cie-2026-s-51-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Construct a back-to-back stem-and-leaf display of weekly snowfall at two resorts, placing Aldeny on the left.", "qpPages": [ 5, 5 ], "msPages": [ 13, 13 ], "answerGuide": "Sort both data sets, use a common stem, order leaves outward correctly, and include a key." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=14", "id": "cie-2026-s-51-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find Aldeny’s median snowfall and interquartile range.", "qpPages": [ 5, 5 ], "msPages": [ 14, 14 ], "answerGuide": "Order Aldeny’s 15 values; identify the median and subtract the lower quartile from the upper quartile." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=14", "id": "cie-2026-s-51-q03c", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give two contextual comparisons of the snowfall distributions at Aldeny and Farnlee.", "qpPages": [ 5, 5 ], "msPages": [ 14, 14 ], "answerGuide": "Compare centre using medians and spread using interquartile ranges, stating which resort has the larger value." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=15", "id": "cie-2026-s-51-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "With replacement, find the chance that the first red marble appears before Jong’s sixth draw.", "qpPages": [ 6, 6 ], "msPages": [ 15, 15 ], "answerGuide": "Use the complement of five consecutive non-red draws, since “before the sixth choice” means one of the first five draws." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=15", "id": "cie-2026-s-51-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "With replacement, find the chance Kai gets the second green marble on draw nine.", "qpPages": [ 6, 6 ], "msPages": [ 15, 15 ], "answerGuide": "Require exactly one green in the first eight draws, then a green on draw nine; use the binomial count." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=16", "id": "cie-2026-s-51-q04c", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the chance Lam gets fewer than three black marbles in ten draws with replacement.", "qpPages": [ 7, 7 ], "msPages": [ 16, 17 ], "answerGuide": "Model the black-marble count as Binomial(10, 5/12) and add probabilities for 0, 1 and 2." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=18", "id": "cie-2026-s-51-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Three letters are chosen from RETRIEVES; find the chance that at least one is E.", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "answerGuide": "Use 1 minus the probability of choosing all three letters from the six non-E letters, without replacement." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=20", "id": "cie-2026-s-51-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of RETRIEVES with R at both ends and the three Es not all consecutive.", "qpPages": [ 9, 9 ], "msPages": [ 20, 20 ], "answerGuide": "Arrange the remaining seven letters with the repeated E accounted for, then subtract the arrangements with all Es together." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=21", "id": "cie-2026-s-51-q05c", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of RETRIEVES with exactly three letters between the two Rs.", "qpPages": [ 9, 9 ], "msPages": [ 21, 21 ], "answerGuide": "Fix the R positions at the required separation, then arrange the remaining multiset of letters." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=22", "id": "cie-2026-s-51-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Estimate how many of 250 Indian elephants have heights between 2.40 m and 3.10 m, given a normal model.", "qpPages": [ 10, 10 ], "msPages": [ 22, 22 ], "answerGuide": "Standardise both bounds, find the central normal probability, multiply by 250 and give a whole-elephant estimate." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=23", "id": "cie-2026-s-51-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Infer the mean height of African elephants from a normal model and the stated upper-tail probability.", "qpPages": [ 11, 11 ], "msPages": [ 23, 23 ], "answerGuide": "Convert the 62% upper tail to a standard-normal quantile and solve the standardisation equation for the mean." }, { "paperId": "cie-2026-s-51", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/51", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2051.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2051.pdf#page=24", "id": "cie-2026-s-51-q06c", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the probability that more than 50 of a sample of 90 African elephants exceed 3.15 m.", "qpPages": [ 11, 11 ], "msPages": [ 24, 25 ], "answerGuide": "Use a binomial model with p=0.62, approximate it by a normal distribution, apply a continuity correction, and standardise." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=9", "id": "cie-2026-s-52-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance that a randomly chosen five-person team from six men and nine women has more men than women.", "qpPages": [ 2, 2 ], "msPages": [ 9, 10 ], "answerGuide": "Count teams with 3, 4 or 5 men and divide by the total number of five-person teams." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=11", "id": "cie-2026-s-52-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count ways to divide 13 members into groups of sizes six, four and three.", "qpPages": [ 2, 2 ], "msPages": [ 11, 11 ], "answerGuide": "Use the multinomial count 13!/(6!4!3!)." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=12", "id": "cie-2026-s-52-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Create the probability distribution for the number of heads from three biased coins, each with head probability one third.", "qpPages": [ 3, 3 ], "msPages": [ 12, 12 ], "answerGuide": "Apply the binomial model for 0, 1, 2 and 3 heads." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=13", "id": "cie-2026-s-52-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance that at least two heads first occur on the seventh repeated three-coin throw.", "qpPages": [ 3, 3 ], "msPages": [ 13, 13 ], "answerGuide": "Use the six initial failures followed by one success, where a throw is successful if it has at least two heads." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=13", "id": "cie-2026-s-52-q02c", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance that the second successful throw occurs on throw six or seven, where success means at least two heads.", "qpPages": [ 3, 3 ], "msPages": [ 13, 13 ], "answerGuide": "Add the negative-binomial probabilities for the second success at trial six and at trial seven." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=13", "id": "cie-2026-s-52-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count three-digit numbers above 200 from the digits 1–7, allowing repetition, that are divisible by five.", "qpPages": [ 4, 4 ], "msPages": [ 13, 13 ], "answerGuide": "The units digit must be 5; choose any of the six permitted hundreds digits and any tens digit." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=14", "id": "cie-2026-s-52-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count even three-digit numbers above 200 formed from digits 1–7 without repetition.", "qpPages": [ 4, 4 ], "msPages": [ 14, 15 ], "answerGuide": "Count by valid last digit and remaining hundreds/tens choices, observing the hundreds digit must be at least 2." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=16", "id": "cie-2026-s-52-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "For a 0.6 daily success rate, find the chance Haz succeeds on at least seven of nine days.", "qpPages": [ 5, 5 ], "msPages": [ 16, 17 ], "answerGuide": "Use a Binomial(9, 0.6) model and sum the probabilities for 7, 8 and 9 successes." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=18", "id": "cie-2026-s-52-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the chance Haz succeeds on fewer than 66 of 120 days.", "qpPages": [ 5, 5 ], "msPages": [ 18, 18 ], "answerGuide": "Use Binomial(120, 0.6) mean and variance, apply the continuity correction at 65.5, then standardise." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=19", "id": "cie-2026-s-52-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance that both sequential pen choices are blue after a green pen is added before Mira chooses.", "qpPages": [ 6, 6 ], "msPages": [ 19, 19 ], "answerGuide": "Multiply the probability Lui takes blue by the conditional probability Mira then takes blue." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=19", "id": "cie-2026-s-52-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability Mira’s pen is red after Lui keeps one pen and a green pen is added.", "qpPages": [ 6, 6 ], "msPages": [ 19, 20 ], "answerGuide": "Split into Lui initially taking red or not red; add the resulting red probabilities for Mira." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=20", "id": "cie-2026-s-52-q05c", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability Lui has red given that Mira has either a blue or green pen.", "qpPages": [ 7, 7 ], "msPages": [ 20, 20 ], "answerGuide": "Form the conditional probability from the joint event and the probability Mira’s colour is blue or green." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=21", "id": "cie-2026-s-52-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a back-to-back stem-and-leaf display for the heights of two rugby teams, with Hippos on the left.", "qpPages": [ 8, 8 ], "msPages": [ 21, 21 ], "answerGuide": "Sort each team’s measurements, align leaves on a shared stem and include a key in centimetres." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=22", "id": "cie-2026-s-52-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the interquartile range of the Hippos’ heights.", "qpPages": [ 8, 8 ], "msPages": [ 22, 22 ], "answerGuide": "Determine the Hippos’ lower and upper quartiles from the ordered values and subtract." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=22", "id": "cie-2026-s-52-q06c", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the replacement player’s height when replacing an 184 cm player increases the 15-player mean by 0.5 cm.", "qpPages": [ 9, 9 ], "msPages": [ 22, 22 ], "answerGuide": "The total height increases by 15×0.5; add that increase to 184 cm." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=23", "id": "cie-2026-s-52-q07a", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation of a normal leaf-length model from two stated cumulative proportions.", "qpPages": [ 10, 10 ], "msPages": [ 23, 23 ], "answerGuide": "Convert both proportions to z-scores and solve the two standardisation equations simultaneously." }, { "paperId": "cie-2026-s-52", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/52", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2052.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2052.pdf#page=24", "id": "cie-2026-s-52-q07b", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Estimate how many of 600 leaves fall within 0.6 cm of the mean under a normal model.", "qpPages": [ 11, 11 ], "msPages": [ 24, 24 ], "answerGuide": "Find the probability between z=−0.6/0.55 and z=0.6/0.55, multiply by 600, then round to a whole leaf." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=7", "id": "cie-2026-s-53-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance the first 5 on a fair eight-sided die appears on throw seven.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "answerGuide": "Use six non-5 outcomes followed by a 5: (7/8)^6×(1/8)." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=7", "id": "cie-2026-s-53-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance the first 5 appears before throw ten.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "answerGuide": "Use the complement of getting no 5 in the first nine throws." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=8", "id": "cie-2026-s-53-q01c", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "In 12 throws, find the chance of getting between one and three 5s inclusive.", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "answerGuide": "Use a Binomial(12, 1/8) model and sum the terms for 1, 2 and 3." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=9", "id": "cie-2026-s-53-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Construct the probability table for X taking −2, 0, 1 and 3 when P(X=x)=k(x+1)^2, with fractions independent of k.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Sum the four weights to determine k, then substitute it to give the four probabilities." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=10", "id": "cie-2026-s-53-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of the discrete random variable X.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Use the distribution to calculate E(X) and E(X²), then evaluate E(X²)−E(X)²." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=11", "id": "cie-2026-s-53-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a histogram for grouped completion times of 150 students.", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "answerGuide": "Calculate frequency densities from frequency divided by class width, then draw adjoining bars to scale with labelled axes." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=12", "id": "cie-2026-s-53-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the mean completion time from grouped frequency data.", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "answerGuide": "Use each class midpoint in the weighted mean Σ(f×midpoint)/150." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=12", "id": "cie-2026-s-53-q03c", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the class interval containing the 40th percentile of completion times.", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "answerGuide": "Locate the 60th observation in the cumulative frequencies and select its class." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=13", "id": "cie-2026-s-53-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Draw a fully labelled tree for Ebere’s coin-dependent choice of two discs without replacement.", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "answerGuide": "Show the coin probabilities, select the appropriate bag for each result, then update disc probabilities after the first draw." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=14", "id": "cie-2026-s-53-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that the two discs Ebere draws have the same colour.", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "answerGuide": "Add the probabilities of the possible same-colour paths on the tree." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=15", "id": "cie-2026-s-53-q04c", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability Ebere tossed heads given that at least one chosen disc is red.", "qpPages": [ 7, 7 ], "msPages": [ 15, 16 ], "answerGuide": "Use Bayes’ rule: probability of heads and at least one red divided by the total probability of at least one red." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=17", "id": "cie-2026-s-53-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability an Indian cobra’s normally distributed length lies between 105 cm and 135 cm.", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "answerGuide": "Standardise both limits and calculate the central normal probability." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=17", "id": "cie-2026-s-53-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Infer the standard deviation of cobra mass from its normal model and the probability mass is below 2.7 kg.", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "answerGuide": "Convert the 0.81 cumulative probability to a z-score and solve for the standard deviation." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=18", "id": "cie-2026-s-53-q05c", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the chance fewer than 60 of 80 cobras have mass below 2.7 kg.", "qpPages": [ 9, 9 ], "msPages": [ 18, 18 ], "answerGuide": "Use a binomial model with p=0.81, approximate by a normal distribution, apply a continuity correction and standardise." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=19", "id": "cie-2026-s-53-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of POSSESSION with all four Ss consecutive while the two Os are separated.", "qpPages": [ 10, 10 ], "msPages": [ 19, 19 ], "answerGuide": "Treat the four Ss as a block, arrange it with the remaining letters, then subtract cases where the Os are adjacent." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=20", "id": "cie-2026-s-53-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count arrangements of POSSESSION with P first, N last and exactly four letters between the Os.", "qpPages": [ 10, 10 ], "msPages": [ 20, 20 ], "answerGuide": "Fix P and N at the ends, place the Os in the required pair of positions, then arrange the remaining multiset." }, { "paperId": "cie-2026-s-53", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/53", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2053.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2053.pdf#page=21", "id": "cie-2026-s-53-q06c", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Select five letters from POSSESSION, requiring at least two Ss and at least one O.", "qpPages": [ 11, 11 ], "msPages": [ 21, 21 ], "answerGuide": "Enumerate feasible counts of S and O, then choose the remaining letters from the other letter types." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=9", "id": "cie-2026-s-55-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the standard deviation of 60 values using the supplied sums of deviations from c.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Apply the population standard-deviation formula using Σ(x−c) and Σ(x−c)²." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=9", "id": "cie-2026-s-55-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the mean of the 60 values and the supplied deviation sum to find c.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Write Σ(x−c)=60(mean−c) and solve for c." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=10", "id": "cie-2026-s-55-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a cumulative-frequency curve for 160 pupils’ puzzle times.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Plot cumulative frequencies at their upper time bounds and draw a smooth increasing curve from the origin." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=10", "id": "cie-2026-s-55-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the interquartile range from the cumulative-frequency curve.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Read the times at cumulative frequencies 40 and 120 and subtract the lower quartile from the upper quartile." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=11", "id": "cie-2026-s-55-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the unknown cake probability for water given the drink probabilities and overall cake probability.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Set the weighted sum of the conditional cake probabilities equal to 0.505 and solve for x." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=11", "id": "cie-2026-s-55-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability Jules chooses juice given that she does not have cake.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Divide P(juice and no cake) by P(no cake), using the given conditional probabilities." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=12", "id": "cie-2026-s-55-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the chance fewer than seven of nine residents shop locally when each has probability 0.8.", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "answerGuide": "Use a Binomial(9,0.8) model and sum outcomes 0 through 6, or take the complement of 7 through 9." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=14", "id": "cie-2026-s-55-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the chance more than 90 of 120 residents shop locally.", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "answerGuide": "Use the binomial mean and variance, apply a continuity correction at 90.5, then calculate the normal upper tail." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=15", "id": "cie-2026-s-55-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Construct the distribution for the absolute difference between two fair four-sided spinner results.", "qpPages": [ 8, 8 ], "msPages": [ 15, 15 ], "answerGuide": "List all 16 ordered pairs, group them by absolute difference 0–3 and divide frequencies by 16." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=16", "id": "cie-2026-s-55-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of the absolute difference between the spinner results.", "qpPages": [ 8, 8 ], "msPages": [ 16, 16 ], "answerGuide": "Use the distribution to calculate E(X) and E(X²), then subtract E(X)²." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=16", "id": "cie-2026-s-55-q05c", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance the first difference of 3 occurs on spin seven.", "qpPages": [ 9, 9 ], "msPages": [ 16, 16 ], "answerGuide": "Use six non-3 results followed by a result with absolute difference 3." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=17", "id": "cie-2026-s-55-q05d", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the chance the third difference of 3 occurs on spin seven.", "qpPages": [ 9, 9 ], "msPages": [ 17, 17 ], "answerGuide": "Require exactly two successes among the first six spins and a success on the seventh, using the success probability from the table." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=18", "id": "cie-2026-s-55-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability a normally distributed sweet box has mass from 412 g to 424 g.", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "answerGuide": "Standardise both endpoints using mean 420 and standard deviation 6.4, then find the central area." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=18", "id": "cie-2026-s-55-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation of the second brand’s normal mass model from the proportion exceeding 245 g.", "qpPages": [ 10, 10 ], "msPages": [ 18, 18 ], "answerGuide": "Use 42/500 as the upper-tail probability, convert to a z-score, and solve the standardisation equation for σ." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=19", "id": "cie-2026-s-55-q06c", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the chance a third-brand box exceeds 1.2 times the mean when its standard deviation is one quarter of its mean.", "qpPages": [ 11, 11 ], "msPages": [ 19, 19 ], "answerGuide": "Standardise with mean µ and standard deviation 0.25µ; the resulting z-score is fixed by the ratio." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=20", "id": "cie-2026-s-55-q07a", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count CANADIANS arrangements where the three As form one block and the two Ns are not adjacent.", "qpPages": [ 12, 12 ], "msPages": [ 20, 20 ], "answerGuide": "Treat AAA as a block, count all arrangements, and subtract those with the Ns adjacent." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=21", "id": "cie-2026-s-55-q07b", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability a random arrangement of CANADIANS has exactly four letters between its two Ns.", "qpPages": [ 12, 12 ], "msPages": [ 21, 22 ], "answerGuide": "Count arrangements with the Ns in positions five places apart and divide by the total distinct arrangements of the multiset." }, { "paperId": "cie-2026-s-55", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 1", "year": 2026, "session": "May/June", "paperCode": "9709/55", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2055.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2055.pdf#page=22", "id": "cie-2026-s-55-q07c", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Count ways to split CANADIANS into groups of sizes four, three and two, with all As together and both Ns together in one group each.", "qpPages": [ 13, 13 ], "msPages": [ 22, 22 ], "answerGuide": "Enumerate the valid placements of the AAA block and NN block among the labelled group sizes, then count the remaining letters." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=9", "id": "cie-2026-s-61-q01", "questionNumber": 1, "questionPart": null, "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that the combined mass of ten randomly selected apples falls within the stated interval.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Add the independent normal masses to obtain the total mean and variance, standardise both bounds, and subtract the two cumulative probabilities." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=9", "id": "cie-2026-s-61-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State the null and one-sided alternative hypotheses for an increase in the faulty rate.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Define p as the faulty proportion and write H₀: p = 0.02 against H₁: p > 0.02." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=9", "id": "cie-2026-s-61-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Describe a Type II error for this quality-control test.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "In context, this is failing to detect an increase when the true faulty proportion has increased." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=9", "id": "cie-2026-s-61-q02c", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the Type II error probability when the actual faulty proportion is 0.12.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Under p = 0.12, model the faulty count as Bin(50, 0.12); the test fails to reject when at most three are faulty, so sum probabilities for 0 to 3." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=10", "id": "cie-2026-s-61-q03ai", "questionNumber": 3, "questionPart": "3(a)(i)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the model for the number of incorrect characters in a 500-page book.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Use the constant rate 5 per 100 pages to obtain a Poisson mean of 25." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=10", "id": "cie-2026-s-61-q03aii", "questionNumber": 3, "questionPart": "3(a)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Give and justify a normal approximation for the book’s error count.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Approximate Po(25) by N(25, 25), noting the Poisson mean is sufficiently large." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=10", "id": "cie-2026-s-61-q03aiii", "questionNumber": 3, "questionPart": "3(a)(iii)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the approximation to find the probability that the error count exceeds 35.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Apply a continuity correction at 35.5, standardise with mean and variance 25, and find the upper tail." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=11", "id": "cie-2026-s-61-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use an approximation to determine the page-count threshold associated with a lower-tail probability of 0.01 for fewer than 18 errors.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "The MS sets the normal lower-tail boundary at 17.5 and solves the 0.01 quantile, yielding n = 607 as the smallest qualifying page count. The printed QP says “largest”; that condition has no finite maximum, so the wording conflicts with the MS." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=11", "id": "cie-2026-s-61-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the expected score on one spinner turn.", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "answerGuide": "Multiply each score by its probability and sum the products to obtain 2.8." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=12", "id": "cie-2026-s-61-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the stated probability for the mean of 300 spins to determine the interval half-width k.", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "answerGuide": "Apply the Central Limit Theorem to the sample mean, use variance 1.16/300, and equate the central probability 0.2 to the corresponding normal interval." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=12", "id": "cie-2026-s-61-q04c", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain why the Central Limit Theorem is needed for the sample-mean calculation.", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "answerGuide": "The single-spin population is not normally distributed, so the sampling distribution is justified by the large sample size." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=13", "id": "cie-2026-s-61-q04d", "questionNumber": 4, "questionPart": "4(d)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at the 5% level whether the spinner’s mean score differs from 2.8.", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "answerGuide": "Use a two-sided test with the sample-mean standard error √(1.16/300), compare the standardised result with the 5% critical value, and state a contextual conclusion." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=13", "id": "cie-2026-s-61-q05ai", "questionNumber": 5, "questionPart": "5(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that X takes a value from 1 to 3 inclusive when X is Poisson with mean 2.5.", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "answerGuide": "Add the Poisson probabilities for X = 1, 2 and 3 using λ = 2.5." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=14", "id": "cie-2026-s-61-q05aii", "questionNumber": 5, "questionPart": "5(a)(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use the supplied equality involving X and X + Y to determine a probability for their sum.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "answerGuide": "Set the Poisson point probabilities for X = 5 and X + Y = 5 equal to solve for Y’s mean; then use the resulting summed mean to calculate P(X + Y = 1)." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=14", "id": "cie-2026-s-61-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show that an appropriate approximation reproduces the stated binomial probability to two, but not three, decimal places.", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "answerGuide": "Approximate Bin(200, 0.02) by Po(4), calculate P(W = 2), and compare rounding at both requested precisions." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=15", "id": "cie-2026-s-61-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the parameter of the given probability density function by normalising it.", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "answerGuide": "Integrate the density over its support, set the result equal to 1, and solve for the parameter." }, { "paperId": "cie-2026-s-61", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/61", "variant": "Variant 1", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2061.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2061.pdf#page=16", "id": "cie-2026-s-61-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the mean of the continuous random variable using integration by parts.", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "answerGuide": "Evaluate the expectation integral ∫x f(x) dx by parts, apply the support limits, and simplify." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=9", "id": "cie-2026-s-62-q01", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates of the population mean and variance from ten observations.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Find the sample mean and use the unbiased variance formula with denominator n − 1." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=9", "id": "cie-2026-s-62-q02", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that one normal train time exceeds twice the sum of the other two independent journey times.", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "answerGuide": "Form C − 2(A + B); combine means and variances for independent normal variables, then standardise the probability that the result is positive." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=9", "id": "cie-2026-s-62-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses for testing whether the Poisson population mean at a junction has increased from 2.4.", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "answerGuide": "Write H₀: μ = 2.4 against the one-sided alternative H₁: μ > 2.4." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=10", "id": "cie-2026-s-62-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the smallest observed count that would lead to rejection at the 5% level.", "qpPages": [ 5, 5 ], "msPages": [ 10, 11 ], "answerGuide": "Use upper-tail probabilities under Po(2.4); compare the tails from 5 and 6 upward with 0.05 and choose the first qualifying cutoff." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=11", "id": "cie-2026-s-62-q03c", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State the Type I error probability for the test.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Use the null-model probability of the selected rejection region, following the cutoff from part (b)." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=11", "id": "cie-2026-s-62-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the sum of two independent Poisson variables is less than four.", "qpPages": [ 6, 6 ], "msPages": [ 11, 11 ], "answerGuide": "Add the Poisson means to obtain λ = 3.8, then sum the probabilities for totals 0 through 3." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=12", "id": "cie-2026-s-62-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Given that the Poisson total is below four, find the conditional probability that D equals one.", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "answerGuide": "Divide P(D = 1 and E ≤ 2) by P(D + E < 4), using independence to evaluate the numerator." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=13", "id": "cie-2026-s-62-q04c", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the probability that the total of 100 Poisson observations exceeds 190.", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "answerGuide": "The total is Po(170); approximate by N(170, 170), use the continuity-corrected boundary 190.5, and find the upper tail." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=14", "id": "cie-2026-s-62-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the two possible success counts that produce the stated width of an approximate 95% proportion interval.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "answerGuide": "Set the normal-approximation interval width equal to 0.165 and solve the resulting quadratic for x." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=14", "id": "cie-2026-s-62-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Find the confidence percentage given the probability that exactly one of two intervals contains the true proportion.", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "answerGuide": "Let y be the interval coverage; use 2y(1 − y) = 0.1128 and select the solution at least 0.90." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=15", "id": "cie-2026-s-62-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether the mean customer visit time has fallen below 12.4 minutes.", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "answerGuide": "Use the one-sided normal test statistic for the sample mean, compare with the 5% lower-tail critical value, and conclude in context." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=15", "id": "cie-2026-s-62-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain whether a Central Limit Theorem argument is needed in the preceding test.", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "answerGuide": "It is not needed because the population distribution is already normal." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=16", "id": "cie-2026-s-62-q06c", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the integer sample sizes for which a later one-sided test would provide evidence of a lower mean.", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "answerGuide": "Standardise the mean 11.8 under μ = 12.4, impose the 5% lower-tail criterion, solve for n, and retain integer values meeting the inequality." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=12", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=17", "id": "cie-2026-s-62-q07a", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the density’s support parameter and use it to find the expected value.", "qpPages": [ 12, 12 ], "msPages": [ 17, 17 ], "answerGuide": "Normalise the given density to solve for its endpoint, then integrate x f(x) over the support." }, { "paperId": "cie-2026-s-62", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/62", "variant": "Variant 2", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2062.pdf#page=13", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2062.pdf#page=18", "id": "cie-2026-s-62-q07b", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the exact median of the continuous distribution.", "qpPages": [ 13, 13 ], "msPages": [ 18, 18 ], "answerGuide": "Set the cumulative area up to m equal to one half, solve the resulting quadratic, and reject any root outside the support." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=9", "id": "cie-2026-s-63-q01", "questionNumber": 1, "questionPart": null, "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use 60 successes in 200 spins to construct an approximate 94% confidence interval for the spinner’s success probability.", "qpPages": [ 3, 3 ], "msPages": [ 9, 9 ], "answerGuide": "Use the sample proportion and the normal interval formula with z ≈ 1.881; calculate and report both bounds." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=9", "id": "cie-2026-s-63-q02a", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Use a binomial model to assess the claim that the vegetarian proportion is below 0.30 at the 2.5% level.", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "answerGuide": "Under H₀ use Bin(20, 0.3), calculate the lower-tail probability through two successes, compare it with 0.025, and give a contextual conclusion." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=10", "id": "cie-2026-s-63-q02b", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State the smallest significance level at which the claim would be rejected.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "answerGuide": "Use the lower-tail probability from part (a) as the p-value and express it to three significant figures." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=11", "id": "cie-2026-s-63-q03", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the variance of a variable with the shown linear probability density, given its mean.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Determine the density parameter from E(X) = 2, integrate x²f(x) to find E(X²), and subtract the squared mean." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=12", "id": "cie-2026-s-63-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State a condition required for the junction’s car count to follow a Poisson model.", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "answerGuide": "Give a suitable condition such as arrivals occurring independently, singly, or at random at a constant rate." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=12", "id": "cie-2026-s-63-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability of at least four arrivals in a randomly selected three-minute interval.", "qpPages": [ 6, 6 ], "msPages": [ 12, 12 ], "answerGuide": "Use Po(9.6) for the three-minute count and subtract the probabilities for 0, 1, 2 and 3 from 1." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=12", "id": "cie-2026-s-63-q04c", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Given at least four arrivals, find the probability of exactly five arrivals in the interval.", "qpPages": [ 7, 7 ], "msPages": [ 12, 12 ], "answerGuide": "Divide P(N = 5) under Po(9.6) by the probability of N ≥ 4." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=13", "id": "cie-2026-s-63-q04d", "questionNumber": 4, "questionPart": "4(d)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Approximate the probability of more than 60 arrivals in a randomly selected 20-minute interval.", "qpPages": [ 7, 7 ], "msPages": [ 13, 13 ], "answerGuide": "Use Po(64) ≈ N(64, 64), apply the continuity correction at 60.5, and calculate the upper tail." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=13", "id": "cie-2026-s-63-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates of the mean and variance from the regional snake-length sample summaries.", "qpPages": [ 8, 8 ], "msPages": [ 13, 13 ], "answerGuide": "Use Σx/n for the mean and the n − 1 variance formula with the supplied Σx²; show the variance rounds to 64.483." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=14", "id": "cie-2026-s-63-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test the researcher’s claim of a lower regional mean at the 2% significance level.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "answerGuide": "Use the sample mean and known variance 64.483 to form a lower-tail z statistic; compare with the 2% critical value and state the contextual decision." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=14", "id": "cie-2026-s-63-q05c", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Use the test decision to decide whether a Type I error is possible here.", "qpPages": [ 9, 9 ], "msPages": [ 14, 14 ], "answerGuide": "Since the null hypothesis was not rejected, the test did not make the rejection required for a Type I error." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=15", "id": "cie-2026-s-63-q05d", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the Type II error probability when the true regional mean is 38.7 cm.", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "answerGuide": "Use the critical sample-mean boundary from the 2% test and calculate, under mean 38.7 and variance 64.483, the probability of failing to reject." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=15", "id": "cie-2026-s-63-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a randomly selected batch contains more of ingredient A than ingredient B by volume.", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "answerGuide": "Form X − Y; combine the independent normal means and variances, then calculate P(X − Y > 0)." }, { "paperId": "cie-2026-s-63", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/63", "variant": "Variant 3", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2063.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2063.pdf#page=16", "id": "cie-2026-s-63-q06b", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the mass-per-litre constant given the probability that the batch’s total ingredient mass is below 70 kg.", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "answerGuide": "Combine the independent amounts to obtain the total-mass mean and variance in terms of m, standardise the 0.3 lower-tail probability, and solve." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=9", "id": "cie-2026-s-65-q01a", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the exact binomial model for the diseased-plant count in the sample.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Use n = 250 and disease probability 1/500." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=9", "id": "cie-2026-s-65-q01b", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State a suitable approximation for the diseased-plant count.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "Use a Poisson approximation with mean np = 0.5." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=9", "id": "cie-2026-s-65-q01c", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the approximation to find the chance that more than two sampled plants are diseased.", "qpPages": [ 2, 2 ], "msPages": [ 9, 9 ], "answerGuide": "For Po(0.5), subtract P(X = 0) + P(X = 1) + P(X = 2) from 1." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=2", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=10", "id": "cie-2026-s-65-q01d", "questionNumber": 1, "questionPart": "1(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use an approximation to find the probability that fewer than 20 diseased plants are found in a day.", "qpPages": [ 2, 2 ], "msPages": [ 10, 10 ], "answerGuide": "Approximate the daily Poisson count by N(32, 32), apply a continuity correction at 19.5, and calculate the lower tail." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=3", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=10", "id": "cie-2026-s-65-q02", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that three regular coffees have greater combined volume than two large coffees.", "qpPages": [ 3, 3 ], "msPages": [ 10, 10 ], "answerGuide": "Form the difference 3R − 2L; combine means and variances for independent normal volumes, then find the probability that the difference is positive." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=4", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=11", "id": "cie-2026-s-65-q03a", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the two possible dog-owner counts that give the stated width of the approximate 94% proportion interval.", "qpPages": [ 4, 4 ], "msPages": [ 11, 11 ], "answerGuide": "Set the interval width using z ≈ 1.881 and n = 150, then solve the quadratic for the count a." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=11", "id": "cie-2026-s-65-q03b", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Given that at least one of two 94% intervals covers the true proportion, find the chance that both do.", "qpPages": [ 5, 5 ], "msPages": [ 11, 11 ], "answerGuide": "Use independence of interval coverage events: compute P(both) = 0.94² and divide by P(at least one) = 1 − 0.06²." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=5", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=12", "id": "cie-2026-s-65-q03c", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate unbiased estimates of the mean and variance of daily dog-walking distance from the sample summaries.", "qpPages": [ 5, 5 ], "msPages": [ 12, 12 ], "answerGuide": "Find the sample mean and use the unbiased sample-variance formula with denominator n − 1." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=6", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=13", "id": "cie-2026-s-65-q04a", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show that the median of the variable with a linearly increasing density is half its support endpoint.", "qpPages": [ 6, 6 ], "msPages": [ 13, 13 ], "answerGuide": "Use total area to relate the density height and endpoint, then set the area from zero to the median equal to one half of the total." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=7", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=14", "id": "cie-2026-s-65-q04b", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Use the given expectation to determine the endpoint of the linear-density distribution.", "qpPages": [ 7, 7 ], "msPages": [ 14, 14 ], "answerGuide": "Integrate x f(x) over the support, equate the result to 4, and solve for the positive endpoint." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=14", "id": "cie-2026-s-65-q05a", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the smallest significance level for the stated lower-tailed test of a car-ownership proportion.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "answerGuide": "Under the null proportion 0.2, use Bin(40, 0.2) and calculate P(X < 4), the test’s rejection probability." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=8", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=14", "id": "cie-2026-s-65-q05b", "questionNumber": 5, "questionPart": "5(b)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State the probability of a Type I error for the car-ownership test.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "answerGuide": "Under the null hypothesis, the Type I error probability is the significance level found in part (a)." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=9", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=15", "id": "cie-2026-s-65-q05c", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the Type II error probability when the true car-ownership proportion is 0.05.", "qpPages": [ 9, 9 ], "msPages": [ 15, 15 ], "answerGuide": "Under Bin(40, 0.05), failure to reject occurs with at least four owners; calculate that upper-tail probability." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=10", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=15", "id": "cie-2026-s-65-q06a", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that the mean wingspan of 25 swans lies between the two stated values.", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "answerGuide": "The sample mean has standard error 6.9/√25; standardise both bounds and subtract the cumulative probabilities." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=16", "id": "cie-2026-s-65-q06bi", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 5, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test at the 5% level whether the regional mean swan wingspan differs from 234 cm.", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "answerGuide": "Use a two-sided z test with known standard deviation 6.9 and n = 25; compare the test statistic with the 5% critical value and conclude in context." }, { "paperId": "cie-2026-s-65", "board": "CIE", "qualification": "Cambridge International AS & A Level Mathematics", "syllabus": "9709", "unit": "Probability & Statistics 2", "year": 2026, "session": "May/June", "paperCode": "9709/65", "variant": "Variant 5", "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "qpUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Question%20Paper%20%2065.pdf#page=11", "msUrl": "https://pastpapers.co/cie/A-Level/Mathematics-9709/2026-May-June/9709%20Mathematics%20June%202026%20Mark%20Scheme%20%2065.pdf#page=16", "id": "cie-2026-s-65-q06bii", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Explain whether a Central Limit Theorem argument was needed for the preceding test.", "qpPages": [ 11, 11 ], "msPages": [ 16, 16 ], "answerGuide": "It was not needed because the population distribution of wingspans is normal." }, { "id": "ial-2024-10-WMA11-01-q01a", "paperId": "ial-2024-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2024, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the line through the two given points.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-que-20241010.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the gradient from the coordinate differences, then use point-gradient form and rearrange to the requested form.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": "882989930D3710F6EDD5FE6384F4C009E9AB2559346E50E755C14CF6E8732327", "markScheme": "85B0DB85BC90DB4D0A67DE3BFCDB0D916B069730EC8B869BC46A6A62AD6DD365", "examinerReport": "135F558C97B2C3B8BB090C510F5CEF37969536E91AB8BC52284CD82471ED1251" } }, { "id": "ial-2024-10-WMA11-01-q01b", "paperId": "ial-2024-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2024, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the perpendicular bisector line through the midpoint of the segment.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-que-20241010.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the midpoint, use the negative reciprocal gradient and substitute the midpoint into the line equation.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": "882989930D3710F6EDD5FE6384F4C009E9AB2559346E50E755C14CF6E8732327", "markScheme": "85B0DB85BC90DB4D0A67DE3BFCDB0D916B069730EC8B869BC46A6A62AD6DD365", "examinerReport": "135F558C97B2C3B8BB090C510F5CEF37969536E91AB8BC52284CD82471ED1251" } }, { "id": "ial-2024-10-WMA11-01-q02i", "paperId": "ial-2024-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2024, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Simplify the given expression involving powers and surds.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-que-20241010.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply index laws and rationalize or combine surd terms before reducing the expression.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": "882989930D3710F6EDD5FE6384F4C009E9AB2559346E50E755C14CF6E8732327", "markScheme": 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paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the vertical translation and state the required minimum point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-que-20241010.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=18", "qpPages": [ 22, 23 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the translated maximum to find the constant, then apply the cosine minimum location and shift its y-coordinate.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=20", "qpPages": [ 24, 25 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the normal gradient to the tangent gradient, evaluate the derivative at the stated x-value and solve for the parameter.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": "882989930D3710F6EDD5FE6384F4C009E9AB2559346E50E755C14CF6E8732327", "markScheme": "85B0DB85BC90DB4D0A67DE3BFCDB0D916B069730EC8B869BC46A6A62AD6DD365", "examinerReport": "135F558C97B2C3B8BB090C510F5CEF37969536E91AB8BC52284CD82471ED1251" } }, { "id": "ial-2024-10-WMA11-01-q08b", "paperId": "ial-2024-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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"msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=21", "qpPages": [ 26, 28 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand if useful, then apply the power rule and simplify.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": "882989930D3710F6EDD5FE6384F4C009E9AB2559346E50E755C14CF6E8732327", "markScheme": "85B0DB85BC90DB4D0A67DE3BFCDB0D916B069730EC8B869BC46A6A62AD6DD365", "examinerReport": "135F558C97B2C3B8BB090C510F5CEF37969536E91AB8BC52284CD82471ED1251" } }, { "id": "ial-2024-10-WMA11-01-q09c", "paperId": "ial-2024-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2024, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "differentiation", "functions", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of a second point where the tangent has the same gradient as at the known point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-que-20241010.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=21", "qpPages": [ 26, 28 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this equal-gradient task caused difficulty; many candidates stopped after finding only the known tangent gradient.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate, equate the two gradients, factor the resulting equation and select the other valid point.", "sourceEvidencePages": { "examinerReport": [ 12, 13 ] }, "sourceFiles": { "questionPaper": "WMA11_QP.pdf", "markScheme": "WMA11_MS.pdf", "examinerReport": "WMA11_ER.pdf" }, "sourceSha256": { "questionPaper": "882989930D3710F6EDD5FE6384F4C009E9AB2559346E50E755C14CF6E8732327", "markScheme": "85B0DB85BC90DB4D0A67DE3BFCDB0D916B069730EC8B869BC46A6A62AD6DD365", "examinerReport": "135F558C97B2C3B8BB090C510F5CEF37969536E91AB8BC52284CD82471ED1251" } }, { "id": "ial-2024-10-WMA11-01-q09d", "paperId": "ial-2024-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2024, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "differentiation", "functions", "algebra" ], "primaryTopicId": "differentiation", "summary": "Map the specified point under the stated horizontal translation and vertical stretch.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-que-20241010.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-rms-20250116.pdf#page=21", "qpPages": [ 26, 28 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the horizontal translation to the x-coordinate and the vertical scale factor to the y-coordinate, following the order stated.", "sourceFiles": { "questionPaper": "WMA11_QP.pdf", 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use each adjacent pair of x-values as a strip, apply the trapezium formula and round the estimate as requested.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { 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was more demanding because candidates had to recognize the extra exact integral and add it to the numerical estimate.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Separate the known tabulated integral from the added term, integrate that term exactly over the limits and combine the results.", "sourceEvidencePages": { "examinerReport": [ 4 ] }, "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q02a", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", 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"WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the supplied condition on the early terms to find the recurrence constant.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson notes that some candidates added five terms rather than the required first four when finding the recurrence constant.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Write out the relevant terms, form the stated sum or condition and solve the resulting linear equation.", "sourceEvidencePages": { "examinerReport": [ 5 ] }, "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q02c", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use the recurrence pattern to evaluate the requested long sum.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports mixed success and omissions of the extra terms when candidates used the repeating sequence pattern for the long sum.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Group the repeating terms into a short cycle, count complete cycles and account for any remaining terms.", "sourceEvidencePages": { "examinerReport": [ 5 ] }, "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q03a", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Use the remainder condition to establish a relation between polynomial coefficients.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the remainder theorem at the divisor root and equate the result to the given remainder.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q03b", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the two integer coefficients using the additional factor condition.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the root of the known factor, combine with the relation from part (a), and solve simultaneously.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q03c", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the quotient after dividing the completed polynomial by a linear factor.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that candidates were often unsure what the quotient meant and rarely identified the quotient correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Perform polynomial division or factorization and identify the quotient separately from any remainder.", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "sourceFiles": { 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the terms as powers of x and differentiate term by term.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q04b", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the x-coordinate of the stationary point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the first derivative equal to zero and solve within the curve’s domain.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q04c", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the second derivative and use it to classify the stationary point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate again, substitute the stationary x-coordinate, and use the sign of the second derivative to justify the classification.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q04d", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "State where the curve is decreasing.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=14", "qpPages": [ 10, 11 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sign of the first derivative on the intervals separated by the stationary value.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q05a", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the given binomial through the first three powers of x.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial coefficients and collect the constant, x and x-squared terms in simplest form.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q05b", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the specified constant term to determine the non-zero parameter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=16", "qpPages": [ 12, 13 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports this part as substantially more difficult because many candidates did not identify all contributions to the constant term.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Substitute the expansion into the related expression, identify every contribution to the constant term and solve for the parameter.", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q06", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation using logarithm laws.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=18", "qpPages": [ 14, 15 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the logarithm rules to combine terms, solve the resulting algebraic equation and reject values outside the logarithm domains.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": "C96BA0E5348B8CB02CEF52965A85242D8AC91E0CD0B71EC411405816960989D3", "examinerReport": "BF9B2302147D8BB60467C73CDA7D134EAFEB2B455B65751469C735CB62924A83" } }, { "id": "ial-2024-10-WMA12-01-q07a", "paperId": "ial-2024-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2024, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Show the monthly arithmetic payment at the specified month.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=20", "qpPages": [ 16, 19 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the month index into the arithmetic term formula and retain the currency precision required.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": "6466B308A15572D7DC9CA12C88174FC80E3C6CF4DF6F761728ABE3E4A6729B24", "markScheme": 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3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Compare totals for the arithmetic and geometric schemes over the full period.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-que-20241016.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-rms-20250116.pdf#page=20", "qpPages": [ 16, 19 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the geometric ratio from the first and final payments, calculate each series total and round their difference as requested.", "sourceFiles": { "questionPaper": "WMA12_QP.pdf", "markScheme": "WMA12_MS.pdf", "examinerReport": "WMA12_ER.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-que-20241022.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-rms-20250116.pdf#page=19", "qpPages": [ 20, 23 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Transform the minimum value and reverse the inequality when multiplying by a negative factor.", "sourceFiles": { "questionPaper": "WMA13_QP.pdf", "markScheme": "WMA13_MS.pdf", "examinerReport": "WMA13_ER.pdf" }, "sourceSha256": { "questionPaper": "D6E3AAE65983951F911E6D0EB4B464C38A54D1992049C24A6F86738E6715BCCA", "markScheme": "995DBB6E8BF7C786120BAF2301ACA0CF9758DBCC9A5FA4990DCCC1C29F2FE38D", "examinerReport": "026FDF1C7A01F9636B1452F89002D9AC325EDE5DB1B388AB54BDE11EF3180CEF" } }, { "id": 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"primaryTopicId": "exponentials", "summary": "State the initial heart rate from the exponential model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-que-20241022.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-rms-20250116.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute zero for time.", "sourceFiles": { "questionPaper": "WMA13_QP.pdf", "markScheme": "WMA13_MS.pdf", "examinerReport": "WMA13_ER.pdf" }, "sourceSha256": { "questionPaper": "D6E3AAE65983951F911E6D0EB4B464C38A54D1992049C24A6F86738E6715BCCA", "markScheme": "995DBB6E8BF7C786120BAF2301ACA0CF9758DBCC9A5FA4990DCCC1C29F2FE38D", "examinerReport": 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"numerical-methods" ], "primaryTopicId": "exponentials", "summary": "Find the time of the modelled maximum heart rate.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-que-20241022.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-rms-20250116.pdf#page=21", "qpPages": [ 24, 27 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies the exponential rearrangement and logarithm steps for finding the modelled maximum as a stumbling block.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Differentiate the rate function, set the derivative to zero, rearrange the exponential equation and solve for time.", 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"integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Show the equation of the normal at the specified point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-que-20241022.pdf#page=28", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-rms-20250116.pdf#page=23", "qpPages": [ 28, 32 ], "msPages": [ 23, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the tangent derivative at the point, take its negative reciprocal and use point-gradient form.", "sourceFiles": { "questionPaper": "WMA13_QP.pdf", "markScheme": "WMA13_MS.pdf", "examinerReport": "WMA13_ER.pdf" }, "sourceSha256": { "questionPaper": "D6E3AAE65983951F911E6D0EB4B464C38A54D1992049C24A6F86738E6715BCCA", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-que-20241029.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-rms-20250116.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use their direction vectors in the dot-product formula and simplify exactly.", "sourceFiles": { "questionPaper": "WMA14_QP.pdf", "markScheme": "WMA14_MS.pdf", "examinerReport": "WMA14_ER.pdf" }, "sourceSha256": { "questionPaper": "AEFE7B38C80603244EBC6403D942FB8A4F8568B7C0D3F7122CB5C10B5CB329AE", "markScheme": "15DA938CA14871B976F42F55EC04B36B8B22096FD7012867778404D4304EFCA7", "examinerReport": "3541E6C04E9CB39B8DAE45056A7F760F417C7E546D731F7334A5B335F88C0594" } }, { "id": 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null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use area as the integral of y dx, substitute dy/dt and dx/dt correctly, then apply the double-angle identity.", "sourceFiles": { "questionPaper": "WMA14_QP.pdf", "markScheme": "WMA14_MS.pdf", "examinerReport": "WMA14_ER.pdf" }, "sourceSha256": { "questionPaper": "AEFE7B38C80603244EBC6403D942FB8A4F8568B7C0D3F7122CB5C10B5CB329AE", "markScheme": "15DA938CA14871B976F42F55EC04B36B8B22096FD7012867778404D4304EFCA7", "examinerReport": "3541E6C04E9CB39B8DAE45056A7F760F417C7E546D731F7334A5B335F88C0594" } }, { "id": "ial-2024-10-WMA14-01-q10b", "paperId": "ial-2024-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2024, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 5, "topicIds": [ 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=8", "qpPages": [ 4, 7 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about one support, combine the beam and child weights and solve for the centre-of-mass coordinate.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q02b", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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"WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q03a", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show the position vector of a ship moving at constant velocity.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the velocity from the two observations, then add velocity times elapsed time to the initial position.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q03b", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the ship’s speed from its velocity components.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take the magnitude of the constant velocity vector and convert the distance and time units.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q03c", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show that the other ship passes through the same point.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find its position at the stated time and substitute that time into the first ship’s position vector.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q04a", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show the steady speed from the cyclist’s speed-time graph and distance.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the area under the graph to the journey distance and solve for the constant speed.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q04b", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the first cyclist’s acceleration.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the change in speed by the acceleration time.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q04c", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch both cyclists’ speed-time histories over the stated interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Plot the initial rest and acceleration segments and then the constant-speed portions with correct start times.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q04d", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration duration that makes the second cyclist catch the first.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls this catch-up calculation challenging; a common mistake was to treat the shared distance as the same for both cyclists.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Equate the total distances from the areas under both speed-time graphs and solve for the acceleration time.", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q05a", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Write an equation of motion for the lighter hanging particle.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve vertically and include its weight, tension and acceleration with a consistent positive direction.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q05b", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension in the two-particle system.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the second particle’s equation of motion and solve the simultaneous equations.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q05c", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the resultant force on the pulley from the two string segments.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports this part was poorly answered; many candidates used an incorrect sum of particle weights instead of resolving the string forces.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve the equal tensions in their directions and calculate the resultant magnitude.", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q05d", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the speed of the descending particle when it reaches the ground after the string breaks.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies the final motion after the string broke as a discriminator because candidates had to account for both stages of motion.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the initial acceleration to find speed at the break point, then apply suvat over the remaining fall.", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q06a", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the least horizontal force that can hold the particle in equilibrium on the rough plane.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve parallel and perpendicular to the plane, use limiting friction and test the relevant friction direction.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along the plane to find acceleration and apply suvat from rest.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q07a", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Show the first return time of a vertically projected ball.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=17", "qpPages": [ 22, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply suvat for the return to launch height and use the positive non-zero time.", "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q07b", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the magnitude of the impulse at the first ground impact.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=17", "qpPages": [ 22, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says finding the second ball’s height and equating the two heights required careful time handling and was often unsuccessful.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the speed immediately before impact, use the rebound height to find the upward speed and calculate the change in momentum.", "sourceEvidencePages": { "examinerReport": [ 8 ] }, "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": "6AF79EE684C8C371852563145DCE7F25529F625EF3DD882B1D8C733CD5E9E55A", "markScheme": "D5573F0B2BC277CC841C495E1F20B4954185D3F80C828AEB0DC1469A02C0CD5E", "examinerReport": "D142CB19E7E3C9977B4E0E3C333A34203AD8343DC36E81CC32A747E1F70E176C" } }, { "id": "ial-2024-10-WME01-01-q07c", "paperId": "ial-2024-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2024, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find when the rebounding ball and a second projected ball first reach the same height.", "qpUrl": "https://www.alevelcopilot.com/whfile/1b/03/df/b5/a2/65/02/6b/b2/da/36/9f/9e/d2/92/7d.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-rms-20250116.pdf#page=17", "qpPages": [ 22, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says the final equal-height task required a complete setup with two different motion times; many candidates did not complete it.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Write the vertical displacement of each ball with the correct time origin, equate the heights and solve for the required time.", "sourceEvidencePages": { "examinerReport": [ 8 ] }, "sourceFiles": { "questionPaper": "WME01_QP.pdf", "markScheme": "WME01_MS.pdf", "examinerReport": "WME01_ER.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each velocity component and use the initial position vector to find the constants.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q01b", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration magnitude at the specified time.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the velocity components, substitute the time and calculate the vector magnitude.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q01c", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the particle velocity points in the specified vector direction.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the component ratio equal to the target direction ratio and solve for the non-negative time.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q02", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2", "marks": 7, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the velocity vector after a two-dimensional impulse using the impulse and energy data.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that many candidates found this impulse-and-energy problem very challenging, especially in recovering the correct velocity vector.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the impulse-momentum relation for the component changes and the kinetic-energy increase to determine the unknown velocity components.", "sourceEvidencePages": { "examinerReport": [ 5 ] }, "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q03a", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the van’s instantaneous acceleration while towing the trailer at constant engine power.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use power equals driving force times speed, then apply Newton’s second law to the combined system.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q03b", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find how far the trailer travels after the towbar breaks.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the work-energy principle to the trailer, accounting for gravity and resistance until it stops.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q04a", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass offset of a uniform trapezoidal lamina.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the shape into simple laminae, calculate their areas and centroid distances, then take the area-weighted mean.", "sourceFiles": { 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use composite area and centre-of-mass moments about the suspension point, then relate the centre-of-mass line to the hanging angle.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q05a", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show the horizontal range formula for a projectile returning to its launch height.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve initial velocity into components, use the vertical return condition to find flight time and substitute into horizontal motion.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q05b", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the horizontal distance beyond the pole to the landing point.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the flight-time or range formula with the stated launch speed and angle, then subtract the known distance.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q05c", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile’s clearance above the pole as it passes its horizontal position.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the time to reach the pole’s horizontal coordinate and use the vertical displacement equation.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q06a", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the normal reaction at the peg in terms of the beam’s weight scale.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says static rigid bodies are a challenge and notes that clear force diagrams and equations were often missing in the first part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Take moments about the ground contact, use the beam and package weights and the given geometry.", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q06b", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction at limiting equilibrium.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says the rigid-body equilibrium work was challenging, with unclear force diagrams and inconsistent force equations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Resolve vertically and horizontally to determine the normal reaction and friction force, then use the limiting-friction relation.", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "sourceFiles": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=18", "qpPages": [ 22, 24 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply momentum conservation and the impact law, then impose the condition that its post-impact velocity has the original sign.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": 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"difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the impulse in terms of the post-impact velocity, combine with momentum and the impact law and simplify.", "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WME02-01-q07c", "paperId": "ial-2024-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2024, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the impulse in a second collision after the other particle rebounds from a wall.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-que-20241024.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-rms-20250116.pdf#page=18", "qpPages": [ 22, 24 ], "msPages": [ 18, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies the final part of the successive-collisions question as a discriminator between grades.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the wall restitution to obtain the rebound speed, then apply momentum and restitution to the later particle collision.", "sourceEvidencePages": { "examinerReport": [ 8 ] }, "sourceFiles": { "questionPaper": "WME02_QP.pdf", "markScheme": "WME02_MS.pdf", "examinerReport": "WME02_ER.pdf" }, "sourceSha256": { "questionPaper": "4D0924F7F8C40A4A67EF4FC01DFF77474171634E416BCC25686991F2864AEEFC", "markScheme": "E8359F56AD0193ABDAD8599FDD2F2C6C189B46D396390CFFE293FA415AFD7E44", "examinerReport": "856A24EF6C13D14CF8323F919117EA11DE1638D1215A052574E491C7593B7525" } }, { "id": "ial-2024-10-WST01-01-q01a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Read the modal running time for one film group from a back-to-back stem-and-leaf diagram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the key and identify the most frequent value on the specified side.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": 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"answerGuide": "Locate the required ordered observations in the stem-and-leaf data and match them to the table.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q01c", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the mean and standard deviation to one decimal place.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the grouped data and the appropriate summary formulae, then round both statistics to one decimal place.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q01d", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Evaluate a standardized skewness measure and describe the shape.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the mean, mode and standard deviation into the measure and interpret its sign.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q01e", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Compare one feature of the two running-time distributions using a statistic.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Choose a relevant summary statistic from each group and state the supported difference.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q02a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the required sum of squares and calculate the product-moment correlation coefficient.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the centred sum of squares, substitute into the PMCC formula, and give the value to the requested accuracy.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q02b", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the correlation coefficient in context.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says interpreting the correlation coefficient was more demanding; many candidates did not explain what its value meant.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "State the direction and strength of the linear association indicated by the coefficient.", "sourceEvidencePages": { "examinerReport": [ 4 ] }, "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q02c", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Decide whether the correlation supports a linear regression model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson identifies the follow-up model-suitability interpretation as more demanding because candidates did not relate the correlation to linear regression.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Use the strength and direction of the coefficient as evidence for or against a linear relationship.", "sourceEvidencePages": { "examinerReport": [ 4 ] }, "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q02d", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the regression line predicting weight from length.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the regression slope from the summary statistics and use the sample means to determine the intercept.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q02e", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the regression model to estimate a rabbit’s weight.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=6", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the given length into the regression line and report the predicted weight.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q03a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete a three-set Venn diagram from magazine readership counts.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Place the exclusive counts first, find the shared intersection, then fill remaining regions and the outside group.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q03bi", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that an adult reads all three magazine types.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the central Venn-region count by the total sample.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q03bii", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that an adult does not read one specified magazine type.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the appropriate regions outside that set and divide by the total.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q03c", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find a conditional probability involving travel and sport magazine readership.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Restrict the denominator to adults who read travel magazines and divide the joint count by that total.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q04a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability of a motorbike journey exceeding a distance threshold.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=9", "qpPages": [ 14, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardize the threshold using the given Normal mean and standard deviation and evaluate the upper tail.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q04b", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the distance associated with a specified upper-tail probability.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=9", "qpPages": [ 14, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the corresponding Normal quantile and transform it back to the distance scale.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q05a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Explain why a histogram is suitable for the work-time data.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify that the measured variable is continuous.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q05b", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the number of photographers in a new time interval from the histogram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given interval count to determine the scale, then compare the relevant histogram areas.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q05c", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median work time from the grouped histogram.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate half the total frequency and interpolate within the class interval.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the histogram’s shape and the estimated mean with the symmetry expected under a Normal model.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q06a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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"markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q06b", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Recover the probability distribution from the cumulative table.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take successive differences of the cumulative probabilities.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q06c", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Determine two face probabilities from the expected score of a second die.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use normalization and the given expectation to form two linear equations and solve for both constants.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q06d", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Show the second moment for the second die’s score.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply each squared score by its probability and sum.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q06e", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the variance after applying a linear transformation to the score.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use Var(a+bY)=b²Var(Y), first obtaining the variance from the first and second moments.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q06f", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(f)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the two independent die scores sum to the target.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the score pairs that meet the sum and add their independent joint probabilities.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": 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"primaryTopicId": "probability", "summary": "Show the probability that the two selected counters have different colours.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-que-20241018.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-rms-20250116.pdf#page=12", "qpPages": [ 24, 27 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the red-then-black and black-then-red path probabilities and simplify.", "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": 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"answerGuide": "Divide the black-then-red path probability by the total unlike-colour probability.", "sourceEvidencePages": { "examinerReport": [ 8 ] }, "sourceFiles": { "questionPaper": "WST01_QP.pdf", "markScheme": "WST01_MS.pdf", "examinerReport": "WST01_ER.pdf" }, "sourceSha256": { "questionPaper": "1C1BC434799967F89AF2B96289A73D43F90E54C15A8C0DE45FE4FB1DACE4F3B8", "markScheme": "92814C863F60D7D14689C4D343B48743E19807D93E23C3BF143B3A5EC8D94A38", "examinerReport": "00D2A8C62415A17B9987A521D39FC8685674BB0BE877046D07690A5CCA77FA45" } }, { "id": "ial-2024-10-WST01-01-q08a", "paperId": "ial-2024-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2024, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the mean to rate times area and use the Poisson probability formula.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q01b", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability of a face-mask count above a threshold.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=2", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Thin the Poisson process by the mask proportion, then evaluate the required upper tail.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", 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probability statement about a sample of students.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=4", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the number of correct first answers with a binomial variable, express the cumulative-probability difference and solve for the integer cutoff.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": 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"year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State two assumptions supporting the customer-arrival model.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Describe independent arrivals and a constant average rate over the interval.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the null rate to the historical value and use a two-sided alternative.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q03d", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the critical region for the rate test at the stated significance level.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the null Poisson distribution to determine the lower and upper tail cutoffs.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q03e", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the actual significance level of the critical region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the null probabilities in the rejection tails.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q03f", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Interpret the observed customer count using the critical region.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=8", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check whether the observed count falls in the rejection region and state the contextual conclusion.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q04ia", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Determine the endpoints of a uniform distribution from a tail probability and variance.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the uniform tail length and variance formula to form equations for the two endpoints.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": 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"difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express each probability as a length proportion over the interval and solve the resulting linear equation.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q04ii", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that two randomly cut wire pieces form squares with side lengths differing by more than the threshold.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=12", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the condition into an interval for the random cut point and divide its total length by the wire length.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q05a", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Show the cumulative distribution function over the first interval.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the density from the lower support endpoint to x and simplify.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q05b", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the cumulative distribution function across its full support.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each piece in turn, preserve the boundary values at the joins and give the zero and one regions.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q05c", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find a probability over an interval that crosses the density support.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=16", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the cumulative function at both endpoints and subtract.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q06a", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 7, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of the total value of three coins selected from two boxes.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "List the possible coin combinations, multiply the independent selection probabilities and combine outcomes with the same total.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q06b", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of the median coin value.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=20", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=11", "qpPages": [ 20, 23 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "For each possible median, sum the probabilities of the ordered selections that produce it.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q07a", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Use the total area of the piecewise density to verify its initial slope parameter.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the triangular area under the density equal to one and solve for the slope.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q07b", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the two remaining constants in the piecewise density.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use symmetry of the density to determine the second slope, then apply continuity or the endpoint condition to find the remaining constant.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2024-10-WST02-01-q07c", "paperId": "ial-2024-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2024, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Show the variance by integrating the first and second moments.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20241101.pdf#page=24", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson notes that this moment calculation required clear algebraic integration; many candidates made long inaccurate calculations or relied too heavily on a calculator.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Use symmetry for the mean, integrate x-squared times each density segment and subtract the square of the mean.", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says only the strongest candidates completed both quartiles correctly, often because they did not use the density’s symmetry.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20250116.pdf", "tagConfidence": "reviewed", "answerGuide": "Set the cumulative area equal to one quarter and three quarters, integrate on the appropriate segment and solve.", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20250116.pdf#page=12", "qpPages": [ 24, 28 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the probability from the density or compare the endpoints with the quartiles, then show it exceeds one half.", "sourceFiles": { "questionPaper": "WST02_QP.pdf", "markScheme": "WST02_MS.pdf", "examinerReport": "WST02_ER.pdf" }, "sourceSha256": { "questionPaper": "F35E09B699FEB10FD97FB1B4B8ACD7D030563083EBAD323AB034C3D13DDCFDF6", "markScheme": "F0F769898A708E5F2AB8F98BC3AE4FC7D7A02B325A7F808B19FB3576288DADA1", "examinerReport": "E3B8DFAAADE04DAF6A64DB2FEB2B067F174D6B91FF9FFE0876BAC1D74DAED5D6" } }, { "id": "ial-2026-6-WMA11-01-q01", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level 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"coordinate-geometry", "summary": "Find the area of a triangle defined by two intersecting lines and their y-intercepts.", "answerGuide": "Find the relevant coordinates, then use a base-height calculation or the coordinate area formula.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q03a", "paperId": "ial-2026-6-WMA11-01", "board": 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"msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q03c", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the smallest angle of the triangle to one decimal place.", "answerGuide": "Use the sine or cosine rule and choose the angle opposite the shortest side.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q04a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel 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"https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=13", "qpPages": [ 8, 11 ], "msPages": [ 13, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q05a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Rewrite the curve expression in the requested transformed form.", "answerGuide": "Complete the square or use index laws to express the function in the required form.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=17", "qpPages": [ 12, 13 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q05b", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Find the coordinates of the maximum point of the curve.", "answerGuide": "Use the transformed form to read the maximum coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=17", "qpPages": [ 12, 13 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q05c", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Define a shaded region bounded by the curve, a line and the axes using inequalities.", "answerGuide": "State the inequalities for each boundary and choose the correct sides from the diagram.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=17", "qpPages": [ 12, 13 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q06a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Determine a cubic function from its contact points and intercepts.", "answerGuide": "Use the repeated and simple roots to form a factorised cubic, then use the y-intercept to find its scale factor.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=19", "qpPages": [ 14, 17 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q06b", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find exact coordinates where a line meets a cubic at two additional points.", "answerGuide": "Equate the cubic and line, factor out the known intersections, then solve the remaining factor exactly.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=19", "qpPages": [ 14, 17 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q07a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Write equations from the area and perimeter of a sector.", "answerGuide": "Use sector area one half r squared theta and perimeter 2r plus r theta.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=22", "qpPages": [ 18, 19 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q07b", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "algebra", "trigonometry" ], "primaryTopicId": "algebra", "summary": "Solve the sector equations exactly for the radius and angle.", "answerGuide": "Substitute one equation into the other, solve the quadratic, use r greater than 2, then recover theta.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=22", "qpPages": [ 18, 19 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q088a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "differentiation", "algebra", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent gradient parameter at P, then find the normal equation at P.", "answerGuide": "Use the tangent condition at x = 1/4 to determine k, then use the negative reciprocal gradient and point P to write the normal.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=24", "qpPages": [ 20, 23 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q08b", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Recover a function from its derivative and a point on the curve.", "answerGuide": "Integrate the derivative, include a constant, and use the point to determine it.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=24", "qpPages": [ 20, 23 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q09a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "State the x-coordinate of a minimum point on a trigonometric curve.", "answerGuide": "Use the sine minimum condition and select the point in the stated interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=26", "qpPages": [ 24, 25 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q09b", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Count roots of a trigonometric equation over a long interval.", "answerGuide": "Use periodicity and count roots in one period, checking interval endpoints.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=26", "qpPages": [ 24, 25 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q09c", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "State the coordinates of a minimum point on a transformed sine curve.", "answerGuide": "Set the sine factor to its minimum, then evaluate the corresponding coordinates.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=26", "qpPages": [ 24, 25 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q10a", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch a rational curve and mark where it crosses a coordinate axis.", "answerGuide": "Identify domain restrictions and intercepts, then sketch the branches with correct signs and asymptotic behaviour.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=28", "qpPages": [ 26, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q10b", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use a tangent condition to derive an equation for the curve parameter.", "answerGuide": "Equate the curve and line and apply the repeated-root or equal-gradient condition.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=28", "qpPages": [ 26, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA11-01-q10c", "paperId": "ial-2026-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Find exact coordinates of the specified intersection point.", "answerGuide": "Solve the parameter equation, select the case below the x-axis, and substitute into the line and curve.", "qpUrl": "https://www.alevelcopilot.com/whfile/f0/68/f2/15/6e/d5/88/4d/4c/14/a0/61/3e/88/3e/e3.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/0e/9b/b9/73/27/9c/f9/5c/00/20/88/b7/a0/b6/a1/5d.pdf#page=28", "qpPages": [ 26, 28 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA11-QP.pdf", "markScheme": "2026-06-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "CA38854B9C1F291D7A1CFBF79B2DFF9BDFE3C6E1BAEAB1273E053A0FBED051D9", "markScheme": "5EFF1BCF34547EBC41BF3CBDE3C10E4A3321D552F4353686D29B96BDB9C78C71" } }, { "id": "ial-2026-6-WMA12-01-q01a", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Use the remainder theorem to find a polynomial remainder.", "answerGuide": "Evaluate the polynomial at the root of the divisor.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q01b", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Find the quotient after dividing a cubic polynomial by a linear factor.", "answerGuide": "Use polynomial division or coefficient comparison, using the remainder from part (a).", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q02i", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over a specified interval in radians.", "answerGuide": "Use a double-angle identity to form a quadratic in cos x, then retain solutions in the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q02ii", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve a trigonometric equation over a degree interval to one decimal place.", "answerGuide": "Use the tangent addition identity and include every solution in the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q033a", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Test whether the four terms can form arithmetic and geometric sequences, finding u3 where possible.", "answerGuide": "Check consecutive differences for the arithmetic case and consecutive ratios for the geometric case; give the conclusion and u3 for each.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q03b", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Use a period-four condition and finite sum to determine a sequence term.", "answerGuide": "Group the terms into repeated blocks of four, include the remaining terms, and solve the sum equation.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=11", "qpPages": [ 6, 9 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q04i", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Disprove a universal perfect-cube claim with a counterexample.", "answerGuide": "Evaluate the expression for a suitable positive integer and show it lies between consecutive cubes.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q04ii", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 3, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove a quadratic inequality for all real x.", "answerGuide": "Rearrange to a quadratic expression and show it is positive, for example by completing the square.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=13", "qpPages": [ 10, 11 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, 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"https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q05b", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "integration", "coordinate-geometry" ], "primaryTopicId": "integration", "summary": "Estimate the volume of a ramp from its cross-section, platforms and uniform width.", "answerGuide": "Add the curved-section estimate and both platform areas, then multiply by the ramp width.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q05c", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 1, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Decide whether the trapezium-rule volume estimate is an overestimate or underestimate.", "answerGuide": "Use the curve shape between ordinates to justify the direction of the error.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], 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asymptote from the equation, then sketch the branches.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q06ib", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(i)(b)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve an exponential equation using logarithms to three significant figures.", "answerGuide": "Isolate the exponential term, take logarithms and solve for x.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": 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"https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=17", "qpPages": [ 16, 17 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q07a", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 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9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Express the area of a logo made from a rectangle and two quarter-circles in terms of x and y.", "answerGuide": "Use the gap to express the radius in terms of x, then add rectangle and circular areas.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=24", "qpPages": [ 24, 28 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": 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the stationary value, and round x to three significant figures.", "qpUrl": "https://www.alevelcopilot.com/whfile/5c/04/da/fa/be/ae/1a/6a/78/4f/bd/5a/56/07/8f/f9.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/1a/6d/18/1f/3e/6c/70/b4/22/f3/b0/5f/3b/f9/43/f9.pdf#page=24", "qpPages": [ 24, 28 ], "msPages": [ 24, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA12-QP.pdf", "markScheme": "2026-06-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "78691E5D7EA1847CF79EF1B1B14A76F43F4DB0C80DAD7B9BC3C6F4D76F5580DE", "markScheme": "B7AF92BB7CE479893D6F49A1F9796854202062A95FDF65BDFF1D91A64662493A" } }, { "id": "ial-2026-6-WMA12-01-q09d", "paperId": "ial-2026-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": 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change.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q011c", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", 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"C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q02a", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "State the range of an exponential function.", "answerGuide": "Use the positive exponential term and horizontal asymptote.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], 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range restriction.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q02c", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard 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"ial-2026-6-WMA13-01-q03c", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the image of a point under y=f(2x)+3.", "answerGuide": "Apply a horizontal scale factor of one half and then translate the y-coordinate up by three.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=10", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=10", "qpPages": [ 10, 11 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", 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"differentiation", "summary": "Use implicit differentiation to find where a curve tangent meets the x-axis.", "answerGuide": "Differentiate implicitly, substitute the given y-coordinate, find the tangent equation and set y=0.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=16", "qpPages": [ 16, 17 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q07a", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find when the car-price model reaches the specified price.", "answerGuide": "Isolate the exponential, take natural logarithms and round the time to two decimals.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=18", "qpPages": [ 18, 21 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q07b", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the rate of change of the car price at t=5.", "answerGuide": "Differentiate the exponential model and substitute t=5.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=18", "qpPages": [ 18, 21 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q077c", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the initial van price to determine m and compare the initial car and van prices.", "answerGuide": "Substitute t = 0 into the van model to find m, then compare the two modelled initial prices in consistent units.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=18", "qpPages": [ 18, 21 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q07d", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find when the car and van price models are equal using algebra.", "answerGuide": "Set the models equal, form and solve a quadratic in an exponential, then take logarithms.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=18", "qpPages": [ 18, 21 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q08a", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Prove an identity involving secant, cosecant and double angles.", "answerGuide": "Apply the sine double-angle identities and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q08b", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Convert a trigonometric equation to a quadratic in cos 2x.", "answerGuide": "Use the identity from part (a), double-angle and secant identities, then collect terms.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q08c", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation on a restricted interval.", "answerGuide": "Solve the quadratic in cos 2x, retain valid values, and list permitted x values.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q09a", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the sum of a rational term and an exponential term.", "answerGuide": "Use the logarithm rule for the linear denominator and integrate the exponential term.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q09b", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the maximum point of an inverted-V graph defined by an absolute-value function.", "answerGuide": "Set the expression inside the absolute value to zero to obtain the vertex, then read its coordinates from the function.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q09c", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use an intersection and y-intercept to determine two model parameters.", "answerGuide": "Substitute the y-intercept to find one parameter and use the intersection to find the other.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA13-01-q09d", "paperId": "ial-2026-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(d)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Find the area bounded by an exponential curve, another graph and the axes.", "answerGuide": "Identify intervals, integrate each boundary function, combine signed areas and round to three significant figures.", "qpUrl": "https://www.alevelcopilot.com/whfile/7b/a2/89/9b/43/10/da/cd/13/b8/d3/96/0a/66/24/e2.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/c1/09/f8/2d/64/0d/84/b6/60/f7/6c/1c/f1/95/2c/73.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA13-QP.pdf", "markScheme": "2026-06-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "C8BAC8A2123B27CDE6BD88123F9C081420F409B4747640CA2CE020A0D00AE409", "markScheme": "EEFA18988034AD62D5E0B68D88440C6764452E96DFE0ACA557229670BAC11A78" } }, { "id": "ial-2026-6-WMA14-01-q01a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Express a rational function in partial fractions.", "answerGuide": "Set up partial fractions, clear denominators, and solve for the constants.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q01b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use partial fractions and binomial series to find the first three terms in ascending powers.", "answerGuide": "Expand each partial-fraction term, multiply by its coefficient and collect powers.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q01c", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "State the range of x for which the binomial expansions are valid.", "answerGuide": "Apply the modulus condition on each expansion and intersect the ranges.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q02a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Explain why two three-dimensional lines are not parallel.", "answerGuide": "Compare the direction vectors and state that they are not scalar multiples.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q02b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "proof", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Prove by contradiction that two nonparallel lines do not intersect.", "answerGuide": "Assume an intersection, equate components and show the parameter values are inconsistent.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=6", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q03a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at a specified point on a parametric curve.", "answerGuide": "Evaluate the point at t=2, find dy/dx as (dy/dt)/(dx/dt), then use point-slope form.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q03b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Eliminate the parameter to express the curve as a function of x.", "answerGuide": "Solve the x relation for t, substitute into y, and identify constants and domain.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q03c", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the range of the function obtained from the parametric curve.", "answerGuide": "Use the stated x-domain and the resulting function endpoints or turning values.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q04a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find the time to fill a container from its volume-depth model and constant inflow.", "answerGuide": "Evaluate the volume at full depth and divide by the inflow rate.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q04b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "differentiation", "integration", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the exact rate of change of water depth at a specified depth.", "answerGuide": "Differentiate V with respect to h, use dV/dt=(dV/dh)(dh/dt), and substitute the rate and depth.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=15", "qpPages": [ 12, 13 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q05a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the angle between two given vectors in a triangle.", "answerGuide": "Use the dot-product formula and convert the inverse-cosine result to degrees.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=17", "qpPages": [ 14, 17 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q05b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "trigonometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find triangle area from its side vectors and included angle.", "answerGuide": "Use one half the product of side magnitudes and the sine of the included angle.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=17", "qpPages": [ 14, 17 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q05c", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find vector AD when AB is extended and CD is perpendicular to the extension.", "answerGuide": "Write AD as a scalar multiple of AB, use the perpendicular dot-product condition with AC-AD, and solve.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=14", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=17", "qpPages": [ 14, 17 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q06a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Use implicit differentiation to find dy/dx for the given curve.", "answerGuide": "Differentiate each term, use the chain rule on y terms, collect dy/dx terms and simplify.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=20", "qpPages": [ 18, 21 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q06b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the point on the curve with greatest x-coordinate.", "answerGuide": "At the rightmost point the tangent is vertical, so set the denominator of dy/dx to zero; combine with the curve equation and select the greatest x.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=18", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=20", "qpPages": [ 18, 21 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q07i", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 6, "topicIds": [ "integration", "trigonometry", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate an exponential-trigonometric integral by parts.", "answerGuide": "Integrate by parts and use the resulting recurrence to isolate the original integral.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=23", "qpPages": [ 22, 25 ], "msPages": [ 23, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q07ii", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 7, "topicIds": [ "integration", "trigonometry", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate an exact integral using the given trigonometric substitution.", "answerGuide": "Transform the integrand and limits, then integrate the resulting algebraic expression exactly.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=22", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=23", "qpPages": [ 22, 25 ], "msPages": [ 23, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q088a", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "exponentials", "algebra", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find the parameter k from the initial area, then find the rate of change of the algae-covered area at t = 10.", "answerGuide": "Use S = 18 at t = 0 in the original model to find k, then differentiate S with respect to t and evaluate at t = 10.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=26", "qpPages": [ 26, 28 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q08b", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 6, "topicIds": [ "differentiation", "integration", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Solve a differential equation for algae growth using an initial condition.", "answerGuide": "Separate variables, integrate, apply S=18 at t=0, and rearrange for S.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=26", "qpPages": [ 26, 28 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WMA14-01-q08c", "paperId": "ial-2026-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 1, "topicIds": [ "exponentials", "functions" ], "primaryTopicId": "exponentials", "summary": "Find the maximum pond area implied by the improved growth model.", "answerGuide": "Use the horizontal limiting value of the solution.", "qpUrl": "https://www.alevelcopilot.com/whfile/be/3c/6d/c9/92/06/5a/27/a7/1f/06/41/bb/38/db/68.pdf#page=26", "msUrl": "https://www.alevelcopilot.com/whfile/76/1c/75/23/69/2a/e6/52/5b/15/d4/d6/68/3b/bc/b9.pdf#page=26", "qpPages": [ 26, 28 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WMA14-QP.pdf", "markScheme": "2026-06-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "FFF7FC0C770B016C8819D2766C6543F931C05927798E1CC36EE3EFFE4AA0C0D5", "markScheme": "28A21FD1EFA34C8B8E18D1224CC1355B91C2A315CE302086FF4C1FE12DC3E4BC" } }, { "id": "ial-2026-6-WME01-01-q01a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the common speed after two railway trucks collide and couple.", "answerGuide": "Choose a positive direction and apply conservation of momentum with opposite initial directions.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q01b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the impulse exerted on one truck during the collision.", "answerGuide": "Use impulse as the change in momentum of truck B and the post-collision speed.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q02a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find a supporting-string tension for a non-uniform rod in equilibrium.", "answerGuide": "Take moments about the other support and use the rod weight at its centre of mass.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q02b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the support tension after a particle is attached and the rod is on the point of tilting.", "answerGuide": "Use the limiting pivot and take moments with the rod weight and particle.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q03a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Read the ball displacement at the final time from a displacement-time graph.", "answerGuide": "Read the ordinate at t=0.5 seconds.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q03b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find total distance travelled from a displacement-time graph.", "answerGuide": "Add absolute displacement changes over each straight-line segment.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q03c", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Sketch the velocity-time graph corresponding to the displacement-time graph.", "answerGuide": "Use each displacement segment gradient as velocity, preserving the interval and sign.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q03d", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Find the magnitude and units of impulse at a specified time.", "answerGuide": "Find the velocities on either side from graph gradients, then calculate mass times change in velocity.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q04a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the initial speed of a ball projected upward from a height before it hits the ground.", "answerGuide": "Use a constant-acceleration equation with consistent signs.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q04b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find total time above a height threshold after repeated bounces with halved speeds.", "answerGuide": "Calculate the time above the threshold for each qualifying bounce and sum the intervals.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q05", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the force needed to keep a particle in limiting equilibrium on a rough inclined plane.", "answerGuide": "Resolve parallel and perpendicular to the plane, use limiting friction mu R, and solve for X.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q06a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of a resultant force vector.", "answerGuide": "Take the square root of the sum of squared components.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q06b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the angle between a resultant force and a given direction vector.", "answerGuide": "Use the dot product and vector magnitudes to find the angle.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q06c", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry", "algebra" ], "primaryTopicId": "mechanics", "summary": "Find two forces with stated directions whose sum is the given resultant.", "answerGuide": "Represent each force as a scalar multiple of its direction vector and solve component equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q07a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Show the position vector of a boat as a function of time.", "answerGuide": "Use constant velocity and the known position at t=1.5 to find the position at t.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q07b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry", "algebra" ], "primaryTopicId": "kinematics", "summary": "Find how long a boat remains within a specified distance of the origin.", "answerGuide": "Set the squared position magnitude to at most 81, solve the time interval and find its length.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q07c", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find when the boat is northwest of the origin.", "answerGuide": "Use sign and relative-size conditions on the east and north components.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q07d", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 6, "topicIds": [ "kinematics", "coordinate-geometry", "proof" ], "primaryTopicId": "kinematics", "summary": "Show the distance from the origin when the boat is perpendicular to its direction of travel.", "answerGuide": "Set position dot velocity to zero, find time, substitute into the position vector and evaluate its magnitude.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=14", "qpPages": [ 24, 27 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q08a", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the friction force on a particle on an inclined plane.", "answerGuide": "Use friction coefficient times normal reaction, found by resolving perpendicular to the plane.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q08b", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Write an equation of motion for a particle on a smooth horizontal plane.", "answerGuide": "Apply Newton second law horizontally using tension and acceleration.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q08c", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Show the tension in a two-particle connected system.", "answerGuide": "Write equations of motion for both particles, use shared acceleration and solve.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q08d", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 1, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "State the consequence of the string being inextensible.", "answerGuide": "State that connected particles have equal acceleration magnitudes along the string.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q08e", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(e)", "marks": 1, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "State the direction of the force exerted on the pulley by the string.", "answerGuide": "Combine the two tension directions as vectors and give the resultant direction in terms of alpha.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WME01-01-q08f", "paperId": "ial-2026-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(f)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the magnitude of the resultant force on the pulley.", "answerGuide": "Resolve both string tensions into components and calculate the resultant magnitude.", "qpUrl": "https://www.alevelcopilot.com/whfile/05/81/17/ff/57/1c/7f/9c/94/76/2c/63/d8/05/bd/39.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/1a/89/40/d6/bc/40/cc/3b/1e/80/75/95/bb/9d/e7/36.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WME01-QP.pdf", "markScheme": "2026-06-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "C8445CB40D6691D137B9B37C45F6562D432977D89E7F0D2E4958E4D3A2FA9331", "markScheme": "9CAA9FBA070E0170E26B5BBB1887F21490914906491B44366636199F3BBBF06E" } }, { "id": "ial-2026-6-WST01-01-q01a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "State the expectation of a discrete random variable.", "answerGuide": "Apply the expectation formula or use the given distribution.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q01b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the second moment of a discrete random variable.", "answerGuide": "Sum x squared times its probability for all possible values.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=2", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q02a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Complete a probability tree for a three-game process whose difficulty changes after wins and losses.", "answerGuide": "Add the conditional win and loss probabilities at each game level.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q02b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of winning all three games.", "answerGuide": "Multiply the conditional probabilities along the all-win branch.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q02c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of exactly two wins in three games.", "answerGuide": "Add the path probabilities for every branch with exactly two wins.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=4", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=7", "qpPages": [ 4, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q03a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the width and height of a histogram bar from its scale and frequency density.", "answerGuide": "Use the reference bar to determine the scale, then calculate the target width and height.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q03b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median from grouped data by linear interpolation.", "answerGuide": "Locate the median class and interpolate using cumulative frequency and class width.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q033c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Show the grouped-data mean to two decimal places, then estimate the standard deviation.", "answerGuide": "Calculate the estimated mean from the class midpoints and frequencies, then use the grouped-data variance formula with the same midpoints.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q03d", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain why grouped-data statistics are estimates.", "answerGuide": "State that the exact individual values are not known.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q03e", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 1, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Describe the skewness of the grouped distribution.", "answerGuide": "Compare mean and median or use the supplied moment estimate, with a matching conclusion.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=8", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q04a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Determine an unknown probability constant from a discrete distribution.", "answerGuide": "Set probabilities to sum to one, solve the quadratic and reject the invalid root.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q04b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expectation of a linear transformation of a random variable.", "answerGuide": "Calculate E(X) and use E(2X+4)=2E(X)+4.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q04c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the variance of a linear transformation of a discrete random variable.", "answerGuide": "Find E(X squared), calculate Var(X), then use Var(2X+4)=4Var(X).", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q04d", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the variance after changing the sign and shift of a random variable.", "answerGuide": "Use invariance of variance to shifts and the squared scale factor.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=12", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q055a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the given sample means and variances to find the required sums of squares for x and y.", "answerGuide": "Apply the variance identity to the supplied means and variances for the 10 paired observations.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q055b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the corrected sum of squares for y and the corrected cross-product sum for x and y.", "answerGuide": "Centre the supplied raw sums about the means to obtain Syy and Sxy.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q05c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation coefficient.", "answerGuide": "Divide Sxy by the square root of Sxx times Syy.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q05d", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe the relationship indicated by the correlation coefficient.", "answerGuide": "Interpret the positive coefficient as a positive linear association.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q05e", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Predict how correlation changes when the mean point is added to the data.", "answerGuide": "Recognise that the added point equals the means, so corrected sums and correlation do not change.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=16", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q06a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Shade the Venn diagram region for the complement of A intersect complement of B.", "answerGuide": "Shade the area outside both events.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q066b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Express P(C) and P(D given C) in terms of x from the Venn diagram.", "answerGuide": "Add the regions belonging to C, then divide the C and D intersection probability by P(C) for the conditional probability.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q06c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Express the probability of C union D in terms of x and y.", "answerGuide": "Add the disjoint regions in C union D.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q06d", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "probability", "algebra" ], "primaryTopicId": "probability", "summary": "Use a relationship between P(C) and P(D) to solve for two region probabilities.", "answerGuide": "Use total probability and the stated relationship to form simultaneous equations.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q06e", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine whether C and D are independent and justify the answer.", "answerGuide": "Compare P(C intersection D) with P(C)P(D), or compare a conditional and marginal probability.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=20", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q07a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Find the least-squares regression line of practical marks on theory marks.", "answerGuide": "Calculate gradient S_tp/S_tt, find the intercept from sample means, and state p=a+bt.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q07b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the gradient of the regression line in context.", "answerGuide": "Explain the predicted practical-mark increase for each additional theory-test mark.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q07c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "data-analysis", "algebra" ], "primaryTopicId": "data-analysis", "summary": "Use the regression line to estimate both missing practical-test marks.", "answerGuide": "Substitute each theory mark into the regression line and round both estimates.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q07d", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Choose which regression estimate is more reliable and explain why.", "answerGuide": "Identify the estimate based on interpolation within the observed theory-mark range.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=24", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q088a", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the requested lower-tail and between-values probabilities for the normal tin-weight distribution.", "answerGuide": "Standardise the relevant cutoffs, then use the normal cumulative probabilities and subtract for the interval.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q08b", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the symmetric interval half-width containing 98 percent of a normal distribution.", "answerGuide": "Use the 0.99 standard normal quantile and multiply by the standard deviation.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2026-6-WST01-01-q08c", "paperId": "ial-2026-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "distributions", "algebra" ], "primaryTopicId": "distributions", "summary": "Find the mean and standard deviation from two normal-tail probabilities.", "answerGuide": "Convert both conditions to standard-normal equations and solve simultaneously.", "qpUrl": "https://www.alevelcopilot.com/whfile/f2/ba/4d/63/7d/95/04/f7/0b/22/ae/c5/e4/92/9f/2c.pdf#page=28", "msUrl": "https://www.alevelcopilot.com/whfile/06/51/ed/da/58/71/ce/9c/ac/b0/ec/73/55/e0/7b/28.pdf#page=16", "qpPages": [ 28, 32 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-06-WST01-QP.pdf", "markScheme": "2026-06-WST01-MS.pdf" }, "sourceSha256": { "questionPaper": "D14C4595A20AB09AF93893C3FE17F8B908AC60D3C031CEEEC89935DDE7FAB22E", "markScheme": "41F6B66CE3F24D6B21D212690E321C21A7A107BD6AD6C199E684417EC77F7C7C" } }, { "id": "ial-2025-10-WMA11-01-q01a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given curve and express the derivative in simplest form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the power rule to each term, including the square-root term, then simplify the resulting derivative." }, { "id": "ial-2025-10-WMA11-01-q01b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent line to the curve at the given point, in integer-coefficient form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the derivative at the point to get the tangent gradient, then use the point-gradient form and rearrange to y = mx + c." }, { "id": "ial-2025-10-WMA11-01-q02", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the product of the two given polynomial factors and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=10", "qpPages": [ 4, 4 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand the factors first, integrate the resulting polynomial term by term, and include a constant of integration." }, { "id": "ial-2025-10-WMA11-01-q03a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the largest angle in a triangle whose side lengths have ratio 2:4:5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the cosine rule with the side opposite the largest angle, then give the angle to one decimal place." }, { "id": "ial-2025-10-WMA11-01-q03b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the triangle area to find its shortest side, given the same 2:4:5 side ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=12", "qpPages": [ 6, 6 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the sides as 2t, 4t and 5t. Use the area formula with the included angle from part (a) to solve for t, then calculate 2t." }, { "id": "ial-2025-10-WMA11-01-q04i", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(i)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the given equation involving powers of 3 without relying on a calculator.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite the fractions and surd as powers of 3, equate exponents, and solve the resulting linear equation." }, { "id": "ial-2025-10-WMA11-01-q04ii", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(ii)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find all real roots of the given rational equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=14", "qpPages": [ 8, 8 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Clear the denominator while keeping its excluded values in mind, form a polynomial in x squared, solve for x squared, then retain only valid real roots." }, { "id": "ial-2025-10-WMA11-01-q05a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Rewrite the quadratic in completed-square form and identify its constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor out the leading coefficient from the quadratic and complete the square inside the bracket." }, { "id": "ial-2025-10-WMA11-01-q05b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the x-coordinate of the second intersection of the parabola with the line through the origin and its minimum point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the completed-square form to locate the minimum point, find the line through that point and the origin, then solve the line-curve intersection and exclude the minimum point." }, { "id": "ial-2025-10-WMA11-01-q05c", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Describe the shaded region between the curve, line and axis using inequalities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=16", "qpPages": [ 12, 12 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the x-boundaries from the origin to the minimum point, then bound y between the line and the parabola over that interval." }, { "id": "ial-2025-10-WMA11-01-q06a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Sketch the given rational curve, showing its axis intercepts and vertical asymptote.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find where the numerator is zero for the x-intercept, evaluate the curve at x = 0 for the y-intercept, and set the denominator to zero for the asymptote; place the branches consistently." }, { "id": "ial-2025-10-WMA11-01-q06b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the parameter range for which the given line intersects the rational curve at least once.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the line and curve, form the resulting quadratic, and require a real solution using its discriminant; combine with the stated positivity condition on the parameter." }, { "id": "ial-2025-10-WMA11-01-q07a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Show that the larger circle has the stated radius, using the given sector area and angle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=22", "qpPages": [ 20, 20 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the sector-area formula to find the smaller circle radius, then add the given radial gap PQ to obtain OQ." }, { "id": "ial-2025-10-WMA11-01-q07b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the total surface area of the pool formed by the two joined sectors.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=22", "qpPages": [ 20, 20 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the area of the major sector using its radius and angle, add the smaller sector area, and round to the nearest square metre." }, { "id": "ial-2025-10-WMA11-01-q07c", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the total perimeter of the pool outline to two significant figures.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=22", "qpPages": [ 20, 20 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the two exposed arc lengths and the two straight radial edges, using radians for the sector arcs." }, { "id": "ial-2025-10-WMA11-01-q08a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact value of the constant in the derivative, using the given tangent at x = √2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=24", "qpPages": [ 24, 24 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "The tangent gradient is the derivative value at √2. Equate the stated derivative to the line gradient and solve for the constant." }, { "id": "ial-2025-10-WMA11-01-q08b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal to the curve at x = √2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=24", "qpPages": [ 24, 24 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the negative reciprocal of the tangent gradient for the normal gradient, then use the point on the given tangent at x = √2 to form the line equation." }, { "id": "ial-2025-10-WMA11-01-q08c", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Recover the curve function from its derivative, in simplest form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=24", "qpPages": [ 24, 24 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate each term in the derivative and use the known point on the curve from the tangent to determine the constant of integration." }, { "id": "ial-2025-10-WMA11-01-q09a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Identify a trigonometric expression for the graphed function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=26", "qpPages": [ 28, 28 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the amplitude, period and direction of the graph at the origin; match these features to a sine or cosine expression." }, { "id": "ial-2025-10-WMA11-01-q09bi", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(i)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Give the image of the marked point after the specified horizontal translation of the graph.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=26", "qpPages": [ 28, 28 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the horizontal shift to the point’s x-coordinate and leave its y-coordinate unchanged." }, { "id": "ial-2025-10-WMA11-01-q09bii", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)(ii)", "marks": 2, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Give the image of the marked point after the specified vertical scaling and reflection.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=26", "qpPages": [ 28, 28 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Multiply the point’s y-coordinate by the stated factor, including its negative sign, and keep the x-coordinate unchanged." }, { "id": "ial-2025-10-WMA11-01-q10a", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find algebraically the x-coordinate of the second intersection of the curve with its normal at x = 0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=28", "qpPages": [ 30, 30 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate to find the tangent gradient at x = 0, take its negative reciprocal for the normal, form its equation, and solve the curve-line intersection after removing the known point." }, { "id": "ial-2025-10-WMA11-01-q10b", "paperId": "ial-2025-10-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "October", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the exact positive x-coordinate where the tangent is parallel to the normal from part (a).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P1_MS.pdf#page=28", "qpPages": [ 30, 30 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative equal to the normal-line gradient and solve for the positive root, keeping the result exact." }, { "id": "ial-2025-10-WMA12-01-q01a", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the requested later term of the arithmetic sequence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the first-term and common-difference formula for the specified term." }, { "id": "ial-2025-10-WMA12-01-q01b", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum of the first 100 terms of the arithmetic sequence.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the arithmetic-series sum formula with 100 terms and common difference −3." }, { "id": "ial-2025-10-WMA12-01-q02", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 6, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Determine the three unknown coefficients in the first four terms of the binomial expansion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Expand through the cubic term, compare coefficients in order, and solve for the constants." }, { "id": "ial-2025-10-WMA12-01-q03a", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use logarithms to find the curve parameter from the given point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the point into the exponential equation, take logarithms, and round as requested." }, { "id": "ial-2025-10-WMA12-01-q03bi", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(i)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Approximate the specified integral using all tabulated ordinates and the trapezium rule.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the common strip width and the trapezium-rule endpoint and interior weights." }, { "id": "ial-2025-10-WMA12-01-q03bii", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)(ii)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Use the trapezium-rule estimate to evaluate the related integral.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=11", "qpPages": [ 6, 6 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the stated scaling relationship to the numerical integral estimate." }, { "id": "ial-2025-10-WMA12-01-q04a", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "algebra", "differentiation" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder from the polynomial division.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Perform the division to obtain the quotient; use the remainder theorem or the constant remainder from the division to obtain the remainder." }, { "id": "ial-2025-10-WMA12-01-q04b", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "algebra", "differentiation" ], "primaryTopicId": "algebra", "summary": "Verify the stated linear factor, then prove the cubic equation has exactly one real root.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the factor theorem at the stated root, then factor out the known linear factor and use the remaining quadratic's discriminant to rule out further real roots." }, { "id": "ial-2025-10-WMA12-01-q04c", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "algebra", "differentiation" ], "primaryTopicId": "algebra", "summary": "Find the interval where the cubic function is decreasing.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate, factor the derivative, and identify where it is negative." }, { "id": "ial-2025-10-WMA12-01-q05i", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over the stated degree interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rewrite tangent using sine and cosine, factor the equation, and list every admissible angle to one decimal place." }, { "id": "ial-2025-10-WMA12-01-q05ii", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(ii)", "marks": 4, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the second trigonometric equation on the stated radian interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a double-angle identity to reduce the equation, then select roots in the interval." }, { "id": "ial-2025-10-WMA12-01-q06a", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle whose diameter endpoints are given.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the midpoint for the centre, use the diameter to obtain the radius, then write the circle equation." }, { "id": "ial-2025-10-WMA12-01-q06b", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find both possible equations of an origin-centred circle tangent to the first circle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=20", "qpPages": [ 18, 18 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the distance between centres and the sum or difference of radii for external or internal contact." }, { "id": "ial-2025-10-WMA12-01-q07a", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Differentiate the curve equation and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=23", "qpPages": [ 22, 22 ], "msPages": [ 23, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate each power term using the power rule." }, { "id": "ial-2025-10-WMA12-01-q07b", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Use the stationary-point condition to determine the positive constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=23", "qpPages": [ 22, 22 ], "msPages": [ 23, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the stated x-coordinate into the derivative and set it equal to zero." }, { "id": "ial-2025-10-WMA12-01-q07c", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Find the shaded area by algebraic integration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=23", "qpPages": [ 22, 22 ], "msPages": [ 23, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the curve, use the stationary-point height for the horizontal boundary, and evaluate the bounded area." }, { "id": "ial-2025-10-WMA12-01-q08i", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Prove the formula for the sum of a finite geometric series.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=26", "qpPages": [ 26, 26 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the sum and its ratio multiple, subtract, and rearrange to the required expression." }, { "id": "ial-2025-10-WMA12-01-q08iia", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Verify the remaining volume after six years of annual evaporation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=26", "qpPages": [ 26, 26 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model each year by multiplying by 0.92 and calculate the sixth-year amount." }, { "id": "ial-2025-10-WMA12-01-q08iib", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)(b)", "marks": 3, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the combined liquid volume across the 40 barrels.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=26", "qpPages": [ 26, 26 ], "msPages": [ 26, 27 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Sum the geometric sequence of year-end volumes and round to the nearest litre." }, { "id": "ial-2025-10-WMA12-01-q09i", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(i)", "marks": 5, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the logarithmic equation, accounting for the logarithm domains.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=28", "qpPages": [ 30, 30 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms, convert to an algebraic equation, solve, and reject any inadmissible root." }, { "id": "ial-2025-10-WMA12-01-q09iia", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Derive the stated relation between the two positive variables from the logarithmic identity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=28", "qpPages": [ 30, 30 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine logarithms into a product, equate arguments, then rearrange for the target variable." }, { "id": "ial-2025-10-WMA12-01-q09iib", "paperId": "ial-2025-10-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "October", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(ii)(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "State the full restriction on the variable and justify it.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P2_MS.pdf#page=28", "qpPages": [ 30, 30 ], "msPages": [ 28, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Check positivity and the denominator or logarithm domain in the derived expression." }, { "id": "ial-2025-10-WMA13-01-q01a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "State the range of the logarithmic function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the minimum of the quadratic inside the logarithm to determine its range." }, { "id": "ial-2025-10-WMA13-01-q01b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the inverse of the rational function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the output equal to the function, rearrange for the input, and state the inverse domain restriction." }, { "id": "ial-2025-10-WMA13-01-q01c", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Evaluate the specified composite function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the inner function first, then substitute that value into the outer function." }, { "id": "ial-2025-10-WMA13-01-q01d", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the exact parameter satisfying the exponential-function condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the exponential input into the function relation and solve using logarithms." }, { "id": "ial-2025-10-WMA13-01-q02a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Show that a root lies inside the given interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Evaluate the function at both endpoints, show a sign change, and cite continuity." }, { "id": "ial-2025-10-WMA13-01-q02b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Rearrange the root equation into the required fixed-point form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the function equal to zero and isolate the exponential term before taking the required root." }, { "id": "ial-2025-10-WMA13-01-q02c", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "numerical-methods", "exponentials" ], "primaryTopicId": "numerical-methods", "summary": "Use the iteration formula to calculate the third iterate, then estimate the root to four decimal places.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the supplied formula three times from the given initial value; use the resulting iterate for the four-decimal root estimate." }, { "id": "ial-2025-10-WMA13-01-q03a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find the straight-line model for log price against time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the two points to calculate the gradient and intercept." }, { "id": "ial-2025-10-WMA13-01-q03b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Rewrite the model as an exponential function of time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert from the base-10 logarithmic equation and identify both constants." }, { "id": "ial-2025-10-WMA13-01-q03c", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find the year when the rate of increase reaches the specified value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=12", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the exponential model, equate its rate to the target, and solve for time." }, { "id": "ial-2025-10-WMA13-01-q04a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Express the trigonometric function as a single shifted sine curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match sine and cosine coefficients to the amplitude and phase shift, using the required angle range." }, { "id": "ial-2025-10-WMA13-01-q04b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact minimum of the transformed sine function and the smallest input where it occurs.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the amplitude and vertical shift to obtain the minimum value; set the sine angle to its minimum and choose the least admissible input." }, { "id": "ial-2025-10-WMA13-01-q05a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the modelled population at the start of the study.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute time zero into the population model." }, { "id": "ial-2025-10-WMA13-01-q05b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "differentiation", "summary": "Differentiate the population model with respect to time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the quotient or product rule and simplify the derivative." }, { "id": "ial-2025-10-WMA13-01-q05c", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "differentiation", "summary": "Show the exponential condition at the time of maximum population.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the derivative to zero and rearrange the exponential factor." }, { "id": "ial-2025-10-WMA13-01-q05d", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the maximum population and round to a whole number.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the exponential condition for time, then substitute that time into the model." }, { "id": "ial-2025-10-WMA13-01-q06i", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(i)", "marks": 2, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "integration", "summary": "Differentiate the logarithmic expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the chain rule to the logarithm of the quadratic." }, { "id": "ial-2025-10-WMA13-01-q06ii", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "integration", "summary": "Find the greatest integer parameter satisfying the definite-integral inequality.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the logarithmic derivative, substitute the limits, and solve the resulting inequality for the integer." }, { "id": "ial-2025-10-WMA13-01-q07", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": null, "marks": 7, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Find the parameter range that gives the curve two distinct stationary points.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=20", "qpPages": [ 22, 22 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the exponential-polynomial product, set the derivative to zero, and require the resulting quadratic to have two distinct real roots." }, { "id": "ial-2025-10-WMA13-01-q08a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Use the known intersection to derive the relation between the absolute-value constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=22", "qpPages": [ 26, 26 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the coordinates into the correct branch of the absolute-value graph." }, { "id": "ial-2025-10-WMA13-01-q08b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Use the second intersection to determine both constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=22", "qpPages": [ 26, 26 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the second point, combine with the first relation, and solve simultaneously." }, { "id": "ial-2025-10-WMA13-01-q08c", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "State the vertex coordinates of the absolute-value graph.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=22", "qpPages": [ 26, 26 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the zero of the expression inside the modulus and the vertical constant." }, { "id": "ial-2025-10-WMA13-01-q08d", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 6, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Find the remaining intersection coordinates algebraically.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=22", "qpPages": [ 26, 26 ], "msPages": [ 22, 24 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the cubic and absolute-value branches, factor the resulting polynomial, and select the appropriate intersection." }, { "id": "ial-2025-10-WMA13-01-q09a", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the expression using only double-angle sine and cosine.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=25", "qpPages": [ 30, 30 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use double-angle identities and simplify each term." }, { "id": "ial-2025-10-WMA13-01-q09b", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Derive the stated quadratic in the double-angle sine.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=25", "qpPages": [ 30, 30 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the result from part (a), multiply through to remove cotangent, and use the Pythagorean identity." }, { "id": "ial-2025-10-WMA13-01-q09c", "paperId": "ial-2025-10-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "October", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the original equation over the stated degree interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P3_MS.pdf#page=25", "qpPages": [ 30, 30 ], "msPages": [ 25, 26 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Solve the quadratic in sin(2x), then use the interval to obtain all admissible angles." }, { "id": "ial-2025-10-WMA14-01-q01a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 5, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first four terms of the binomial expansion in simplest form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Factor out the constant and apply the generalized binomial expansion through cubic order." }, { "id": "ial-2025-10-WMA14-01-q01b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "State the interval in which the expansion is valid.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply the convergence condition on the magnitude of the expansion variable." }, { "id": "ial-2025-10-WMA14-01-q01c", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use the small-variable approximation to identify the three constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Match the constant, linear, and quadratic terms after truncating the expansion." }, { "id": "ial-2025-10-WMA14-01-q02", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the exact volume generated by rotating the bounded region about the x-axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=11", "qpPages": [ 4, 4 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the disk method with radius from the curve, then integrate between the stated limits." }, { "id": "ial-2025-10-WMA14-01-q03a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Eliminate the parameter to obtain the Cartesian curve and its constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Rearrange one parametric equation for the parameter, substitute into the other, and collect powers." }, { "id": "ial-2025-10-WMA14-01-q03b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "State the range of the resulting function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=13", "qpPages": [ 6, 6 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the given parameter domain and the transformed expression to determine endpoint behavior." }, { "id": "ial-2025-10-WMA14-01-q04a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find dy/dx for the implicit curve in terms of x and y.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate both sides implicitly using product and chain rules, then collect derivative terms." }, { "id": "ial-2025-10-WMA14-01-q04b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the positive coordinates of the point with the stated gradient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=15", "qpPages": [ 10, 10 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute the gradient condition into the derivative, combine with the curve equation, and choose the positive solution." }, { "id": "ial-2025-10-WMA14-01-q05", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the parameter for which the two lines intersect and give the coordinates of the intersection.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate corresponding coordinates to solve for the parameter, then substitute it into either line to find the point." }, { "id": "ial-2025-10-WMA14-01-q06", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": null, "marks": 7, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Evaluate the definite trigonometric integral using the given substitution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=19", "qpPages": [ 16, 16 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the limits and integrand into the new variable, integrate, and evaluate exactly." }, { "id": "ial-2025-10-WMA14-01-q07a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact cosine of the angle between the stated sides.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=21", "qpPages": [ 18, 18 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the dot product divided by the product of the vector magnitudes." }, { "id": "ial-2025-10-WMA14-01-q07b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact area of the parallelogram and the integer constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=21", "qpPages": [ 18, 18 ], "msPages": [ 21, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the product of side lengths and the sine of the included angle." }, { "id": "ial-2025-10-WMA14-01-q08a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Solve the differential equation and apply the initial volume.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=24", "qpPages": [ 20, 20 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Separate variables, integrate, and use the initial condition to determine the constant." }, { "id": "ial-2025-10-WMA14-01-q08b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Decide whether the model ever reaches the stated container capacity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=24", "qpPages": [ 20, 20 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the horizontal limiting volume with the capacity." }, { "id": "ial-2025-10-WMA14-01-q09a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "integration", "summary": "Use parametric differentiation to find the exact y-coordinate for the given gradient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=26", "qpPages": [ 22, 22 ], "msPages": [ 26, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Form dy/dx from dy/dt divided by dx/dt, set it to the given gradient, and substitute the resulting parameter." }, { "id": "ial-2025-10-WMA14-01-q09b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 9, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "integration", "summary": "Express the shaded area as the required parameter integral, then evaluate it exactly in the requested logarithmic form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=26", "qpPages": [ 22, 22 ], "msPages": [ 26, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use area as the integral of y times dx/dt with the transformed limits; integrate algebraically, substitute the limits, and simplify the logarithms." }, { "id": "ial-2025-10-WMA14-01-q10a", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Complete the contradiction proof about an integer whose square is odd.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=30", "qpPages": [ 26, 26 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assume an even representation, square it, and show this contradicts oddness." }, { "id": "ial-2025-10-WMA14-01-q10b", "paperId": "ial-2025-10-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "October", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove by contradiction that the stated integer equation has no solutions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_P4_MS.pdf#page=30", "qpPages": [ 26, 26 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assume integers exist, derive that the square is odd, use the earlier result, then show the resulting parity contradiction." }, { "id": "ial-2025-10-WME01-01-q01a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the distance from the second support that keeps the beam in equilibrium.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical force balance and moments about a support, with the uniform beam weight acting at its midpoint." }, { "id": "ial-2025-10-WME01-01-q01b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the greatest attached package weight before the first rope breaks.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the support tension to its limiting value and apply force and moment equilibrium." }, { "id": "ial-2025-10-WME01-01-q02a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find one post-collision speed from the impulse received by the second particle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Relate impulse to the change in momentum of that particle." }, { "id": "ial-2025-10-WME01-01-q02b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find both admissible values of the incoming speed factor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of momentum and restitution or the given velocity relations, then solve for the two cases." }, { "id": "ial-2025-10-WME01-01-q03a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration vector from the two applied forces.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the force vectors and divide the resultant by the particle mass." }, { "id": "ial-2025-10-WME01-01-q03b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the speed after three seconds from the initial velocity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use constant-acceleration vector kinematics and take the magnitude of the velocity." }, { "id": "ial-2025-10-WME01-01-q03c", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the relation between the two unknown force components.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the components of the resultant to the common scalar multiple." }, { "id": "ial-2025-10-WME01-01-q03d", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the two possible additional force vectors from the resultant magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the component relation with the prescribed resultant magnitude and solve both signs." }, { "id": "ial-2025-10-WME01-01-q04a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the greatest height reached by the particle projected from above the ground.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use vertical-motion equations with zero velocity at the maximum height." }, { "id": "ial-2025-10-WME01-01-q04b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the two vertically moving particles collide.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write their heights as functions of time and equate positions at the common time." }, { "id": "ial-2025-10-WME01-01-q04c", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the first particle’s speed just before impact.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use its constant-acceleration velocity equation at the collision time." }, { "id": "ial-2025-10-WME01-01-q05a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the force needed to make the box just start sliding.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve horizontally and vertically, use limiting friction, then solve for the applied force." }, { "id": "ial-2025-10-WME01-01-q05b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration after the applied force is increased.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Recalculate the normal reaction and friction, then apply Newton’s second law." }, { "id": "ial-2025-10-WME01-01-q05c", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the speed after the box travels the stated distance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use constant-acceleration kinematics from rest." }, { "id": "ial-2025-10-WME01-01-q06a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the car's acceleration on the first segment and its deceleration on the final segment.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Read the gradient of the velocity-time graph on each stated interval, using the magnitude of the negative gradient for deceleration." }, { "id": "ial-2025-10-WME01-01-q06b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the car’s acceleration-time graph over the full journey.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert each straight-line velocity segment into its constant gradient." }, { "id": "ial-2025-10-WME01-01-q06c", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled by the car.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the area under the velocity-time graph using the appropriate triangles and rectangles." }, { "id": "ial-2025-10-WME01-01-q06d", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the motorcycle acceleration that lets it catch the car at 20 seconds.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Equate the two travelled distances over the relevant time intervals and solve for the constant acceleration." }, { "id": "ial-2025-10-WME01-01-q07a", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Write an equation of motion for the hanging block.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=20", "qpPages": [ 24, 24 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces vertically in the direction of acceleration." }, { "id": "ial-2025-10-WME01-01-q07b", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the string tension while the other block moves up the rough incline.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=20", "qpPages": [ 24, 24 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve perpendicular and parallel to the plane, include friction, and use the shared acceleration." }, { "id": "ial-2025-10-WME01-01-q07c", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Explain how the solution uses the inextensible-string condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=20", "qpPages": [ 24, 24 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that the connected blocks have equal acceleration magnitudes." }, { "id": "ial-2025-10-WME01-01-q07d", "paperId": "ial-2025-10-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "October", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the force magnitude exerted on the pulley by the two string sections.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M1_MS.pdf#page=20", "qpPages": [ 24, 24 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the two tension vectors at the known angle and express the result as a multiple of mg." }, { "id": "ial-2025-10-WME02-01-q01a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the acceleration of the car and trailer system at the stated speed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use power equals driving force times speed, then apply Newton’s second law to the combined system." }, { "id": "ial-2025-10-WME02-01-q01b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the towbar tension at the stated instant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=8", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Newton’s second law to either vehicle using the acceleration from part (a)." }, { "id": "ial-2025-10-WME02-01-q02a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the unknown component of the particle’s velocity after the impulse.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use impulse as change in momentum, then apply the magnitude condition." }, { "id": "ial-2025-10-WME02-01-q02b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the angle between the initial direction and the impulse.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the impulse vector and the dot product or an equivalent trigonometric ratio." }, { "id": "ial-2025-10-WME02-01-q03a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration vector at the specified time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the velocity vector componentwise." }, { "id": "ial-2025-10-WME02-01-q03b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the exact position vector as a function of time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Integrate the velocity components and use the given position to determine both constants." }, { "id": "ial-2025-10-WME02-01-q03c", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the exact speed when the velocity has the stated direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set the velocity components in the given ratio, solve for time, then calculate the velocity magnitude." }, { "id": "ial-2025-10-WME02-01-q04a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass distance from the common diameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Combine the signed moments of the joined and removed laminae about the diameter." }, { "id": "ial-2025-10-WME02-01-q04b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the equilibrium angle of the freely suspended template.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the centre-of-mass offset and the geometry relative to the pivot." }, { "id": "ial-2025-10-WME02-01-q04c", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the attached particle weight ratio for the stated equilibrium.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the pivot, including the template weight and applied horizontal force." }, { "id": "ial-2025-10-WME02-01-q05a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the contact distance using the right-triangle geometry.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Pythagoras to the centre, radius, and contact point." }, { "id": "ial-2025-10-WME02-01-q05b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction at the rod’s contact with the hemisphere.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Take moments about the ground contact point and use the geometric ratios." }, { "id": "ial-2025-10-WME02-01-q05c", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction at the ground contact.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve forces along and perpendicular to the rod or coordinate axes, then form friction divided by normal reaction." }, { "id": "ial-2025-10-WME02-01-q06a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Derive the speed of the second particle after the collision.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply conservation of momentum and Newton’s restitution relation." }, { "id": "ial-2025-10-WME02-01-q06b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the kinetic-energy loss coefficient for the given restitution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the total initial and final kinetic energies and simplify their difference." }, { "id": "ial-2025-10-WME02-01-q06c", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find the restitution coefficient for the collision with the wall.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=15", "qpPages": [ 18, 18 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the impulse-momentum relation for the wall impact and compare rebound and approach speeds." }, { "id": "ial-2025-10-WME02-01-q07a", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the work done against friction along the ramp.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the normal reaction and friction, then multiply the friction force by the distance." }, { "id": "ial-2025-10-WME02-01-q07b", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the initial projection speed using work and energy.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Balance initial kinetic energy against the rise in gravitational potential and work against friction." }, { "id": "ial-2025-10-WME02-01-q07c", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find how long the particle is above the level of the ramp top.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Resolve the initial velocity vertically and solve the vertical displacement equation." }, { "id": "ial-2025-10-WME02-01-q07d", "paperId": "ial-2025-10-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "October", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 6, "topicIds": [ "energy-momentum", "kinematics" ], "primaryTopicId": "energy-momentum", "summary": "Find the horizontal distance from the foot of the ramp to the landing point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_M2_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the full flight time to ground level, calculate horizontal travel, and add the ramp’s horizontal projection." }, { "id": "ial-2025-10-WST01-01-q01a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the corrected cross-product and weight sums.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the supplied totals to calculate the centered sums." }, { "id": "ial-2025-10-WST01-01-q01b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product-moment correlation coefficient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Divide the centered cross-product by the square root of the product of centered sums of squares." }, { "id": "ial-2025-10-WST01-01-q01c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the regression equation for weight on height.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find the gradient from centered sums, then calculate the intercept from the sample means." }, { "id": "ial-2025-10-WST01-01-q01d", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the regression gradient in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Describe the average change in predicted weight for each centimetre of height." }, { "id": "ial-2025-10-WST01-01-q01e", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the regression equation to estimate weight at each of the two stated heights.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Substitute each height into the regression equation and evaluate both predictions." }, { "id": "ial-2025-10-WST01-01-q01f", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(f)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Choose the more reliable of the two predictions and explain why.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare each height with the observed data range and distinguish interpolation from extrapolation." }, { "id": "ial-2025-10-WST01-01-q02a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the lower and upper quartiles from the ordered weights.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated quartile convention on the ordered observations." }, { "id": "ial-2025-10-WST01-01-q02b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show which observation is the only outlier.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the interquartile-range fences and compare the observations with both limits." }, { "id": "ial-2025-10-WST01-01-q02c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a box plot for the first club’s weights on the common scale.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the five-number summary and mark the whiskers and quartiles correctly." }, { "id": "ial-2025-10-WST01-01-q02d", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe and interpret the skew of the second club’s distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=8", "qpPages": [ 6, 6 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the median’s position and tail lengths to identify skew and compare spread around the median." }, { "id": "ial-2025-10-WST01-01-q03a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Give the probability function for the fair-die score.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Assign equal probability to each of the six possible outcomes." }, { "id": "ial-2025-10-WST01-01-q03b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Name the distribution of the score.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the discrete uniform distribution." }, { "id": "ial-2025-10-WST01-01-q03c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected value of the linear transformation of the score.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=9", "qpPages": [ 10, 10 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use linearity of expectation with the mean score." }, { "id": "ial-2025-10-WST01-01-q04a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine the two unknown probabilities from total probability and the mean.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use probabilities summing to one and the expectation equation, then solve simultaneously." }, { "id": "ial-2025-10-WST01-01-q04b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Give the cumulative distribution values at the listed points.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Accumulate probability masses from the smallest support value upward." }, { "id": "ial-2025-10-WST01-01-q04c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Evaluate the requested event probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the inequality into the qualifying support values and add their probabilities." }, { "id": "ial-2025-10-WST01-01-q04d", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the variance of the transformed random variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the linear transformation rule for variance or calculate the first two moments." }, { "id": "ial-2025-10-WST01-01-q05a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the histogram bar dimensions for the last class.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the common frequency-density scale and preserve the given width-to-height ratio." }, { "id": "ial-2025-10-WST01-01-q05b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median from grouped data by interpolation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Locate the median class and interpolate within it using cumulative frequency." }, { "id": "ial-2025-10-WST01-01-q05ci", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)(i)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the mean apple weight from the coded data.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the coded mean back to the original weight scale." }, { "id": "ial-2025-10-WST01-01-q05cii", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)(ii)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the variance of the original weight data.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=13", "qpPages": [ 16, 16 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert the coded variance using the square of the scale factor." }, { "id": "ial-2025-10-WST01-01-q06a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use independence to find the missing Venn-region probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Write the marginal and intersection probabilities and apply the independence equation." }, { "id": "ial-2025-10-WST01-01-q06b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the remaining region probability from the conditional probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express the conditional probability as a ratio over the complement event and solve." }, { "id": "ial-2025-10-WST01-01-q06c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine the last unlabelled region probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use that the disjoint regions sum to one." }, { "id": "ial-2025-10-WST01-01-q06d", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probabilities of the two stated events using the Venn diagram.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=15", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Identify the regions included in each expression, then add their probabilities, respecting complements and intersections." }, { "id": "ial-2025-10-WST01-01-q07a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a probability between two values for the first normal variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Standardise each boundary and subtract the corresponding cumulative probabilities." }, { "id": "ial-2025-10-WST01-01-q07b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find a conditional probability for the first normal variable.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Calculate the numerator interval probability and divide by the given conditioning probability." }, { "id": "ial-2025-10-WST01-01-q07c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the mean and standard deviation from two normal probabilities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=16", "qpPages": [ 22, 22 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Convert both probabilities to standard-normal quantiles and solve the resulting simultaneous equations." }, { "id": "ial-2025-10-WST01-01-q08a", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Show the overall probability of arriving late.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Add the three route-specific late probabilities weighted by route probabilities." }, { "id": "ial-2025-10-WST01-01-q08b", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability of travelling by car given lateness.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=18", "qpPages": [ 26, 26 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Apply Bayes’ theorem using the route and late probabilities." }, { "id": "ial-2025-10-WST01-01-q08c", "paperId": "ial-2025-10-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "October", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the tree diagram with the route and lateness probabilities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S1_MS.pdf#page=18", "qpPages": [ 26, 28 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Fill each branch with its conditional probability and use complements for on-time branches." }, { "id": "ial-2025-10-WST02-01-q01a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Suggest a suitable sampling frame for club members.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a complete, current list or register of all members." }, { "id": "ial-2025-10-WST02-01-q01b", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the sampling units.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State that each individual club member is one unit." }, { "id": "ial-2025-10-WST02-01-q01c", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give an advantage and a disadvantage of random sampling compared with a census.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Mention reduced time or cost, and the possibility that the sample is not representative." }, { "id": "ial-2025-10-WST02-01-q02a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Use the summaries to assess a Poisson model for daily complaints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Compare the sample mean and variance and relate their similarity to the Poisson assumption." }, { "id": "ial-2025-10-WST02-01-q02b", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Calculate the probabilities of at least four complaints and of more than two but fewer than seven complaints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the upper-tail complement for the first event; for the second, subtract the cumulative probability through two from that through six." }, { "id": "ial-2025-10-WST02-01-q02c", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Test whether the mean complaint rate exceeds the stated value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=7", "qpPages": [ 4, 4 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State hypotheses, calculate the relevant upper-tail probability for the five-day total, and compare with the significance level." }, { "id": "ial-2025-10-WST02-01-q03a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Explain what a sampling distribution describes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Describe the possible values of a statistic across samples and their probabilities." }, { "id": "ial-2025-10-WST02-01-q03b", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "List the possible unordered combinations of the three sampled coins.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Enumerate the combinations by the number and denominations of each coin." }, { "id": "ial-2025-10-WST02-01-q03c", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 1, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability attached to the first sample-mean value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the stated probability for the corresponding all-one-pound sample." }, { "id": "ial-2025-10-WST02-01-q03d", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 6, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Calculate the remaining probabilities in the sampling distribution of the sample mean.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the binomial combination counts for each mix of coin types." }, { "id": "ial-2025-10-WST02-01-q03e", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of the modal coin value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=9", "qpPages": [ 8, 8 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Map each possible coin combination to its modal value and combine probabilities." }, { "id": "ial-2025-10-WST02-01-q04a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the constant that makes the cumulative function valid.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use continuity and the requirement that the cumulative distribution reaches one." }, { "id": "ial-2025-10-WST02-01-q04b", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the probability of the specified single point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use continuity of the distribution to identify the point probability." }, { "id": "ial-2025-10-WST02-01-q04c", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the sum of the two interval probabilities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use differences of cumulative distribution values at each pair of endpoints." }, { "id": "ial-2025-10-WST02-01-q04d", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the two pieces of the probability density function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=11", "qpPages": [ 12, 12 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Differentiate the corresponding cumulative-function branches over their intervals." }, { "id": "ial-2025-10-WST02-01-q05a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the probability that more than four flowers are yellow.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Model the count with a binomial distribution and use the upper tail." }, { "id": "ial-2025-10-WST02-01-q05b", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the smallest sample size giving the required chance of at least one yellow flower.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the complement probability of no yellow flowers, take logarithms, and round up." }, { "id": "ial-2025-10-WST02-01-q05c", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Approximate the stated binomial count probability for 100 flowers.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use a normal approximation with continuity correction and subtract the standardized tail probabilities." }, { "id": "ial-2025-10-WST02-01-q05d", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Test whether the red-flower probability exceeds the stated value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=12", "qpPages": [ 16, 16 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Set one-sided binomial hypotheses, calculate the tail probability or critical region, and decide at the given level." }, { "id": "ial-2025-10-WST02-01-q06a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "Find the expected area of the rectangle with fixed perimeter and random side length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=14", "qpPages": [ 20, 20 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Express area as a function of the side, use the uniform density, and integrate over its range." }, { "id": "ial-2025-10-WST02-01-q06b", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "distributions", "summary": "Find the probability that the rectangle’s area exceeds the stated comparison area.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=14", "qpPages": [ 20, 20 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Translate the inequality into a range for the uniform variable and calculate its interval probability." }, { "id": "ial-2025-10-WST02-01-q07a", "paperId": "ial-2025-10-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "October", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Verify the median of the first continuous distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Oct%202025/IAL_MATHS_2025_Oct_S2_MS.pdf#page=15", "qpPages": [ 22, 22 ], "msPages": [ 15, 16 ], "hardestPart": null, 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"probability" ], "primaryTopicId": "distributions", "summary": "Determine E(W", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-que-20230524.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-rms-20230817.pdf#page=11", "qpPages": [ 18, 18 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the constant density on [a,b] to express probabilities as interval lengths, then apply the given mean or variance condition.", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "WST02-01-MS.pdf", "examinerReport": "WST02-01-ER.pdf" }, "sourceSha256": { "questionPaper": "8f0fbc8f11d6c5130cd672cb60838cd3d9017695ec05722e315eb7df57307edc", "markScheme": 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null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Find each expected count from row total × column total ÷ grand total, then compare the chi-squared statistic with the correct degrees of freedom and critical value.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2023-6-WST03-01-q-3-a", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use expectation to test unbiasedness, subtract the target parameter to find bias, and use independence when combining variances.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": 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null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use expectation to test unbiasedness, subtract the target parameter to find bias, and use independence when combining variances.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2023-6-WST03-01-q-3-e", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Use the unbiasedness condition to prove the stated variance formula for a linear estimator.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=10", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=8", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "hardestPart": { "label": "3(e)", "marks": 3 }, "studyPriority": 4, "difficultyEvidence": "The report says this show-that part caused difficulty and careless algebra errors were common.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20230817.pdf", "tagConfidence": "reviewed", "answerGuide": "Use independence to add the variances of the weighted observations, 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"distributions", "summary": "Demonstrate that for 0  a  4", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=14", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=9", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use the total probability or boundary conditions to find the constants, then integrate or differentiate the CDF as appropriate.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": 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"tagConfidence": "reviewed", "answerGuide": "Use estimate ± critical z-value × standard error, then use the interval endpoints to address the comparison asked.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2023-6-WST03-01-q-5-b", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Determine a 95% confidence interval for μ, in terms of y–", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=18", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=10", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "Use estimate ± critical z-value × standard error, then use the interval endpoints to address the comparison asked.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": 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"https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=10", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "hardestPart": { "label": "5(c)(i)–(ii)", "marks": 7 }, "studyPriority": 1, "difficultyEvidence": "Question 5 was the most discriminating question; candidates struggled to find the two non-overlap regions and to standardize the difference of sample means correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20230817.pdf", "tagConfidence": "reviewed", "answerGuide": "Find the distribution of the difference of independent sample means, express non-overlap as either tail beyond the sum of the interval half-widths, then standardize and double the one-tail probability.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, 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"msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the estimator, distribution, or hypotheses being used, calculate the statistic or probability from the supplied data, and finish with the contextual conclusion.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2023-6-WST03-01-q-6-b", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Carry out the specified significance test and report its conclusion in context.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State H0 and H1, calculate the test statistic, compare it with the critical value or p-value, then conclude in context.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2023-6-WST03-01-q-6-c", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Decide whether a normal population distribution must be assumed for the large-sample comparison of means, with a reason.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 12 ], "hardestPart": { "label": "6(c)", "marks": 2 }, "studyPriority": 2, "difficultyEvidence": "The report says parts (c) and (d) challenged students; for part (c), many incorrectly assumed normality was necessary instead of invoking the central limit theorem for large samples.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20230817.pdf", "tagConfidence": "reviewed", "answerGuide": "State that a normal population model is not required because both samples are large enough for the central limit theorem to justify the sampling distribution of the mean.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" }, "sourceEvidencePages": { "examinerReport": [ 4, 5 ] } }, { "id": "ial-2023-6-WST03-01-q-6-d", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State an assumption used in carrying out the two-sample test.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=22", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=11", "qpPages": [ 22, 22 ], "msPages": [ 11, 12 ], "hardestPart": { "label": "6(d)", "marks": 1 }, "studyPriority": 3, "difficultyEvidence": "The report says part (d) challenged students; many named independence alone and missed the assumption needed for both groups’ sample variance estimates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20230817.pdf", "tagConfidence": "reviewed", "answerGuide": "State that each sample variance is being used as an estimate of its corresponding population variance, for both groups.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" }, "sourceEvidencePages": { "examinerReport": [ 5 ] } }, { "id": "ial-2023-6-WST03-01-q-7-a", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Determine P(X < 40)", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=13", "qpPages": [ 26, 26 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the estimator, distribution, or hypotheses being used, calculate the statistic or probability from the supplied data, and finish with the contextual conclusion.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2023-6-WST03-01-q-7-b", "paperId": "ial-2023-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2023, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Evaluate the probability requested in this part using the distribution and parameters stated in the question.", "qpUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-que-20230614.pdf#page=26", "msUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-rms-20230817.pdf#page=13", "qpPages": [ 26, 26 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "answerGuide": "State the estimator, distribution, or hypotheses being used, calculate the statistic or probability from the supplied data, and finish with the contextual conclusion.", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "WST03-01-MS.pdf", "examinerReport": "WST03-01-ER.pdf" }, "sourceSha256": { "questionPaper": "25fce562e4e24f17636053f9a86c84d9515fc1bb2813657388391f6dbc5397b0", "markScheme": "ad5e14534ee48a5f9c7cdcda16c56cfe0882ca33245415e90e98e4e6439a6f3a", "examinerReport": "6589cfc246bcc3b72f4fbb3cfe24732dd96dde73de91a05d4ffd2a0e49d202db" } }, { "id": "ial-2026-1-WDM11-01-q01a", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain why binary search cannot be applied to the unsorted sports list.", "answerGuide": "Binary search requires an ordered list; this list is not in alphabetical order.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q01b", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Put the sports list in a suitable order and trace binary search for Football.", "answerGuide": "Sort the initial letters alphabetically, show each chosen middle pivot and discard the half that cannot contain F; conclude that F is absent.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=6", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q02", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra’s algorithm to find the quickest route from C to K in the weighted network.", "answerGuide": "Label vertices in increasing tentative-distance order, update routes through each labelled vertex, then give the predecessor route to K and its total time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=7", "qpPages": [ 3, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q03ia", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(a)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain how to modify Kruskal’s algorithm when a particular arc must appear in the minimum spanning tree.", "answerGuide": "Add the required arc first, then continue Kruskal by considering remaining arcs in increasing weight and rejecting any that create a cycle.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q03ib", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the least-weight spanning tree that includes arc BJ using the adapted Kruskal method.", "answerGuide": "Start with BJ, add the lightest cycle-free arcs in order, and show the accepted and rejected arcs until the spanning tree is complete.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q03ic", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the weight of the required spanning tree.", "answerGuide": "Add the weights of the arcs in the spanning tree from part (b).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q03iia", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Minimise a closed route that traverses every network arc at least once, starting and ending at A.", "answerGuide": "Use the route-inspection method: identify the four odd vertices, compare all three pairings, repeat the cheapest paired paths, and add that extra length to the total network weight.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q03iib", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find how much shorter the minimum route from F to J is than the closed route from part (a).", "answerGuide": "Pair the four odd vertices for an open route from F to J, find the least added pairing cost, then subtract it from the part (a) added cost.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q04a", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain why x lies between 45 and 57 from the two bubble-sort passes.", "answerGuide": "Use the adjacent comparisons shown in pass two: 44 does not swap with x, and x does not swap with 58.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q04b", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Carry out the next left-to-right pass of bubble sort.", "answerGuide": "Compare each adjacent pair from left to right, swapping whenever the left value is larger, and write the full list after the pass.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q04c", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine x using the bin lower bound and first-fit decreasing conditions.", "answerGuide": "Use the total-weight lower-bound condition to restrict x, then apply first-fit decreasing at capacity 160 to the second-bin placement and solve for x.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=9", "qpPages": [ 5, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q05", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 8, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate the table-production profit model as a linear-programming problem.", "answerGuide": "Let x,y,z be the small, medium and large table counts; maximise total profit, then translate wood, time, percentage and ratio conditions into simplified integer-coefficient inequalities, with non-negativity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q06a", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the two nearest-neighbour routes starting and finishing at A and their lengths.", "answerGuide": "At each step choose a nearest unvisited town, resolving the tie at the final choice in both ways; return to A and total each route.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q06b", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Choose the better nearest-neighbour upper bound.", "answerGuide": "Select the route with the smaller total length and state that a tour gives an upper bound.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q06c", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Prim’s algorithm after deleting A to find a lower bound for the tour length.", "answerGuide": "Build the minimum spanning tree on the remaining towns starting at B, add the two shortest edges from A, and state the resulting lower bound.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q06d", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the smallest interval containing the optimal tour length.", "answerGuide": "Use the strongest lower bound and the shortest tour upper bound found so far as interval endpoints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=11", "qpPages": [ 7, 7 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q07a", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the activity table with immediately preceding activities.", "answerGuide": "Read each activity’s incoming arcs from the network and enter all immediate predecessors.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q07b", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the early and late event times on the activity network.", "answerGuide": "Perform a forward pass using maximum predecessor times, then a backward pass from the project completion time using minimum successor times.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q07c", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the critical activities.", "answerGuide": "Identify activities whose earliest and latest event-time difference gives zero float.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q07d", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate a lower bound for the minimum number of workers.", "answerGuide": "Divide the total activity duration by the minimum project duration and round up to the next whole worker.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q07e", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw a feasible minimum-duration schedule using the minimum number of workers.", "answerGuide": "Place all activities within their earliest/latest windows, respect every predecessor relation, and check that four workers suffice throughout the project.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q08a", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the graph of the linear-programming constraints and identify the feasible region.", "answerGuide": "Draw each boundary line from two points, shade the excluded half-planes, and label the remaining feasible region R.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q08b", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the vertex method to find the exact optimal vertex.", "answerGuide": "Find the feasible-region vertices by solving boundary intersections, evaluate P=6x+5y at each, and select the greatest value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q08c", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the optimal integer solution and objective value.", "answerGuide": "Check feasible integer points near the continuous optimum, reject points outside the constraints, and compare their objective values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WDM11-01-q08d", "paperId": "ial-2026-1-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2026, "session": "January", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the range of k for which the optimal vertex remains unchanged for P=kx+y.", "answerGuide": "Compare the objective-line gradient with the slopes of the two active constraints at the optimal vertex and state the limiting inequalities for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_QP.pdf#page=9", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_D1_MS.pdf#page=14", "qpPages": [ 9, 9 ], "msPages": [ 14, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WDM11-QP.pdf", "markScheme": "2026-01-WDM11-MS.pdf" }, "sourceSha256": { "questionPaper": "239E1C7FC71CD7E714625A44F2CF5EBA4E7B64308DFFB73F86D3B51B1A0705CE", "markScheme": "0465F28614EE2B8B5CAEB063813708DFA08C77206D32306C376BD98E9933FA1F" } }, { "id": "ial-2026-1-WMA11-01-q01a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given function and simplify.", "answerGuide": "Rewrite fractional powers using indices, differentiate each term by the power rule, and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q01b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the indefinite integral of the given function.", "answerGuide": "Integrate each power term separately and include the constant of integration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q02a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Show that substitution from the linear equation gives the stated quadratic.", "answerGuide": "Substitute y=x+4 into the second equation, expand, and rearrange to the required quadratic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q02b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve the simultaneous equations exactly in the requested surd form.", "answerGuide": "Solve the quadratic for x using a non-calculator method, then use y=x+4 and pair each x-value with its y-value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q03i", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the range of z satisfying the exponential inequality.", "answerGuide": "Rewrite both sides with a common base, compare exponents, and state the resulting interval or union of intervals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q03ii", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "algebra", "coordinate-geometry" ], "primaryTopicId": "algebra", "summary": "Represent the compound inequality on the supplied graph.", "answerGuide": "Draw the vertical boundary x=−2 and shade the regions satisfying both curve/line inequalities, including the correct boundary treatment.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=10", "qpPages": [ 6, 7 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q04a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the line through A and B in integer-coefficient form.", "answerGuide": "Calculate the gradient from the two points, use point-slope form, and rearrange to ax+by+c=0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q04b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find where the perpendicular through A meets the x-axis.", "answerGuide": "Use the negative reciprocal gradient through A, set y=0, and solve for the x-coordinate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q04c", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the area of triangle ABC.", "answerGuide": "Use the coordinates to find a base and perpendicular height, then apply one half base times height.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=12", "qpPages": [ 8, 11 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q05a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the specified angle in the triangle and show its value to three significant figures.", "answerGuide": "Use the sine rule with the known side-angle pair to find the required sine, then use the angle sum in radians.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q05b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the total area of the sector-and-triangle logo.", "answerGuide": "Add the sector area using its angle in radians to the triangle area using one half ab sin C.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q05c", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Find the perimeter of the logo.", "answerGuide": "Add the sector arc length to the exposed straight sides, finding the missing triangle side with the cosine rule.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=15", "qpPages": [ 12, 15 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q06a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Deduce both required roots from the graph of the transformed function.", "answerGuide": "Apply the stated input transformation to the known x-intercepts, then solve for each corresponding x-value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=17", "qpPages": [ 16, 19 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q06b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the cubic function in factorised form.", "answerGuide": "Use the simple root and repeated root from the graph, then substitute the y-intercept to determine the scale factor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=17", "qpPages": [ 16, 19 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q06c", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the exact x-coordinates where the cubic meets the given line.", "answerGuide": "Equate the factorised cubic and line, factor out the known intersection, then solve the remaining quadratic exactly.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=17", "qpPages": [ 16, 19 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q07a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 5, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Complete the square for the gradient function.", "answerGuide": "Differentiate the cubic, factor the quadratic coefficient, and complete the square to identify p, q and r.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=19", "qpPages": [ 20, 21 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q07b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "State the minimum gradient and the x-coordinate where it occurs.", "answerGuide": "Read the minimum value and vertex x-coordinate from the completed-square form of the gradient function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=19", "qpPages": [ 20, 21 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q07c", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent equation at the point where the gradient is minimum.", "answerGuide": "Use the x-coordinate to evaluate the curve point, then use the minimum gradient in point-slope form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=19", "qpPages": [ 20, 21 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q08a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Verify the given tangent point and find the other possible x-coordinate.", "answerGuide": "Check the proposed x-value in the derivative condition, then solve the tangent-gradient equation for the other positive root.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q08b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the normal equation at P.", "answerGuide": "Take the negative reciprocal of the tangent gradient and use point-slope form through P.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q08c", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "integration", "differentiation" ], "primaryTopicId": "integration", "summary": "Recover f(x) from its derivative and the point P.", "answerGuide": "Integrate f′(x), use the known point to find the integration constant, and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=20", "qpPages": [ 22, 25 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q09a", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Identify the trigonometric function from its graph.", "answerGuide": "Read the amplitude and phase from the marked maximum/minimum positions and write the corresponding cosine expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q09b", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 3, "topicIds": [ "functions", "trigonometry" ], "primaryTopicId": "functions", "summary": "Sketch the specified parabola and count the solutions of the equation over the stated interval.", "answerGuide": "Draw the downward-opening quadratic with the correct vertex and intercepts, then count intersections over one period and use periodicity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA11-01-q09c", "paperId": "ial-2026-1-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2026, "session": "January", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Sketch the tangent graph and count the solutions of the equation over the stated interval.", "answerGuide": "Draw the tangent branches with their zeros and asymptotes, then count intersections over the repeating interval, including the endpoints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P1_MS.pdf#page=22", "qpPages": [ 26, 28 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA11-QP.pdf", "markScheme": "2026-01-WMA11-MS.pdf" }, "sourceSha256": { "questionPaper": "F511E0C60506CE3DFE9C40F505D472FF11545FE6D4556E0B778A572ABB3899FE", "markScheme": "B18C440CECF2D4D230A93F87C86E4E2479082D885E382214B8D2998CEB597018" } }, { "id": "ial-2026-1-WMA12-01-q01a", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Expand the binomial through the x-cubed term in ascending powers.", "answerGuide": "Use the binomial theorem on (3+kx)^7 and simplify the constant through cubic terms.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q01b", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find k from the relationship between the x-squared and x-cubed coefficients.", "answerGuide": "Equate the x² coefficient to three times the x³ coefficient and solve for nonzero k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q02a", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "Estimate the bounded area using the trapezium rule.", "answerGuide": "Use all six tabulated ordinates with equal width 3 in the trapezium-rule formula.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q02b", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "numerical-methods", "integration" ], "primaryTopicId": "numerical-methods", "summary": "State how to improve the trapezium-rule estimate.", "answerGuide": "Use more ordinates to create narrower strips over the same interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q02c", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "integration", "numerical-methods" ], "primaryTopicId": "integration", "summary": "Use the estimate from part (a) to approximate the related integral.", "answerGuide": "Recognise the given integrand as a scaled version of the curve area and apply the corresponding constant factor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q03i", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Use a factor and the factor theorem to find the remainder for the second divisor.", "answerGuide": "Set f(−2)=0 to find k, then evaluate f(5) for the remainder on division by x−5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q03ii", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Find the quotient and remainder on division by a linear polynomial.", "answerGuide": "Use polynomial long division or coefficient comparison to determine the quadratic quotient and constant remainder.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q04", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": null, "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the positive parameter values that satisfy the given definite-integral condition.", "answerGuide": "Integrate the algebraic expression term by term, apply the stated limits, equate to the given value, and solve the resulting quadratic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q05a", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the centre and radius of the circle.", "answerGuide": "Complete the square in x and y, then read the centre coordinates and take the positive square root of the squared radius.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q05b", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find k so the two circles touch.", "answerGuide": "Use the distance between centres and set it equal to the sum or difference of the radii according to the stated touching configuration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q06i", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(i)", "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove the difference of squares for consecutive positive odd integers is divisible by 8.", "answerGuide": "Write the consecutive odd numbers as 2n+1 and 2n−1, subtract their squares, and factor out 8.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q06ii", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)", "marks": 4, "topicIds": [ "differentiation", "proof" ], "primaryTopicId": "differentiation", "summary": "Prove the cubic curve has no stationary points.", "answerGuide": "Differentiate, then show the derivative quadratic has no real roots, for example by a negative discriminant or completing the square.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q07i", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)", "marks": 4, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Solve the exponential equation exactly in the requested logarithmic form.", "answerGuide": "Isolate the exponential term, take logarithms to base 2, and rearrange into the requested rational-plus-log form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q07ii", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the logarithmic equation using log laws.", "answerGuide": "Combine logarithms, convert to an algebraic equation, solve the quadratic, and reject any root outside the original logarithm domains.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=14", "qpPages": [ 16, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q08i", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the tangent equation on the specified degree interval.", "answerGuide": "Isolate tan(2θ−30°), use the tangent-period solutions, and retain all θ in the interval to one decimal place.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q08ii", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve the trigonometric equation over 0≤x<2π.", "answerGuide": "Rewrite tan x using sin x/cos x, use sin²x=1−cos²x to form a quadratic in cos x, then list valid angles in the interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q09a", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the year-five amount under the constant-decrease model.", "answerGuide": "Use the arithmetic sequence with 14 equal annual changes from year 1 to year 15, then calculate the fifth term.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q09b", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 2, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the total mined over years 1 through 15 under model A.", "answerGuide": "Use the arithmetic-series sum with first and last terms and 15 terms.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q09c", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 3, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Find the year-ten amount under the constant-percentage model.", "answerGuide": "Find the common ratio from the first and fifteenth terms, then evaluate the tenth term and round to the nearest 10 tonnes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q09d", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(d)", "marks": 2, "topicIds": [ "sequences", "exponentials" ], "primaryTopicId": "sequences", "summary": "Find the limiting total under model B.", "answerGuide": "Use the geometric series to infinity with the first term and common ratio, checking that the ratio is below 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q10a", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Find the exact x-coordinate of the stationary point.", "answerGuide": "Differentiate the curve, set dy/dx=0, and solve the resulting equation for the stationary x-coordinate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=20", "qpPages": [ 26, 28 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA12-01-q10b", "paperId": "ial-2026-1-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2026, "session": "January", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find k when the two shaded regions have equal area.", "answerGuide": "Integrate the curve algebraically to obtain the relevant area expressions, equate them, and solve exactly for k.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P2_MS.pdf#page=20", "qpPages": [ 26, 28 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA12-QP.pdf", "markScheme": "2026-01-WMA12-MS.pdf" }, "sourceSha256": { "questionPaper": "3405BE43318165EB19C7B57847CA2273D452A83D06818D2570E3CA0D9960E50C", "markScheme": "8050BC9720098CEAF3727FCFC9E6F14EFA610E15E5D78736F2EF78A8C62AAD6C" } }, { "id": "ial-2026-1-WMA13-01-q01a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Show the root exists in the given interval.", "answerGuide": "Evaluate g at both endpoints, show opposite signs, and cite continuity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q01b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Rearrange the root equation into the stated iteration form.", "answerGuide": "Isolate the sine term, apply inverse sine, and identify the constant p.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q01c", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "numerical-methods", "trigonometry" ], "primaryTopicId": "numerical-methods", "summary": "Calculate the next iterate and use repeated iteration to estimate the root.", "answerGuide": "Substitute into the iteration formula, continue until the values agree to four decimal places, and state the estimate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q02i", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(i)", "marks": 2, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate y=x² ln(2x).", "answerGuide": "Apply the product rule to x² and ln(2x), using the chain rule for the logarithm.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q02ii", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(ii)", "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate the exponential quotient and express it in the required factorised form.", "answerGuide": "Use the quotient rule, simplify the exponential and polynomial factors, and compare with the requested constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q03i", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)", "marks": 2, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate the expression in simplest form.", "answerGuide": "Rewrite the power expression and integrate using the power rule.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q03iia", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(a)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Divide the cubic by the stated quadratic and find quotient and remainder.", "answerGuide": "Use polynomial division or coefficient comparison to obtain a linear quotient and linear remainder.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q03iib", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)(b)", "marks": 4, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Evaluate the definite integral using the decomposition from part (a).", "answerGuide": "Use the quotient/remainder decomposition, integrate each term, and substitute both limits.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q04a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find the heart rate for a mammal of mass 3 kg.", "answerGuide": "Substitute m=3 into the logarithmic model, evaluate the powers, and calculate h.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q04b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Rewrite the model in the form h=pm^q.", "answerGuide": "Use logarithm laws to isolate log10 h and exponentiate, then identify p and q.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q04c", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Interpret the parameter p in context.", "answerGuide": "Set mass m=1 kg in h=pm^q; p is the modelled resting heart rate for a 1 kg mammal.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=11", "qpPages": [ 10, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q05a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the composite function ff(x) as a single fraction.", "answerGuide": "Substitute f(x) into f, combine the resulting rational expression over a common denominator, and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q05b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Evaluate the inverse function at 12.", "answerGuide": "Solve f(x)=12 for x, applying the stated domain restrictions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q05c", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "functions", "exponentials" ], "primaryTopicId": "functions", "summary": "Find exact a-values satisfying the equation involving g(a).", "answerGuide": "Substitute the defined functions, solve the resulting equation algebraically, and enforce the logarithm domain.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q06a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Show the trigonometric equation can be rearranged to the stated tangent equation.", "answerGuide": "Use angle and double-angle identities, collect sine and cosine terms, and divide by the appropriate cosine factor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q06b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve the transformed trigonometric equation on the stated interval.", "answerGuide": "Use the tangent solutions from part (a), account for the period and interval, and round to one decimal place.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=13", "qpPages": [ 16, 19 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q07a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "State the vertex coordinates of the quadratic graph.", "answerGuide": "Read the vertex from the completed-square form of the given quadratic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q07b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve f(x)=8.", "answerGuide": "Substitute the function expression, rearrange to a quadratic, and solve for both x-values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q07c", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Find m-values for which y=mx meets the graph exactly once.", "answerGuide": "Set the line equal to the quadratic and use the discriminant condition for exactly one intersection, respecting the stated range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q07d", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find a and b from the transformed vertex.", "answerGuide": "Track how y=af(x+b) transforms the original vertex horizontally and vertically, then solve for a and b.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=14", "qpPages": [ 20, 23 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q08", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": null, "marks": 4, "topicIds": [ "differentiation", "exponentials" ], "primaryTopicId": "differentiation", "summary": "Differentiate the implicit exponential-trigonometric curve and identify the constants.", "answerGuide": "Take logarithms or use implicit differentiation, apply the chain rule to tan y, and rearrange dy/dx into the requested power-log form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=16", "qpPages": [ 24, 25 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q09a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 6, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Rewrite the trigonometric expression as a single phase-shifted sine.", "answerGuide": "Use compound-angle identities, compare sine and cosine coefficients, and determine the amplitude and phase in the required range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q09b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 1, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "State the range of h.", "answerGuide": "Use the amplitude and vertical shift from part (a) to read the minimum and maximum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q09c", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the x-coordinate of the specified minimum point.", "answerGuide": "Set the phase-shifted sine to its minimum value and choose the largest in-range x-coordinate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=18", "qpPages": [ 26, 29 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q10a", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 2, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find the exact growth rate of population A at t=5.", "answerGuide": "Differentiate the exponential model and substitute t=5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q10b", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Determine P exactly and k to three decimal places for population B.", "answerGuide": "Use the initial value to identify P, then substitute the four-year population and solve for k with logarithms.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA13-01-q10c", "paperId": "ial-2026-1-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2026, "session": "January", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 3, "topicIds": [ "exponentials" ], "primaryTopicId": "exponentials", "summary": "Find when the two population models predict equal populations.", "answerGuide": "Set the two models equal, isolate the exponential term, take logarithms, and round T to one decimal place.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P3_MS.pdf#page=20", "qpPages": [ 30, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA13-QP.pdf", "markScheme": "2026-01-WMA13-MS.pdf" }, "sourceSha256": { "questionPaper": "2BB8131821AF4CD3D552F13D702CFB76907DF61A64F83F121BC6928D467E38ED", "markScheme": "2439E48D2BBA6C8A7741F42807C46134752C7936D39E91CB63D9E61AE7CCEF3B" } }, { "id": "ial-2026-1-WMA14-01-q01a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 5, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find a and n from the first binomial coefficients.", "answerGuide": "Match the x and x² coefficients from the expansion with the given terms, then solve the resulting simultaneous equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q01b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the next coefficient p.", "answerGuide": "Substitute the values of a and n into the x³ binomial coefficient.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q01c", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "State the interval where the expansion is valid.", "answerGuide": "Apply the binomial-series condition |ax|<1 using the value of a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q02", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 5, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the exact definite integral involving a power and logarithm.", "answerGuide": "Use integration by parts with the logarithmic factor as u, integrate the power factor, and apply the limits exactly.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q03a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Factorise x²−4y².", "answerGuide": "Recognise a difference of squares and write the two linear factors.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q03b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove by contradiction that no positive integers satisfy the equation.", "answerGuide": "Factor the difference of squares, use the integer factor possibilities forced by 27, and show that each case makes an integer non-integer or otherwise contradicts positivity/integrality.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=10", "qpPages": [ 8, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q04a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Use u=e^x+4 to transform the integral into the stated rational form.", "answerGuide": "Substitute u=e^x+4, express e^x and dx in terms of u, then simplify the transformed integrand and identify the constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q04b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the resulting logarithmic integral exactly.", "answerGuide": "Use partial fractions in u, integrate logarithmic terms, then substitute u=e^x+4 and apply the limits.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q05a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of P at t=3.", "answerGuide": "Substitute t=3 into both parametric equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q05b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "differentiation" ], "primaryTopicId": "coordinate-geometry", "summary": "Find dy/dx in terms of t.", "answerGuide": "Differentiate x(t) and y(t), then divide dy/dt by dx/dt and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q05c", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the tangent equation at P.", "answerGuide": "Evaluate the parametric gradient at t=3 and use the coordinates from part (a) in point-slope form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q05d", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "functions", "coordinate-geometry" ], "primaryTopicId": "functions", "summary": "Find the domain and range of the Cartesian function.", "answerGuide": "Use the allowed parameter interval and the x(t) mapping for the domain, then use the attainable extrema of y(t) for the range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=14", "qpPages": [ 14, 17 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q06a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the point on the line nearest the origin.", "answerGuide": "Write a general position vector, use perpendicularity of OP to the direction vector (or minimise |OP|²), and substitute the parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q06b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find a so the two lines intersect.", "answerGuide": "Equate corresponding coordinates of the line position vectors, solve the parameter equations, and use them to determine a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q06c", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the coordinates of the intersection point.", "answerGuide": "Substitute the line parameter values into either line’s position vector.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q07a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Find the x-coordinates of P and Q.", "answerGuide": "Substitute y=π/4 into the curve equation, use the exact tangent value, and solve for the two x-values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q07b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the exact gradient dy/dx at P.", "answerGuide": "Differentiate the implicit equation using product, chain and tangent rules, then substitute the coordinates of P and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q07c", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Find the product of the tangent gradient at P and normal gradient at Q.", "answerGuide": "Use the gradients at the symmetric points, take the negative reciprocal for the normal, and multiply.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q08a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Derive the differential equation for the radius as the spherical balloon deflates.", "answerGuide": "Differentiate V=4πr³/3, set dV/dt to a negative constant, and rearrange to dr/dt=−k/r².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q08b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Solve the radius differential equation using the two radius-time conditions.", "answerGuide": "Separate r²dr and dt, integrate to a cubic relation, then use the initial and later radius values to determine the constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q08c", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "integration" ], "primaryTopicId": "integration", "summary": "Find the time T when the balloon volume reaches zero.", "answerGuide": "Set r=0 in the radius-time model and solve for T.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=20", "qpPages": [ 26, 29 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q09a", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Show the integral form for the volume of the solid of revolution.", "answerGuide": "Use V=π∫y² dx, substitute x=6sin t and y=5sin2t, and determine the parameter limits and constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WMA14-01-q09b", "paperId": "ial-2026-1-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2026, "session": "January", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry", "integration" ], "primaryTopicId": "coordinate-geometry", "summary": "Evaluate the exact volume integral.", "answerGuide": "Use sin2t=2sint cost, rewrite powers to integrate algebraically, then evaluate at the stated parameter limits.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_P4_MS.pdf#page=22", "qpPages": [ 30, 32 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WMA14-QP.pdf", "markScheme": "2026-01-WMA14-MS.pdf" }, "sourceSha256": { "questionPaper": "C3FFE490F82E89941CA12347150F31B22B873727E09E0303A0DCB25E600714DD", "markScheme": "E88D97F6FF4E6154BB14CB607EDE8E0783434CFA26DCEDF7F2E86C53878793F9" } }, { "id": "ial-2026-1-WME01-01-q01", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 6, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the speeds of both particles immediately after the collision.", "answerGuide": "Use impulse–momentum to find P’s post-collision speed, then apply conservation of momentum or impulse–momentum to find Q’s speed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q02a", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Show that the speed parameter U is 6.", "answerGuide": "Equate the areas under the three sections of the speed–time graph to 600 m and solve for U.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q02b", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the distance travelled while accelerating from rest to 6 m s⁻¹.", "answerGuide": "Use the triangular area under the speed–time graph up to the time when the speed reaches 6 m s⁻¹.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q02c", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration during the first 12 seconds.", "answerGuide": "Use the change in speed divided by 12 seconds.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q02d", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the deceleration during the final 10 seconds.", "answerGuide": "Use the change from 3U to 2U over the interval from 31 to 41 seconds and give the positive deceleration magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q02e", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the acceleration–time graph for the motion.", "answerGuide": "Draw constant positive acceleration from 0 to 12 s, zero acceleration from 12 to 31 s, and constant negative acceleration from 31 to 41 s; label the levels.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q03a", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Show the relation between p and q from the resultant-force direction.", "answerGuide": "Add the three force vectors, then set the ratio of the i- and j-components equal to the ratio in the stated direction vector and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q03b", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s acceleration when p = 7.", "answerGuide": "Use the relation from part (a) to find q, form the resultant force vector, and apply F = ma with mass 0.5 kg.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q04a", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tensions in the ropes attached at C and A.", "answerGuide": "Take moments about a suitable point to find one tension, then use vertical equilibrium to determine the other.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q04b", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the maximum mass M that keeps the rod in horizontal equilibrium.", "answerGuide": "Take moments about C and use the limiting case where the tension at A becomes zero; solve for M.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=12", "qpPages": [ 12, 15 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q05a", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Show that the speed U at A is 2.1 m s⁻¹ to 2 significant figures.", "answerGuide": "Resolve the normal reaction, use friction to find the constant deceleration, and apply SUVAT from A to B using the 0.75 s travel time and the given speed ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q05b", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the time taken for the particle to move from B to C.", "answerGuide": "Use the constant deceleration from part (a) and SUVAT between the known speeds at B and C.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q05c", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Find the distance AC.", "answerGuide": "Apply SUVAT from A to C, using the time from A to B and the time from B to C, or combine the two displacement intervals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=14", "qpPages": [ 16, 19 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q06a", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the ship’s exact speed.", "answerGuide": "Find the magnitude of its velocity vector (10, 15) km h⁻¹ and express the result exactly.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q06b", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "kinematics" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the boat’s direction as a bearing.", "answerGuide": "Use the east and north components of its velocity to find the angle from north, then express the direction as a three-figure bearing.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q06c", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "coordinate-geometry", "kinematics" ], "primaryTopicId": "coordinate-geometry", "summary": "Show the relative position vector of B from S at time t.", "answerGuide": "Write each position vector at time t and subtract S’s position vector from B’s.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q06d", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 3, "topicIds": [ "coordinate-geometry", "kinematics" ], "primaryTopicId": "coordinate-geometry", "summary": "Show that the boat and ship do not collide.", "answerGuide": "Set each component of the relative position vector to zero and show that the resulting times are unequal.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q06e", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 4, "topicIds": [ "coordinate-geometry", "kinematics" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the separation when the boat is north-east of the ship.", "answerGuide": "Set the east and north components of the relative position equal, solve for t, and find the magnitude of the separation vector.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=16", "qpPages": [ 20, 23 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q07a", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction when the package is in limiting equilibrium.", "answerGuide": "Resolve forces along and perpendicular to the plane for the package and bucket, use limiting friction F = μR, and substitute the exact trigonometric ratio.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=19", "qpPages": [ 24, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME01-01-q07b", "paperId": "ial-2026-1-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2026, "session": "January", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the number of blocks in the bucket and the initial acceleration of P.", "answerGuide": "Use limiting equilibrium to identify the first whole number of blocks that makes P move up, then apply Newton’s second law to find its acceleration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M1_MS.pdf#page=19", "qpPages": [ 24, 28 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME01-QP.pdf", "markScheme": "2026-01-WME01-MS.pdf" }, "sourceSha256": { "questionPaper": "0D947B088F6C500DBA791369047FFC1B6596FC7E42D191434D37EFB527460C30", "markScheme": "2CBE5F5BCF4EBF463EFFFAB0BE0DDCF4B28CC7D67FEAC691D7BD1786EA7203F9" } }, { "id": "ial-2026-1-WME02-01-q01a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "mechanics", "algebra" ], "primaryTopicId": "mechanics", "summary": "Show that the road resistance is 260 N.", "answerGuide": "Use engine power P = Fv to express the driving force in terms of the speed parameter, then apply Newton’s second law along the slope and solve with R = 26λ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q01b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the engine power while the van travels downhill.", "answerGuide": "Resolve forces along the slope with the stated acceleration and resistance, find the driving force, then use P = Fv at 12 m s⁻¹.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=7", "qpPages": [ 2, 5 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q02a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "mechanics", "coordinate-geometry" ], "primaryTopicId": "mechanics", "summary": "Show the centre-of-mass distance for the composite lamina.", "answerGuide": "Find the component areas or mass ratios and their perpendicular distances from AC, then use the weighted-mean formula for the centre of mass.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q02b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find F in terms of W when the lamina is in equilibrium.", "answerGuide": "Substitute d = 4a, locate the combined centre of mass, and take moments about A including the attached particle and horizontal force.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q03a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Show the formula for the string tension in the rod system.", "answerGuide": "Resolve the smooth-contact reactions and take moments about A, then rearrange to the stated relation between T, m, g and θ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q03b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find the greatest distance AE before the string reaches its breaking tension.", "answerGuide": "Set T to its maximum value, substitute tan θ, resolve the reactions, and take moments about A to solve for the particle position.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=11", "qpPages": [ 10, 13 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q04a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find when the particle is instantaneously at rest.", "answerGuide": "Set both components of the velocity vector equal to zero and solve for the positive time that satisfies both.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q04b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find when the acceleration is perpendicular to the given vector.", "answerGuide": "Differentiate the velocity to obtain acceleration, set its dot product with (2i − j) to zero, and solve for t > 0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q04c", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 6, "topicIds": [ "kinematics", "coordinate-geometry" ], "primaryTopicId": "kinematics", "summary": "Find the distance OP at t = 4.", "answerGuide": "Integrate the velocity components from t = 1 to t = 4, add the given position vector at t = 1, then take the position-vector magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q05a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the projectile’s maximum height above the ground.", "answerGuide": "Resolve the initial velocity vertically and use v² = u² + 2as at the top of the trajectory, then add the cliff height.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q05b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics", "trigonometry" ], "primaryTopicId": "kinematics", "summary": "Find the projectile’s direction of motion after 5 seconds.", "answerGuide": "Find the horizontal and vertical velocity components at t = 5, then calculate the direction angle using their ratio and identify the correct quadrant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q05c", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find how long the stone travels faster than its projection speed.", "answerGuide": "Set the speed squared equal to 25² using the horizontal and vertical velocity components, solve for the crossing time, and take the interval after launch for which the speed is greater.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=16", "qpPages": [ 18, 21 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q06a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show the post-collision speed of Q in terms of e.", "answerGuide": "Use conservation of momentum and the coefficient-of-restitution relation, then solve the simultaneous equations for Q’s velocity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q06b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Show that P and Q move in the same direction after impact.", "answerGuide": "Use the post-collision velocity expressions and e > 0 to establish that both velocities have the same sign.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q06c", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Find e from Q’s collision with the wall.", "answerGuide": "Use the impulse–momentum change for Q at the wall with coefficient of restitution 1/2, equate to the given impulse, and solve for e.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=18", "qpPages": [ 22, 25 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q07a", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the coefficient of friction on the inclined plane.", "answerGuide": "Use the work done against friction over distance d to obtain the friction force, then use the normal reaction on A to determine μ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=20", "qpPages": [ 26, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q07b", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Use work–energy to find an expression for V².", "answerGuide": "Apply work–energy as B descends distance d, including the loss of gravitational potential energy, work against friction, and the total kinetic energy of both particles.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=20", "qpPages": [ 26, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME02-01-q07c", "paperId": "ial-2026-1-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2026, "session": "January", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed of A after it has moved a further distance d up the plane.", "answerGuide": "Apply work–energy over the further distance d, using V from part (b), the work against friction, and the change in gravitational potential energy.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M2_MS.pdf#page=20", "qpPages": [ 26, 32 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "2026-01-WME02-QP.pdf", "markScheme": "2026-01-WME02-MS.pdf" }, "sourceSha256": { "questionPaper": "B58CE478AEDFF2DA467E2BE00D6BDAFF02C8E10F66FC5865204A14707C387F53", "markScheme": "A7303DFE45869412752F11D8EC9C3D3CA6DF8D2306D6C4D428BFD0279CFFAC23" } }, { "id": "ial-2026-1-WME03-01-q01a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Prove that the particle executes simple harmonic motion.", "answerGuide": "Differentiate x(t) twice and show that the second derivative equals −k²x.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q01b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "State the amplitude of the motion.", "answerGuide": "Read the amplitude from the coefficient of the cosine displacement function.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q01c", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the maximum speed in terms of k.", "answerGuide": "Use maximum speed = angular frequency × amplitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q01d", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find k from the given period.", "answerGuide": "Use ω = 2π/T and equate the angular frequency in the displacement function to the resulting value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q01e", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the time taken to move directly between A and B.", "answerGuide": "Use the SHM displacement equation to find the two crossing times at x = ±0.2 m in the correct direction, then subtract them.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=8", "qpPages": [ 2, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q02a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the normal reaction when the particle moves in a horizontal circle.", "answerGuide": "Resolve vertically to relate the normal reaction and tension, then use the centripetal-force equation with the geometry of the conical string.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q02b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the upper bound on the angular speed.", "answerGuide": "Set the normal reaction to remain nonnegative and use the tension relation from circular motion to derive the limiting value of ω.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=9", "qpPages": [ 6, 9 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q03a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "kinematics", "differentiation" ], "primaryTopicId": "kinematics", "summary": "Find the acceleration magnitude at x = 3.", "answerGuide": "Use a = v dv/dx, differentiate the given speed function, and substitute x = 3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q03b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Find the exact time when x = 3.", "answerGuide": "Separate variables using dt = dx/v, integrate between x = 1 and x = 3, and use t = 2 at x = 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=10", "qpPages": [ 10, 13 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q04a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 5, "topicIds": [ "mechanics", "integration" ], "primaryTopicId": "mechanics", "summary": "Show the centre of mass of the composite solid lies the stated distance from O.", "answerGuide": "Use the masses and signed axial positions of the hemisphere, cylinder and removed cone; take their weighted mean about O.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=11", "qpPages": [ 14, 17 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q04b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find H when the solid is on the point of tipping.", "answerGuide": "Take moments about the contact edge at the toppling threshold and balance the applied force with the weight acting through the centre of mass.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=11", "qpPages": [ 14, 17 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q05", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the extension of the elastic string in equilibrium.", "answerGuide": "Find the string length from the geometry, use Hooke’s law for its tension, and take moments about B together with the rod’s weight.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q06a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "integration", "mechanics" ], "primaryTopicId": "integration", "summary": "Show the x-coordinate of the lamina’s centre of mass.", "answerGuide": "Use x̄ = (1/A)∫xy dx over the stated region and evaluate the integral between x = −2 and x = 4.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=12", "qpPages": [ 22, 25 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q06b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "integration", "mechanics" ], "primaryTopicId": "integration", "summary": "Show the y-coordinate of the lamina’s centre of mass.", "answerGuide": "Use ȳ = (1/A)∫(y²/2) dx for the area under the curve, apply the given area, and evaluate the limits.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=12", "qpPages": [ 22, 25 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q06c", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Find tan θ when the lamina hangs from B.", "answerGuide": "Use the line from the suspension point B to the centre of mass and resolve its horizontal and vertical offsets.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=12", "qpPages": [ 22, 25 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q07a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed at the specified position.", "answerGuide": "Apply conservation of mechanical energy between release and the point 3a below A, including the change in spring elastic energy and gravitational potential energy.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=13", "qpPages": [ 26, 29 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q07b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the first stopping distance below A.", "answerGuide": "At the turning point set the speed to zero and equate the change in gravitational potential energy to the change in spring energy; solve for the position.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=13", "qpPages": [ 26, 29 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q08a", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 6, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the tension as a function of θ.", "answerGuide": "Use conservation of energy to obtain speed at angle θ, then resolve forces radially and substitute into the centripetal equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=14", "qpPages": [ 30, 36 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q08b", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the speed at the lowest point.", "answerGuide": "Apply conservation of energy from the horizontal starting point to the lowest point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=14", "qpPages": [ 30, 36 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q08c", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the exact speed as the particle passes point C.", "answerGuide": "Use the geometry about the peg to determine the height change from the lowest point, then apply conservation of energy.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=14", "qpPages": [ 30, 36 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WME03-01-q08d", "paperId": "ial-2026-1-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2026, "session": "January", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the tension when P reaches O after contact with the peg.", "answerGuide": "Find the speed at O by energy conservation on the shorter circle about the peg, then use the radial equation to calculate tension.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_M3_MS.pdf#page=14", "qpPages": [ 30, 36 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q01a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the median and interquartile range from the stem-and-leaf plot.", "answerGuide": "Order the 31 marks, locate the median and the lower and upper quartiles, then subtract Q1 from Q3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q01b", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 5, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Draw a box plot and mark any outliers.", "answerGuide": "Calculate the 1.5 × IQR fences, identify marks outside them, then draw the whiskers, box, median and outlier points on the scale.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q01c", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain how the three corrected marks affect the box plot.", "answerGuide": "Recalculate the affected order positions and quartiles after the changes, then state which box-plot features change and which remain unchanged.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q02a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find Sff and Sdf for the paired frequency and depth data.", "answerGuide": "Use the supplied totals and definitions of Sff and Sdf, subtracting the appropriate products of sample means.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q02b", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Identify the response variable.", "answerGuide": "Choose the variable being predicted by the regression relationship and justify it from the context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q02c", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Calculate the product moment correlation coefficient.", "answerGuide": "Use r = Sdf / √(Sdd Sff) with the supplied summary statistics.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q02d", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the correlation coefficient in context.", "answerGuide": "State the direction and strength of the linear association between depth and frequency.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q02e", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the least-squares regression line of frequency on depth.", "answerGuide": "Use the gradient Sdf/Sdd and the sample means to find the intercept; state the line in the requested form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q02f", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(f)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the gradient and intercept of the regression line.", "answerGuide": "Explain that each extra metre of depth changes the modelled frequency by the gradient in kHz per metre, and that the intercept is the modelled frequency at zero depth (the surface).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q03a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probabilities that a person visited only France and Spain, and none of the three countries.", "answerGuide": "Read the two relevant regions from the Venn diagram and divide each count by the total of 120 people.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=8", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q03b", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the three-set Venn diagram.", "answerGuide": "Use the given France total and France–Germany overlap to determine the missing intersections, then use P(F ∪ G) and the overall total to fill the remaining regions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=8", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q03c", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(F′ ∩ G′) and P(F ∩ G | S).", "answerGuide": "For P(F′ ∩ G′), add the regions outside both France and Germany and divide by 120. For P(F ∩ G | S), divide the count in F ∩ G ∩ S by the number who visited Spain.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=8", "qpPages": [ 10, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q04a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find a, d, b and c in the probability distribution and its cumulative distribution function.", "answerGuide": "Read a from the cumulative probability at the first support value, set d to 1 at the maximum value, use total probability a + 3b = 1 to find b, then use the cumulative probability at x = 2 to find c.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q04bi", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)(i)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(12X + 4).", "answerGuide": "Find E(X) from the discrete probability distribution and apply linearity of expectation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q04bii", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)(ii)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Var(12X + 4).", "answerGuide": "Calculate E(X²), obtain Var(X)=E(X²)−E(X)², then use Var(12X+4)=144Var(X).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=9", "qpPages": [ 12, 15 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q05a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Complete the without-replacement tree diagram.", "answerGuide": "Label the first-draw probabilities from the 3:4 ratio and update the remaining red and black counts for each second draw.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q05b", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the total number of counters in the box.", "answerGuide": "Let the counts be 3x and 4x, express the two different-colour probabilities from the tree, equate their sum to 32/63, and solve for x.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q05c", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability the first counter was black given that the second is red.", "answerGuide": "Calculate P(first black and second red) and divide by P(second red), adding both paths that end with a red second draw in the denominator.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=10", "qpPages": [ 16, 19 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q06a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find the probability that X + 2 < 3X − 4.", "answerGuide": "Rearrange the inequality to identify the values of X that satisfy it, then count them among the five equally likely outcomes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=11", "qpPages": [ 20, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q06b", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(X²).", "answerGuide": "Average the squares of the five equally likely values 1, 2, 3, 4 and 5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=11", "qpPages": [ 20, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q06c", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find Var(3 − aX).", "answerGuide": "Use E(X)=3 and the given expectation to determine a, find Var(X) from the discrete uniform distribution, and scale variance by a².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=11", "qpPages": [ 20, 21 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q07a", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show the modelled proportion finishing after 210 minutes.", "answerGuide": "Standardise 210 using the stated normal mean and standard deviation, then find the upper-tail probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=12", "qpPages": [ 22, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q07b", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Estimate the longest time that places Russell in the fastest 15%.", "answerGuide": "Find the 15th percentile of the completion-time distribution using the appropriate normal quantile and convert back to minutes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=12", "qpPages": [ 22, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q07ci", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the conditional probability below μ−5 given Y<μ.", "answerGuide": "Use the symmetry of the normal distribution and the supplied probability at μ+5 to obtain the conditional probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=12", "qpPages": [ 22, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST01-01-q07cii", "paperId": "ial-2026-1-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2026, "session": "January", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show that the standard deviation rounds to 6 minutes.", "answerGuide": "Convert the given normal probability to its standardised z-value and solve 5/σ = z.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S1_MS.pdf#page=12", "qpPages": [ 22, 24 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q01a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Suggest a suitable sampling frame and identify the sampling units.", "answerGuide": "Name a complete list or database of all customers with the internet service as the sampling frame, and identify each customer or subscriber as one sampling unit.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q01b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give one advantage of a sample over a census.", "answerGuide": "State a valid advantage such as lower cost, faster collection or easier processing.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q01c", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Write down a distribution for the number of satisfied customers in the sample.", "answerGuide": "Use a binomial model with n = 50 and p = 0.35.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q01d", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability that more than 40% of the sample are satisfied.", "answerGuide": "More than 40% means at least 21 customers; use the binomial tail or its complement.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=5", "qpPages": [ 2, 3 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Show that k = 1/6.", "answerGuide": "Substitute the upper endpoint into the cumulative distribution and set F(10)=1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(X = 5).", "answerGuide": "Use continuity of the cdf on the stated support to note that a single point has zero probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02c", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(5 < X < 7).", "answerGuide": "Evaluate F(7)−F(5) using the specified cdf branch.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02d", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Specify the probability density function.", "answerGuide": "Differentiate the cdf over each interval and state zero outside the support.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02e", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Sketch the probability density function.", "answerGuide": "Draw the constant density over its nonzero interval, label its height and endpoints, and show zero elsewhere.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02f", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(f)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(X).", "answerGuide": "Use the midpoint of the uniform interval or integrate x times the density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q02g", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(g)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find E(2X²−1).", "answerGuide": "Calculate E(X²) from the density and use linearity: 2E(X²)−1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=6", "qpPages": [ 4, 7 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q03a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability of at least four detections in one 30-minute period and in each of three independent 30-minute periods.", "answerGuide": "Double the 15-minute Poisson mean to 5 and find P(X ≥ 4) for one 30-minute period; cube that probability for three independent periods.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q03b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses for the post-cleaning rate test.", "answerGuide": "State H0 that the rate remains 2.5 per 15 minutes (or 10 per hour) and H1 that it is greater.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q03c", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find the critical region and its probability.", "answerGuide": "Use a one-hour Poisson model with mean 10, find the smallest upper-tail threshold whose probability is below 5%, and state the associated probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q03d", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State the conclusion from 13 detections.", "answerGuide": "Compare 13 with the critical region and give the contextual conclusion about an increased detection rate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=8", "qpPages": [ 8, 11 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q04a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probabilities that exactly one patient and no more than two patients in a sample have the condition.", "answerGuide": "Use X ~ B(30, 0.05): evaluate P(X = 1) and P(X ≤ 2).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=9", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q04b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test the optician’s claim that the proportion has changed.", "answerGuide": "Use a Poisson approximation with mean 150×0.05 = 7.5 for the two-sided count test, find the relevant tail probability or critical region, compare with 5%, and state the conclusion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=9", "qpPages": [ 12, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q05a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "List the possible combinations of the three products.", "answerGuide": "List the distinct product combinations arising from two supplier-A products and one supplier-B product, retaining repeated outcomes where their source differs.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q05b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "List all possible values of the total price T.", "answerGuide": "Add the three product prices for every possible combination and state the distinct totals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q05c", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of T.", "answerGuide": "Multiply the independent selection probabilities for each combination, combine equal totals, and present the probability table.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q05d", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the sampling distribution of the price range R.", "answerGuide": "For each possible product combination calculate max price minus min price, combine equal ranges and total their probabilities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q06a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Sketch the density function for watermelon weight.", "answerGuide": "Draw the stated downward-opening quadratic density on 2≤w≤5 and zero elsewhere, with the support endpoints marked.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q06b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "State the mode of the watermelon weight distribution.", "answerGuide": "Locate the vertex of the quadratic density.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q06c", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Find the mean watermelon weight.", "answerGuide": "Integrate w f(w) over the stated support and evaluate exactly.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q06d", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the standard deviation of watermelon weights.", "answerGuide": "Use the supplied E(W²) and your mean to calculate √(E(W²)−E(W)²), then round to 3 significant figures.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q06e", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 5, "topicIds": [ "distributions", "integration" ], "primaryTopicId": "distributions", "summary": "Show that the median lies between 3.7 kg and 3.8 kg.", "answerGuide": "Integrate the density to obtain F(w), evaluate it at 3.7 and 3.8, and show the values straddle 0.5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q06f", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(f)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use a normal approximation to find the chance of fewer than 20 spoiled watermelons.", "answerGuide": "Scale the Poisson mean from 150 m² to 900 m², approximate by a normal distribution with matching mean and variance, apply a continuity correction, then standardise.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=11", "qpPages": [ 18, 21 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q07a", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find P(X > 16) for the sample of 30 journeys.", "answerGuide": "Use X~B(30,0.45) and take the upper tail above 16.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=13", "qpPages": [ 22, 24 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST02-01-q07b", "paperId": "ial-2026-1-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2026, "session": "January", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 8, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the smallest k satisfying the stated normal-approximation probability.", "answerGuide": "Approximate B(180,0.45) by a normal distribution, apply the appropriate continuity correction for X>k, standardise, use the given tail probability to find the threshold, and round to the smallest integer satisfying it.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S2_MS.pdf#page=13", "qpPages": [ 22, 24 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q01", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": null, "marks": 8, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Test whether aptitude-test result is independent of test centre.", "answerGuide": "Set up the observed two-way table and expected frequencies, calculate the chi-squared statistic, compare with the 5% critical value for one degree of freedom, and state a contextual conclusion.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q02a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State hypotheses for the Spearman rank test.", "answerGuide": "State no positive rank correlation under H0 and positive rank correlation under H1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q02b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Show that the largest possible sum of squared rank differences is 2 after rejection.", "answerGuide": "Use the Spearman statistic and the stated critical threshold to bound Σd², then identify its greatest possible integer value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q02c", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Give a possible ranking for judge Y.", "answerGuide": "Construct a ranking with the required sum of squared differences, for example swapping the two adjacent final ranks.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=7", "qpPages": [ 6, 7 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q03a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Find a 90% confidence interval for the mean packet weight.", "answerGuide": "Recover the sample mean and standard error from the 95% interval, then use the 90% normal critical value to form the new interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q03b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the chance that at least one of three 90% intervals contains μ.", "answerGuide": "Let the number of intervals containing μ be binomial with n=3 and p=0.9, then use the complement of zero successes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=8", "qpPages": [ 8, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q04a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Estimate the binomial parameter p from the sample.", "answerGuide": "Divide the mean number of rotten apples per tray by 10, equivalently divide the total number of rotten apples by the 1,500 apples inspected.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q04b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Show the expected frequency r for exactly five rotten apples.", "answerGuide": "Use the binomial probability for five successes with n=10 and the rounded estimate of p, then multiply by 150.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q04c", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the expected frequency s for six or more rotten apples.", "answerGuide": "Subtract the expected frequencies for 0 through 5 from 150, or sum the binomial tail probabilities and multiply by 150.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q04d", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Explain why the high-count cells are combined.", "answerGuide": "State that the combined expected frequency must be at least 5 for the chi-squared approximation to be valid.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q04e", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(e)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test whether a binomial model is suitable.", "answerGuide": "Combine the required cells, use the supplied partial chi-squared value and the remaining grouped contribution, calculate degrees of freedom, and compare the statistic with the 10% critical value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q05a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-analysis", "distributions" ], "primaryTopicId": "data-analysis", "summary": "Find the unbiased sample mean and variance for professor B’s students.", "answerGuide": "Divide Σx by n for the mean and use [Σx²−(Σx)²/n]/(n−1) for the unbiased variance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q05b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 7, "topicIds": [ "statistical-testing", "distributions" ], "primaryTopicId": "statistical-testing", "summary": "Test the claim that professor A’s mean is more than 10 marks above professor B’s.", "answerGuide": "State one-sided hypotheses for μA−μB, calculate the test statistic using the two sample means and estimated variances, compare with the 5% critical value, and conclude in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q05c", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "State two assumptions for the comparison test.", "answerGuide": "State equal population variances and that the sample means are approximately normal, for example by sufficiently large independent random samples.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q06a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Show that the sample size is 100.", "answerGuide": "Standardise each given sample-mean probability using standard error 6/√n, form two equations in μ and n, and solve simultaneously.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q06b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find μ.", "answerGuide": "Substitute n=100 into either standardised equation and solve for the population mean.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q07a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find an unbiased estimator for a.", "answerGuide": "Use the expectation of the sample mean for a uniform interval [a−1,a+5] and subtract its bias.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=12", "qpPages": [ 20, 21 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q07b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Find the standard error of the unbiased estimator.", "answerGuide": "Use that the estimator differs from the sample mean by a constant, calculate the uniform variance, and divide its standard deviation by √9.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=12", "qpPages": [ 20, 21 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q08a", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Find the mean and variance of T.", "answerGuide": "Use independence and add the means and variances of the two medium weights minus one large weight.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=13", "qpPages": [ 22, 24 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q08b", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability a large cauliflower weighs more than three times a small one.", "answerGuide": "Form L−3S, calculate its mean and variance under independence, then standardise at zero.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=13", "qpPages": [ 22, 24 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2026-1-WST03-01-q08c", "paperId": "ial-2026-1-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2026, "session": "January", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 5, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Find the probability the first of two small cauliflowers is at least 100 g above their sample mean.", "answerGuide": "Rewrite the event using the difference of the two independent normal weights, find its mean and variance, then standardise.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/Jan%202026/IAL_MATHS_2026_Jan_S3_MS.pdf#page=13", "qpPages": [ 22, 24 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply the first‑fit bin packing algorithm to determine how the eleven numbers listed above can be packed into bins of size 14.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 2, 2 ], "msPages": [ 5, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=5", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply a quick sort to obtain the sorted list. You should show the result of each pass and identify your pivots clearly.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 2, 2 ], "msPages": [ 5, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q01b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=5", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply the first‑fit decreasing bin packing algorithm to the sorted list to pack the numbers into bins of size 14.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 2, 2 ], "msPages": [ 5, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q01c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=5", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 1, "questionPart": "1(d)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Explain why the number of bins used in part (c) is optimal.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 2, 2 ], "msPages": [ 5, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q01d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=5", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 1, "questionPart": "1(e)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply the binary search algorithm to try to locate 3.0 in the list of numbers. Clearly indicate how you choose your pivots and which part of the list is rejected at each stage.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 2, 2 ], "msPages": [ 5, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q01e", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=5", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete Diagram 1 in the answer book to show the early event times and the late event times. (ii) State the minimum completion time of the project.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=3", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Work out a lower bound for the number of workers needed to complete the project in the minimum time. You must show your working.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=3", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 2, "questionPart": "2(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Schedule the activities using the minimum number of workers so that the project is completed in the minimum time.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 3, 3 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q02c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=3", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 3, "questionPart": "3(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply Dijkstra’s algorithm to find the shortest time needed to travel from A to J. State the quickest route.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q03a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Hence determine the weight of the minimum spanning tree for the network given in Figure 2. Give a reason for your answer. You do not need to find the minimum spanning tree.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 4, 4 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q03b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply Prim’s algorithm to find a minimum spanning tree for the table of shortest distances. You must state the order in which you select the arcs of your tree.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 6, 6 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 4, "questionPart": "4(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the weight of the minimum spanning tree.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 6, 6 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q04b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 4, "questionPart": "4(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine the length of this route. You must give a reason for your answer.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 7, 7 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=7", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 4, "questionPart": "4(d)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine the roads that need to be traversed twice.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 7, 7 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q04d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=7", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply the nearest neighbour algorithm on the table of shortest distances to find an upper bound for the length of Naoko’s route.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 7, 7 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q04e", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=7", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 4, "questionPart": "4(f)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Apply the values in the table of shortest distances to find a lower bound for the length of Naoko’s route.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 7, 7 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q04f", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=7", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 5, "questionPart": "5(a)", "marks": 7, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Formulate and solve a paper-ordering linear programme with ratio and budget constraints.", "answerGuide": "Translate the supply, ratio and spending limits into constraints, then compare the feasible corner points to find the best order.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine (i) the total number of reams of paper to be ordered, (ii) the number of reams of graph paper to be ordered.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 8, 8 ], "msPages": [ 14, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw the activity network for the project described in the precedence table, using activity on arc and the minimum number of dummies.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 10, 10 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q06a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 6, "questionPart": "6(b)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Give the critical activities of the project.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=11", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "EDFAD0C3245B9E1EAA2DDB670271EE7865D9F8F35FBE5B75DA8480F35ED0E8C2", "markScheme": "82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", 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You must make specific reference to time and activities.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 11, 11 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q06c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=11", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": 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"82585086C5CE742F1EEE9AA2D912A58A1C096AB0A5F549116FF8FB0FD9E617D3", "examinerReport": "5C4C16AB5F1BF89F5317957CE41578F113C29C063148D12D1622801A54C595AD" } }, { "paperId": "ial-2024-6-WDM11-01", "syllabus": "WDM11", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine the inequalities that define R.", "answerGuide": "Show each required step in the stated algorithm or network method, keep the route, schedule or bound visible, and state the requested value.", "qpPages": [ 12, 12 ], "msPages": [ 16, 17 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": 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programme.", "answerGuide": "Sketch the half-plane intersection, list its vertices, and compare the objective value at those vertices as the parameter varies.", "qpPages": [ 12, 12 ], "msPages": [ 16, 17 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WDM11-01-q07b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_D1_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WDM11-01-QP.pdf", "markScheme": "wdm11-01-rms-20240815.pdf", "examinerReport": "wdm11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"305C894969594BBBA46D5C79E7536EECF31DD8EEAA19D256D371AA1CE415802B", "markScheme": "0631B45E30E5B792A93127A40A42BDF8B476EB04E4E433F7017E9192DCE3820F", "examinerReport": "3EE0D66FE17CBA0A7DE728FB656C62372DD011C368688BB78593072292052807" } }, { "paperId": "ial-2024-6-WMA11-01", "syllabus": "WMA11", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Hence find the coordinates of B.", "answerGuide": "Form the relevant coordinate or vector equations, show the key substitutions and algebra, and state the requested coordinates or equation.", "qpPages": [ 18, 18 ], "msPages": [ 17, 17 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01-q07c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "qpUrl": 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"Hence find the range of possible values of A for which C1 meets C2 at 4 distinct points.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 20, 20 ], "msPages": [ 18, 19 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01-q08c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1_MS.pdf#page=18", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01-QP.pdf", "markScheme": "wma11-01-rms-20240815.pdf", "examinerReport": 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2024, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1_MS.pdf#page=20", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01-QP.pdf", "markScheme": "wma11-01-rms-20240815.pdf", "examinerReport": "wma11-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "305C894969594BBBA46D5C79E7536EECF31DD8EEAA19D256D371AA1CE415802B", "markScheme": "0631B45E30E5B792A93127A40A42BDF8B476EB04E4E433F7017E9192DCE3820F", "examinerReport": "3EE0D66FE17CBA0A7DE728FB656C62372DD011C368688BB78593072292052807" } }, { "paperId": "ial-2024-6-WMA11-01", "syllabus": "WMA11", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": 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"Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Solve 5 (x + 3) > 4 (2x –5).", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Write x 2 – 6x + 1 in the form (x + a) 2 + b where a and b are constants. (ii) Hence solve x 2 – 6x + 1 0.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Hence find the values of x that satisfy both 5 (x + 3) > 4 (2x – 5) and x 2 – 6x + 1 0.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q02c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine , giving each term in simplest form. d x.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q03a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Determine the solution of the equation d 2y = 0 d x 2 writing the answer in the form 2 k where k is a constant.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 6, 6 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q03b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Demonstrate that l meets C when (k – 3) x 2 + 6x – 2 = 0.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Hence find the value of k for which l is a tangent to C.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q04b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "quadratics" ], "primaryTopicId": "quadratics", "summary": "Fully factorise 9x 3 – 10x 2 + x.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Hence solve 9 × 27 y – 10 × 9 y + 3 y = 0.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Demonstrate that angle AFB = 0.433 radians to 3 decimal places.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 12, 12 ], "msPages": [ 15, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q06a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine the perimeter of the entrance to the tunnel, ABCDEFA, in metres, to one decimal place.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 12, 12 ], "msPages": [ 15, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 6, "questionPart": "6(c)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine the cross-sectional area of the entrance to the tunnel, ABCDEFA, in m 2, to one decimal place.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 12, 12 ], "msPages": [ 15, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q06c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Determine the distance AB writing your answer as a fully simplified surd.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 16, 16 ], "msPages": [ 17, 18 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=16", "msUrl": 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"Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=19", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "6DD4B91B7EAC009F04A54259AE1CA4A99FA5EF2A95C3FC45C1010B6ACE5FA447", "markScheme": "C1BBF1CC0EA3788E26CCCD1AD611D3F0F736E0923D135175F982C351096F338B", "examinerReport": "D05EFA74EACBEF7865BD1FFB4C4B2E5F8C5222180FBDEFFADC05EAB51736722F" } }, { "paperId": "ial-2024-6-WMA11-01R", "syllabus": "WMA11R", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Give the number of solutions of the equation π (i) tan x = π – x in the interval 0 x 2π 6 π (ii) tan x = π – x in the interval 0 x 100π 6 π (iii) tan x = π + x in the interval 0 x 2π 6.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 24, 24 ], "msPages": [ 21, 21 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q09a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=21", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": 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the line segment PQ.", "answerGuide": "Form the relevant coordinate or vector equations, show the key substitutions and algebra, and state the requested coordinates or equation.", "qpPages": [ 24, 24 ], "msPages": [ 21, 21 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA11-01R-q09c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_MS.pdf#page=21", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA11-01R-QP.pdf", "markScheme": "wma11-01r-rms-20240815.pdf", "examinerReport": "wma11-01r-pef-20240815.pdf" }, "sourceSha256": { 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where tangents to a polynomial curve are parallel.", "answerGuide": "Set the derivative equal to the specified tangent gradient, solve the resulting equation exactly, and retain only values consistent with the stated interval.", "qpPages": [ 26, 26 ], "msPages": [ 22, 24 ], "studyPriority": 1, "difficultyEvidence": "Pearson calls the parallel-tangent part challenging.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01r-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "id": "ial-2024-6-WMA11-01R-q10b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA11/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P1R_QP.pdf#page=26", "msUrl": 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algebra, and state the exact form requested.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Hence find the coefficient of x 3 in the expansion of 1 9 10 x 3 1 x 6 giving the answer in simplest form.", "answerGuide": "Apply the relevant sequence, expansion or index rule, show the intermediate algebra, and state the exact form requested.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q01b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine the value of the first term and the value of the common difference.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Hence find the smallest value of n for which Sn > 8000.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 3, "questionPart": "3(i)", "marks": 5, "topicIds": [ "trigonometry", "exponentials" ], "primaryTopicId": "trigonometry", "summary": "Solve 2 log2 (2 – x) = 4 + log2 (x + 10).", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q03i", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 3, "questionPart": "3(ii)", "marks": 1, "topicIds": [ "trigonometry", "exponentials" ], "primaryTopicId": "trigonometry", "summary": "Determine the value of log a 6 a where a is a positive constant greater than 1.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q03ii", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": 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the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 8, 8 ], "msPages": [ 10, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 4, "questionPart": "4(c)", "marks": 5, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Hence (i) fully factorise f (x), (ii) find the number of real solutions of the equation f (x) = 0 giving a reason for your answer.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 8, 8 ], "msPages": [ 10, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 5, "questionPart": "5(a)", "marks": 4, "topicIds": [ "proof" ], "primaryTopicId": "proof", "summary": "Prove that y > x.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": 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"answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 12, 12 ], "msPages": [ 12, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=12", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Obtain an estimate for 2 x 2 x d x , giving your answer to 2 decimal places. 2.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Estimate 3 . 5 (i) 2 x 2 x d x 2 3 . 5 (ii) 2 x 1 4 x d x 2.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 14, 14 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q06c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "coordinate-geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Determine the coordinates of the centre of C1 (ii) Find the exact value of the radius of C1.", "answerGuide": "Form the relevant coordinate or vector equations, show the key substitutions and algebra, and state the requested coordinates or equation.", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": 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The circle C2 has equation (x 5) ( 8) – 2 + y + 2 = 52 (b) Prove that the circles C1 and C2 neither touch nor intersect.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 18, 18 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q07b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 8, "questionPart": "8(i)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Solve, for 0 < x π, the equation 5 sin x tan x + 13 = cos x giving your answer in radians to 3 significant figures.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 22, 22 ], "msPages": [ 16, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q08i", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 8, "questionPart": "8(ii)(a)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine all possible values for k, giving each answer to 2 decimal places.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 22, 22 ], "msPages": [ 16, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q08iia", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 8, "questionPart": "8(ii)(b)", "marks": 1, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine the maximum temperature inside the greenhouse.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 22, 22 ], "msPages": [ 16, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q08iib", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 8, "questionPart": "8(ii)(c)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine the time of day at which this maximum temperature occurs. 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Give your answer to 3 significant figures.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 30, 30 ], "msPages": [ 19, 20 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01-q10a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_QP.pdf#page=30", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2_MS.pdf#page=19", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01-QP.pdf", "markScheme": "wma12-01-rms-20240815.pdf", "examinerReport": "wma12-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9ACF5CCEDC69C9124F7CFDEB5687ADF4C71FD0E3A3003233BC499E8E721312BE", "markScheme": "A0E9A64301DB2FED020C94963D46B12FBF39762B3FB99367968213D315EDCB30", "examinerReport": "A86F8F9545BCFBA20762819A7787DFD9BCA23C2213D544992C5947681FB93E72" } }, { "paperId": "ial-2024-6-WMA12-01", "syllabus": "WMA12", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Determine the equation of this model in the form N = ab t where N is the number of voles, t years after monitoring began and a and b are constants. 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u2.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 2, 2 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01R-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2R_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2R_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01R-QP.pdf", "markScheme": "wma12-01r-rms-20240815.pdf", "examinerReport": "wma12-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01R-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01R", "variant": "Regional", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2R_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P2R_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA12-01R-QP.pdf", "markScheme": "wma12-01r-rms-20240815.pdf", "examinerReport": "wma12-01r-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "B0241F8F3F6CC7F04792CA2148B27CF958B216CC8C64170DF7696115F0F86557", "markScheme": "59913150CABDD3FFAB17169DA2E56DE57AEEB3804DA086DA36FAB731C65ACBA1", "examinerReport": "198D1DB1095380E4C5725108BACA74A3EE948575CBAEAA0A3A6B6EC2C952CF92" } }, { "paperId": "ial-2024-6-WMA12-01R", "syllabus": "WMA12R", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Apply the given answer of 19.3 to find an approximate value for 5 2 f ( x ) 3 d x 4.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA12-01R-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA12/01R", "variant": "Regional", "qpUrl": 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"Write g (x) in the form C Ax B x 2 where A, B and C are integers to be found.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "integration", "proof" ], "primaryTopicId": "integration", "summary": "Hence use algebraic integration to show that 8 g( x )d x α β ln3 4 where α and β are integers to be found.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 4, 4 ], "msPages": [ 8, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 3, "questionPart": "3(i)", "marks": 3, "topicIds": [ "coordinate-geometry", "functions" ], "primaryTopicId": "coordinate-geometry", "summary": "Sketch the graph of log10 y against log10 x Show on your sketch the coordinates of the points of intersection of the graph with the axes.", "answerGuide": "Form the relevant coordinate or vector equations, show the key substitutions and algebra, and state the requested coordinates or equation.", "qpPages": [ 8, 8 ], "msPages": [ 10, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q03i", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 3, "questionPart": "3(ii)", "marks": 3, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Demonstrate that N = ab t where a and b are constants to be found.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 8, 8 ], "msPages": [ 10, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q03ii", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write f (x) in the form a sin 2 x + b cos 2x + c where a, b and c are integers to be found.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=12", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Apply the answer to part (a) to write f (x) in the form R sin (2x + α) + c π where R > 0 and 0 < α < 2 Give the exact value of R and give the value of α in radians to 3 significant figures.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q04b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=12", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Hence, or otherwise, (i) state the maximum value of f (x) (ii) find the second smallest positive value of x at which a maximum value of f (x) occurs. Give your answer to 3 significant figures.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 10, 10 ], "msPages": [ 12, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=12", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Determine f –1.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "differentiation", "trigonometry" ], "primaryTopicId": "differentiation", "summary": "Apply differentiation to prove that g is an increasing function.", "answerGuide": "Use the relevant identities and interval restrictions, solve every valid branch, and present the answers in the requested form.", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA13-01-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "functions", "exponentials", "differentiation" ], "primaryTopicId": "functions", "summary": "Analyse a composite exponential function, its range and a related rate of change.", "answerGuide": "Find the function's range from its monotonicity and endpoint behaviour, then differentiate the composite expression and substitute the required value.", "qpPages": [ 14, 14 ], "msPages": [ 14, 15 ], "studyPriority": 1, "difficultyEvidence": "Pearson directly identifies part (d) as one of the paper's hardest parts.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "id": "ial-2024-6-WMA13-01-q05d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P3_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA13-01-QP.pdf", "markScheme": "wma13-01-rms-20240815.pdf", "examinerReport": "wma13-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "9EA66F6EFCA90B4B9EC57BDBED4FD3A5C2F1AB82DB479C13F4137213E1C1332D", "markScheme": "7B1A3C8A8924CA033B83E891C06B531AE35CC024FBCB4D2179D3A2B7D6CCF00B", "examinerReport": "911BA39339BB229DF71DD2E09D595048110634B8B0979850224BC45F7F826E86" } }, { "paperId": "ial-2024-6-WMA13-01", "syllabus": "WMA13", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Apply calculus to show that an equation of l is 5x + 2y – 50 = 0.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 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other possible position vector for A.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 14, 14 ], "msPages": [ 15, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA14-01-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "wma14-01-rms-20240815.pdf", "examinerReport": "wma14-01-pef-20240815.pdf" }, "sourceSha256": { 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Give your answer in the form x = f(t).", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 18, 18 ], "msPages": [ 17, 18 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA14-01-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_MS.pdf#page=17", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "wma14-01-rms-20240815.pdf", "examinerReport": "wma14-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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significant figures.", "answerGuide": "Show the algebraic method and key intermediate values, then state the requested exact value or approximation.", "qpPages": [ 18, 18 ], "msPages": [ 17, 18 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA14-01-q07c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_MS.pdf#page=17", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "wma14-01-rms-20240815.pdf", "examinerReport": "wma14-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"primaryTopicId": "proof", "summary": "Use a binomial expansion within a contradiction proof.", "answerGuide": "Expand to the required power, isolate the relevant coefficient or divisibility condition, and show that the assumed integer case cannot hold.", "qpPages": [ 22, 22 ], "msPages": [ 19, 20 ], "studyPriority": 1, "difficultyEvidence": "The report says candidates often earned only setup credit on the sophisticated second part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "id": "ial-2024-6-WMA14-01-q08b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_MS.pdf#page=19", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "wma14-01-rms-20240815.pdf", "examinerReport": "wma14-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "BC1EA5ADC958ACE1E68C543E1577EBA6E3A5699DF46FA7BEB330E4A0DB8E62CA", "markScheme": "4140E8ACF971D2C9049D4D2B274BCC4990EB4C4F520EB4D556E3BA95DE74FCDE", "examinerReport": "3BB37CE7B61538535BEEE55213A01833DC54B6EAF97A6E7FA3170674FBE49BAA" } }, { "paperId": "ial-2024-6-WMA14-01", "syllabus": "WMA14", "questionNumber": 9, "questionPart": "9(a)", "marks": 6, "topicIds": [ "trigonometry", "proof" ], "primaryTopicId": "trigonometry", "summary": "Demonstrate that the volume V of the solid formed is given by ∫ b 1 k d u u 2 a where k, a and b are constants to be found.", "answerGuide": "Use the stated assumptions, justify each algebraic or logical step, and finish with the conclusion required by the question.", "qpPages": [ 26, 26 ], "msPages": [ 21, 24 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WMA14-01-q09a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_QP.pdf#page=26", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_MS.pdf#page=21", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "wma14-01-rms-20240815.pdf", "examinerReport": "wma14-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "BC1EA5ADC958ACE1E68C543E1577EBA6E3A5699DF46FA7BEB330E4A0DB8E62CA", "markScheme": 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_QP.pdf#page=26", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_P4_MS.pdf#page=21", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WMA14-01-QP.pdf", "markScheme": "wma14-01-rms-20240815.pdf", "examinerReport": "wma14-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "BC1EA5ADC958ACE1E68C543E1577EBA6E3A5699DF46FA7BEB330E4A0DB8E62CA", "markScheme": "4140E8ACF971D2C9049D4D2B274BCC4990EB4C4F520EB4D556E3BA95DE74FCDE", "examinerReport": "3BB37CE7B61538535BEEE55213A01833DC54B6EAF97A6E7FA3170674FBE49BAA" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine S in terms of U.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", 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appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q02", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": 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"questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the smallest possible value of M.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", 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"5(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the distance fallen by P in the first 5 seconds.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 12, 12 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { 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"marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the total time from the instant when P is dropped from the helicopter to the instant when P lands on the ground.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 12, 12 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q05c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=11", "hardestPart": null, 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine d in terms of u and g.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 16, 16 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine, in terms of i and j, the velocity of B in km h –1.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine an expression for s in terms of t, i and j.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q07b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Demonstrate that at time t hours after noon, → AB = [(10 – 24t)i + (12t – 15)j] km.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q07c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 7, "questionPart": "7(d)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that the boats will never collide.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q07d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 7, "questionPart": "7(e)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the distance between the boats when the bearing of B from A is 225°.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 20, 20 ], "msPages": [ 15, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q07e", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that the frictional force acting on A as it moves up the plane is 39.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 24, 24 ], "msPages": [ 16, 18 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q08a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Demonstrate that the acceleration of A immediately after its release is g 3.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 24, 24 ], "msPages": [ 16, 18 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q08c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME01-01", "syllabus": "WME01", "questionNumber": 8, "questionPart": "8(d)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine, in terms of h, the total distance travelled by A from when it was released from rest to when it first comes to rest again.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 24, 24 ], "msPages": [ 16, 18 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME01-01-q08d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M1_MS.pdf#page=16", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME01-01-QP.pdf", "markScheme": "wme01-01-rms-20240815.pdf", "examinerReport": "wme01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "A3995DB85D8297E15C4BEE7C508714C3B9894A7AB9D878D81385E1CDB9EF0DEC", "markScheme": "48FFA3D98FA689A6D41E5668A68686134014223F1B07A482CA714DC49DE52AED", "examinerReport": "E916E0143A68F81381AC017803E5A3E90554F4C7E433FBF8E92572390B6600BB" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Determine the total kinetic energy of the two particles before the collision.", "answerGuide": "Set up the relevant energy or momentum equation, define the quantities before and after the event, and state the requested result with units.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "energy-momentum" ], "primaryTopicId": "energy-momentum", "summary": "Determine, in terms of i and j, the impulse received by A in the collision.", "answerGuide": "Set up the relevant energy or momentum equation, define the quantities before and after the event, and state the requested result with units.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q01b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Determine the velocity of B immediately after the collision.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q01c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that k = 4, explaining your method carefully.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the acceleration of P when t = 1.5.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 2, "questionPart": "2(c)", "marks": 7, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the distance of P from O when t = 8.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q02c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that r R 1 k.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q03a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Hence find the range of possible values of k.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q03b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the angle that XY makes with the vertical.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q03c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine, in terms of M, the mass of the lightest particle that could be attached to the template so that it would hang in equilibrium from P with XY horizontal.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q03d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that the magnitude of the frictional force acting on the particle, as it moves 4 mg from A to B, is 35.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine AB in terms of a.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q04b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine, in terms of a and g, the speed of P when it returns to A.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 5, "questionPart": "5(a)", "marks": 6, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Determine, in terms of u and e, the speed of P after the collision.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=12", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Analyse a particle collision with a wall and determine an admissible restitution range.", "answerGuide": "Apply conservation of momentum and the coefficient-of-restitution relation to each impact, then impose the stated direction or speed condition to bound e.", "qpPages": [ 14, 14 ], "msPages": [ 12, 12 ], "studyPriority": 1, "difficultyEvidence": "The report says very few candidates earned full marks on part (b).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 5 ] }, "id": "ial-2024-6-WME02-01-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": 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appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 18, 18 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q06a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that u = 14.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 22, 22 ], "msPages": [ 15, 17 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M2_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME02-01-QP.pdf", "markScheme": "wme02-01-rms-20240815.pdf", "examinerReport": "wme02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "DACFDAC70A1CF1A7A71ED34F92D770F0BBBD703551A7E20943A99AD5EF3DE547", "markScheme": "BE7FA4947DDE2FB1A89D51F4B183FEF354DF0553EC8FE489F366D25F141C09E3", "examinerReport": "C098CC44C925A2ADF5F2B12FD7506B0C8B8E5641D5C22ABC440C0A2616E7A652" } }, { "paperId": "ial-2024-6-WME02-01", "syllabus": "WME02", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Determine the total time, while the ball is in the air, for which the speed of the ball is greater than 7 17 m s− 1.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 22, 22 ], "msPages": [ 15, 17 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME02-01-q07b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", 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apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME03-01-QP.pdf", "markScheme": "wme03-01-rms-20240815.pdf", "examinerReport": "wme03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "7818BF2DB2DD3C0A73C49E5D307FCC73314EAB4001B6F8497949DDC90E09641B", "markScheme": "211E9A97330919C2F3A5E2F88AFFA4C895A0EE84F205AA2E8D20D27242BAE89D", "examinerReport": "3B9BE060770B99296470984E2F4FEA760D1156BBEAE0C7C4685490ACBC2A73A2" } }, { "paperId": "ial-2024-6-WME03-01", "syllabus": "WME03", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the value of k.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 2, 2 ], "msPages": [ 6, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q01b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": 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of the hemisphere.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME03-01-QP.pdf", "markScheme": "wme03-01-rms-20240815.pdf", "examinerReport": "wme03-01-pef-20240815.pdf" }, "sourceSha256": { 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME03-01-QP.pdf", "markScheme": "wme03-01-rms-20240815.pdf", "examinerReport": "wme03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "7818BF2DB2DD3C0A73C49E5D307FCC73314EAB4001B6F8497949DDC90E09641B", "markScheme": "211E9A97330919C2F3A5E2F88AFFA4C895A0EE84F205AA2E8D20D27242BAE89D", "examinerReport": "3B9BE060770B99296470984E2F4FEA760D1156BBEAE0C7C4685490ACBC2A73A2" } }, { "paperId": "ial-2024-6-WME03-01", "syllabus": "WME03", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Demonstrate that v ( x )1 4.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q03a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME03-01-QP.pdf", "markScheme": "wme03-01-rms-20240815.pdf", "examinerReport": "wme03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "7818BF2DB2DD3C0A73C49E5D307FCC73314EAB4001B6F8497949DDC90E09641B", 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13:35.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WME03-01-QP.pdf", "markScheme": "wme03-01-rms-20240815.pdf", "examinerReport": "wme03-01-pef-20240815.pdf" }, "sourceSha256": { 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"primaryTopicId": "kinematics", "summary": "Demonstrate, using algebraic integration, that the distance of the centre of mass of C from its vertex is 3r. 4 [You may assume that the volume of C is πr 3 ] 3.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 12, 12 ], "msPages": [ 12, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=12", "hardestPart": null, 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"ial-2024-6-WME03-01", "syllabus": "WME03", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the exact value of tan α.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 12, 12 ], "msPages": [ 12, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q05c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=12", "hardestPart": null, 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"questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Determine the exact value of tan α.", "answerGuide": "Draw or state the relevant force equations, apply the appropriate law of mechanics, show substitutions, and give the requested result with units.", "qpPages": [ 16, 16 ], "msPages": [ 14, 15 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WME03-01-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_QP.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_M3_MS.pdf#page=14", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { 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"kinematics" ], "primaryTopicId": "kinematics", "summary": "Prove that, before the elastic string 2l goes slack, P moves with simple harmonic motion with period π g.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 20, 20 ], "msPages": [ 16, 18 ], "studyPriority": 1, "difficultyEvidence": "Pearson notes that candidates who ignored the instruction to differentiate with respect to x earned no marks; the report describes the required derivative rearrangement as the route to proving SHM.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "id": "ial-2024-6-WME03-01-q07b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": 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and g, the exact total time, from the instant it is projected from B, 3 that it takes P to travel a total distance of l along the table. 4.", "answerGuide": "Write the relevant displacement, velocity or acceleration relations, use the stated initial conditions, and give the requested time or value with units.", "qpPages": [ 20, 20 ], "msPages": [ 16, 18 ], "studyPriority": 1, "difficultyEvidence": "Pearson says this part challenged candidates at all levels; many could not find the amplitude or complete the time calculation after finding a partial time.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "id": "ial-2024-6-WME03-01-q07c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", 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requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Demonstrate that there is only one outlier amongst the Birch trees.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q01c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, 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distributions of the growth of these Birch trees and the growth of these Maple trees. State the values of any statistics you have used to support your comment.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q01e", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", 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Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine P(2X – 3 > 5).", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Demonstrate that Var(X ) = 4.14.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard 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requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 6, 6 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q02c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=6", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Work out the width and the height of the bar representing the class 30 x < 32.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q03a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 3, "questionPart": "3(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the median of x.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q03b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Demonstrate that the mean of w is 40.8.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q03c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 3, "questionPart": "3(d)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Work out the standard deviation of w.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q03d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 3, "questionPart": "3(e)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give, giving a reason, what effect this would have on the value of (i) the mean of w (ii) the standard deviation of w.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q03e", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Work out the exact value of Sdg and the exact value of Sgg.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": 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"ial-2024-6-WST01-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 4, "questionPart": "4(d)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Give an interpretation, in context, of the gradient of the regression line.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q04d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 4, "questionPart": "4(e)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate the girth of a bear with a length of 2.5 metres, (ii) explain why an estimate for the girth of a bear with a length of 0.5 metres is not reliable.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 14, 14 ], 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standardisation to find the proportion of adults that take less than 18 minutes to complete round one.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": 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"board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 5, "questionPart": "5(c)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Calculate a conditional probability using a Normal model.", "answerGuide": "Standardise the relevant value, use the conditional probability identity, and check the result against the event limits.", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "studyPriority": 1, "difficultyEvidence": "The report explicitly identifies this part as particularly challenging.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "id": "ial-2024-6-WST01-01-q05c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": 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"answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 22, 22 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST01-01-q06a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": 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"https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S1_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST01-01-QP.pdf", "markScheme": "wst01-01-rms-20240815.pdf", "examinerReport": "wst01-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "0B5FF889C6FD0AA272B43083D44ADC8565D8A770D28888CBA1978112201D3380", "markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 6, "questionPart": "6(d)(ii)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "(ii) P(C ½ (S ∩ G)).", "answerGuide": "Express the requested event using the given 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"markScheme": "479DB9C8BAC7FD65F335259A920E015D3130AB80FBC8FB8E07C7D27AE3109CB1", "examinerReport": "83FFFC13C9CC158DEEB31B42941D1BDEA262BC83877269692E223AD0AA981FD8" } }, { "paperId": "ial-2024-6-WST01-01", "syllabus": "WST01", "questionNumber": 6, "questionPart": "6(e)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Use an overlap probability to estimate the size of a population outside a specified activity.", "answerGuide": "Use the supplied overlap information to find the probability of the combined event, then scale by the total group size and subtract the requested subgroup.", "qpPages": [ 22, 22 ], "msPages": [ 11, 12 ], "studyPriority": 1, "difficultyEvidence": "Pearson names part (e) as one of the hardest elements of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst01-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 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"WST02", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine the probability that (i) fewer than 4 customers arrive to buy tyres in the next 10 minutes, (ii) more than 5 customers arrive to buy tyres in the next 10 minutes.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=2", "msUrl": 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or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q01b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": 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State your hypotheses clearly.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q01c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": 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2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": 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"answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q03c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 3, "questionPart": "3(d)", "marks": 3, "topicIds": [ "probability", "statistical-testing" ], "primaryTopicId": "probability", "summary": "Determine the critical region for a two-tailed test of Jian’s belief. You should state the probability in each tail, which should each be as close as possible to 2.5%.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q03d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 3, "questionPart": "3(e)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Give what this suggests about Jian’s belief. Give a reason for your answer.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q03e", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 3, "questionPart": "3(f)", "marks": 7, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test, at the 5% level of significance, whether or not there is evidence that the proportion of shops where the stocktaking system is being used incorrectly is now less than 0.4 You should state your hypotheses and show.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 10, 10 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q03f", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=10", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Demonstrate that P( M = 4 ) = 245.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 4, "questionPart": "4(b)", "marks": 6, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the sampling distribution of M.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q04b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Work out the minimum value of n such that P(T = 0) < 0.15.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 14, 14 ], "msPages": [ 9, 9 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=14", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Determine the probability that at least 2 of these calls are connected to the wrong department on the first attempt.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Apply a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine P(T > 16).", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q05c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "probability", "distributions" ], "primaryTopicId": "probability", "summary": "Combine independent route events with Poisson waiting and uniform time intervals.", "answerGuide": "Find each route's success probability using the appropriate Poisson or uniform model, then combine the independent route probabilities.", "qpPages": [ 18, 18 ], "msPages": [ 10, 10 ], "studyPriority": 1, "difficultyEvidence": "Pearson identifies part (d) as the most difficult part of the paper.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 5, 6 ] }, "id": "ial-2024-6-WST02-01-q05d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=18", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=10", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Demonstrate that 4a + 4b = 1.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 22, 22 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q06a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 6, "questionPart": "6(b)(i)", "marks": 5, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine an equation in terms of a only.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 22, 22 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q06bi", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 6, "questionPart": "6(b)(ii)", "marks": 2, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Hence show that b = 0.1.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 22, 22 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q06bii", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 6, "questionPart": "6(c)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Sketch the probability density function f(x) of X.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 22, 22 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q06c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST02-01", "syllabus": "WST02", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Determine the value of k for which P(X k) = 0.8.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 22, 22 ], "msPages": [ 11, 13 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST02-01-q06d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_QP.pdf#page=22", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S2_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST02-01-QP.pdf", "markScheme": "wst02-01-rms-20240815.pdf", "examinerReport": "wst02-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "3CA924B41FC91157BDDC2A03E6EC073E374DF9840F8E15D45F0AF09E8468217E", "markScheme": "8005CF429C5C51203FD3BBE1FAB44D607079F304E294ADA405541917667D15C0", "examinerReport": "E704099CE3A08A4D78EAF8142D85962F9D6045EFD56D8386DC1B9F00959CA4D8" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Describe how the list of numbers can be used to select a systematic sample of 8 employees.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q01a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give one disadvantage of systematic sampling in this case.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q01b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "probability", "data-analysis" ], "primaryTopicId": "probability", "summary": "Write down the probability that the sample includes both the first name (employee 001) and the last name (employee 400) in the list.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 2, 2 ], "msPages": [ 6, 6 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q01c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=2", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=6", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 2, "questionPart": "2(a)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Work out Spearman’s rank correlation coefficient between Aarush’s order and the actual order. Use a rank of 1 for the highest priced kettle. Show your working clearly.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q02a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Test whether or not there is evidence to suggest that Aarush is able to rank kettles in order of decreasing price. You should state your hypotheses and critical value.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q02b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Explain why Aarush did not use the product moment correlation coefficient in this situation.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q02c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 2, "questionPart": "2(d)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain what effect this has on the rankings for the price.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 4, 4 ], "msPages": [ 7, 7 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q02d", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=4", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=7", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test whether or not μ = 328 State your hypotheses clearly and write down the significance level you have used.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q03a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 3, "questionPart": "3(b)", "marks": 6, "topicIds": [ "statistical-testing", "data-analysis" ], "primaryTopicId": "statistical-testing", "summary": "Work out a 98% confidence interval for μ based on this second sample. You must show all steps in your working. ( ).", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q03b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Work out the probability that more than 3 of these intervals will contain μ.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 8, 8 ], "msPages": [ 8, 8 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q03c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=8", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=8", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 4, "questionPart": "4(a)", "marks": 8, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Apply the data in the table to test, at the 5% level of significance, the manager’s belief. You should state your hypotheses, test statistic, critical value and conclusion clearly.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 12, 12 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q04a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Apply the table to calculate expected frequencies for the group (i) teenagers whose favourite ice cream flavour is vanilla, (ii) adults whose favourite ice cream flavour is chocolate.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 12, 12 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q04b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Write down the number of degrees of freedom for this χ 2 test.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 12, 12 ], "msPages": [ 9, 10 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q04c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=12", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=9", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 5, "questionPart": "5(a)", "marks": 8, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Test, at the 5% level of significance, the manager’s claim. You should state your hypotheses, test statistic, critical value and conclusion clearly.", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 16, 16 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q05a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "statistical-testing" ], "primaryTopicId": "statistical-testing", "summary": "Give two assumptions you have made in carrying out the test in part (a).", "answerGuide": "State the hypotheses, calculate the test statistic, compare it with the stated critical value or probability, and give a conclusion in context.", "qpPages": [ 16, 16 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q05b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 5, "questionPart": "5(c)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Determine an unbiased estimate of the variance for the sample of 46 part-time employees.", "answerGuide": "Use the supplied sample information with the appropriate formula, show the key substitutions, and state the estimate or conclusion in context.", "qpPages": [ 16, 16 ], "msPages": [ 11, 12 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q05c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=16", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=11", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 6, "questionPart": "6(a)", "marks": 4, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Work out the probability that their total weight is more than 3.5 kg.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 20, 20 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q06a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 6, "questionPart": "6(b)", "marks": 5, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Work out the probability that the difference in their weights is more than 0.05 kg.", "answerGuide": "Express the requested event using the given probabilities or distribution, show the calculation, and give the result to the stated accuracy.", "qpPages": [ 20, 20 ], "msPages": [ 13, 14 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q06b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 6, "questionPart": "6(c)", "marks": 7, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Evaluate probabilities for sums and differences of independent Normal variables.", "answerGuide": "For each linear combination, add means and add variances with squared coefficients, then standardise using the correct event and keep independent variables distinct.", "qpPages": [ 20, 20 ], "msPages": [ 13, 14 ], "studyPriority": 2, "difficultyEvidence": "The report says correct responses to Q6(c) were rare; many candidates combined independent quantities incorrectly when forming the final Normal variable.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 6 ] }, "id": "ial-2024-6-WST03-01-q06c", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=20", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=13", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "distributions" ], "primaryTopicId": "distributions", "summary": "Apply the Central Limit Theorem to find an approximate distribution for D Give your answer in terms of n and x where appropriate.", "answerGuide": "Define the relevant random variable and distribution, substitute the given parameters, and state the probability or statistic to the requested accuracy.", "qpPages": [ 24, 24 ], "msPages": [ 15, 16 ], "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "id": "ial-2024-6-WST03-01-q07a", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "source-derived", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "paperId": "ial-2024-6-WST03-01", "syllabus": "WST03", "questionNumber": 7, "questionPart": "7(b)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Use a sample-mean distribution and a confidence bound to determine a sample size.", "answerGuide": "Find the mean and variance of the uniform variable, derive the Normal approximation for the sample mean, then set the lower confidence bound equal to its stated value and solve for n.", "qpPages": [ 24, 24 ], "msPages": [ 15, 16 ], "studyPriority": 1, "difficultyEvidence": "Pearson reports that full marks on Q7(b) were extremely uncommon, often because candidates mismatched the shifted mean from part (a) with the confidence-bound equation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20240815.pdf", "sourceEvidencePages": { "examinerReport": [ 7 ] }, "id": "ial-2024-6-WST03-01-q07b", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "year": 2024, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "qpUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_QP.pdf#page=24", "msUrl": "https://www.pastpaperhub.com/paper/passpapers/IAL/mathematics/2024-may/Mathematics_2024MayJune_S3_MS.pdf#page=15", "hardestPart": null, "tagConfidence": "reviewed", "sourceFiles": { "questionPaper": "WST03-01-QP.pdf", "markScheme": "wst03-01-rms-20240815.pdf", "examinerReport": "wst03-01-pef-20240815.pdf" }, "sourceSha256": { "questionPaper": "743C9E9916DCEA5A1D1490CBF95847B0B15081DF408E7E38DFF869877E9E0F5D", "markScheme": "B2D87E82DCD00AD4E47AA4C2ADE87EB7763AD3F5D06A28B3417A814957FD278D", "examinerReport": "E9B6C866AB0F47F207B228010240389C12CC9218004C987895A59166FF532B13" } }, { "id": "ial-2025-6-WDM11-01-q01a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate a lower bound for the number of vans needed to move the nine items, each van holding at most 600 kg.", "answerGuide": "Add all item masses and divide by 600; round the resulting lower bound up to the next whole van.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q01b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Allocate the nine items to vans using the first-fit bin-packing algorithm.", "answerGuide": "Process the items in the given order, placing each into the first van with enough remaining capacity and opening a new van when needed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=5", "qpPages": [ 2, 2 ], "msPages": [ 5, 5 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q02a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Prim’s algorithm from P to find a minimum spanning tree and show the order of selected arcs.", "answerGuide": "At each step select the shortest available arc connecting the current tree to a new node; record the arcs in order until all seven signposts are connected.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q02b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the minimum spanning tree to obtain an initial upper bound for a route that visits every signpost and returns to P.", "answerGuide": "Find the total weight of the tree and double it to form the initial closed-route upper bound.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q02c", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the nearest-neighbour algorithm from P to find a second upper bound, giving the route and its length.", "answerGuide": "Repeatedly travel from the current node to the nearest unvisited node, then return to P and total the route length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q02d", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Delete T and its arcs and use the tree result to obtain a lower bound for the route length.", "answerGuide": "Remove T from the tree, then add the two shortest arcs connecting T to the remaining network to form the lower-bound calculation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=3", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=6", "qpPages": [ 3, 3 ], "msPages": [ 6, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q03ia", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(a)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Sort the twelve-number list into ascending order with quick sort, showing each pass and the pivots.", "answerGuide": "Choose the specified middle-left or middle-right pivot consistently, partition around each pivot, and repeat on the unsorted sections.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q03ib", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(i)(b)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Show how binary search checks whether 40 appears in the sorted list, identifying pivots and rejected values.", "answerGuide": "Use the middle-right pivot of the sorted list, discard the half that cannot contain 40 after each comparison, and show the final unsuccessful check.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports that a significant minority struggled with the binary-search procedure, especially selecting the pivot and clearly rejecting the correct values.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q03ii", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(ii)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Determine the range of x values consistent with the first three passes of descending bubble sort.", "answerGuide": "Translate the observed swaps and non-swaps into bounds on x, combine the inequalities, and state the resulting range.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q04a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use Dijkstra’s algorithm to find the fastest route from A to H and state the travel time; this single allocation covers the displayed (i) and (ii) prompts.", "answerGuide": "Update working values from the currently labelled node, keep the smallest permanent value at each step, and read the final route and time at H.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q04b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find how much longer the journey becomes when roads EF and GJ are closed.", "answerGuide": "Recalculate the fastest route using the remaining roads, then subtract the original fastest time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=5", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=12", "qpPages": [ 5, 5 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says part (b) was not well answered and was often left blank; some candidates found the new route time but did not calculate the increase.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q05a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 1, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Write the objective for maximizing the total number of rose bushes.", "answerGuide": "State that the objective is to maximize x + y + z.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q05b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Form the budget constraint from the three prices and the £1400 spending limit.", "answerGuide": "Write 12.5x + 10y + 15z ≤ 1400, then divide by 2.5 to obtain the stated integer-coefficient form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q05c", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Form two simplified integer-coefficient constraints for the minimum red share and the required white-to-yellow ratio.", "answerGuide": "Translate “at least 40% red” into x ≥ 0.4(x+y+z) and the colour ratio into z ≥ 1.5y; rearrange both with integer coefficients.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q05d", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "With z fixed at 30, show that the remaining proportion constraint becomes 3x − 2y ≥ 60.", "answerGuide": "Substitute z = 30 into the constraint from part (c), collect the x and y terms, and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q05e", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Graph the three reduced constraints and find the numbers of red and yellow bushes at the optimum.", "answerGuide": "Draw the boundary lines, identify the feasible region, locate its maximizing vertex, and state x and y in the context of rose bushes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=14", "qpPages": [ 6, 6 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports that the final mark was fairly rare: candidates often had the lines but selected the wrong feasible region or did not give the contextual optimum.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q06a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the minimum route-inspection length from A to J and state which roads must be repeated.", "answerGuide": "Identify the four odd vertices F, J, K and M, compare their three possible pairings, repeat the roads in the shortest pairing, and add the repeated distance to the network total.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says surprisingly few candidates earned full marks on part (a); common problems included identifying the odd vertices, calculating the repeated route, or stating the repeated arcs.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q06b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the change in minimum route length after adding the FJ road of length 18 km.", "answerGuide": "Reassess the odd vertices after adding FJ; only K and M need pairing, so compare the new total with the original route length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=7", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=16", "qpPages": [ 7, 7 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q07a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the precedence table for the project’s activity network.", "answerGuide": "For each activity, list the activities that must immediately precede it, following the arrows in the network.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q07b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Complete the activity network with early and late event times.", "answerGuide": "Pass forward through the network taking maximum predecessor times, then pass backward from the finish taking minimum successor times.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=17", "qpPages": [ 8, 8 ], "msPages": [ 17, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q07c", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Calculate a lower bound for the number of workers needed to finish the project in the minimum time.", "answerGuide": "Divide the total duration of all activities by the critical-path completion time and round up to a whole worker.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q07d", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Draw a Gantt cascade chart for the project.", "answerGuide": "Place each activity within its early and late event-time limits, showing the critical activities and allowable floats on the time axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q07e", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 3, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Use the Gantt chart to determine the minimum number of workers, stating the overlapping activities and times.", "answerGuide": "Inspect the two critical overlap intervals: five activities must run together in either interval, so five workers are required.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=18", "qpPages": [ 8, 8 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls part (e) extremely demanding: the modal score was zero, and full reasoning required identifying both overlap intervals and the fifth activity.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q08a", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "State the four remaining inequalities defining the feasible region shown in the graph.", "answerGuide": "Read the direction of each boundary inequality from the shaded excluded region and write all four constraints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=20", "qpPages": [ 10, 10 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q08b", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find integer coefficients a and b from the stated minimum and maximum of P = ax + by at vertices E and C.", "answerGuide": "Use the given vertex coordinates and objective values to form two simultaneous equations, then solve for a and b.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=20", "qpPages": [ 10, 11 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WDM11-01-q08c", "paperId": "ial-2025-6-WDM11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WDM11", "unit": "Decision Mathematics 1 (D1)", "year": 2025, "session": "May/June", "paperCode": "WDM11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "decision-mathematics" ], "primaryTopicId": "decision-mathematics", "summary": "Find the range of positive k for which Q = x + ky is minimized at A and maximized at C.", "answerGuide": "Compare the objective gradient with the adjacent boundary slopes at A and C, or compare Q values at the relevant vertices, then combine the inequalities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_D1_MS.pdf#page=20", "qpPages": [ 10, 11 ], "msPages": [ 20, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says only the most able candidates could make a serious attempt at part (c), and only the strongest gained all four marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wdm11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q01a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "indices" ], "primaryTopicId": "indices", "summary": "Simplify the square root of p into the required form kx^n.", "answerGuide": "Apply the fractional exponent to the coefficient and power of x, then simplify both.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q01b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "indices" ], "primaryTopicId": "indices", "summary": "Write the reciprocal of pq as a simplified single term in x.", "answerGuide": "Multiply p and q, then raise the coefficient and x power to -1; express the result using a negative exponent.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q01c", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "indices" ], "primaryTopicId": "indices", "summary": "Simplify p times q squared into the required form.", "answerGuide": "Square q, multiply its coefficient and power by p, then simplify the coefficient and combine exponents.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q02a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the given fractional-power curve and simplify.", "answerGuide": "Use the power rule term by term, including the fractional exponent, then simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q02b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 3, "topicIds": [ "differentiation", "indices" ], "primaryTopicId": "differentiation", "summary": "Use the curve gradient at x = 2^k to find k.", "answerGuide": "Set the derivative equal to 16 at x = 2^k, rearrange to a power-of-two equation, and equate exponents.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q03", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate (x + 3)^2 divided by 3√x, giving a simplified antiderivative.", "answerGuide": "Expand the numerator over √x into powers of x, integrate each power, and add a constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q04a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 1, "topicIds": [ "trigonometry", "graphs" ], "primaryTopicId": "trigonometry", "summary": "Read the coordinates of the marked positive x-axis point P on y = 4 cos x.", "answerGuide": "Use the degree value where the cosine curve crosses the positive x-axis at the labelled point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q04b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "trigonometry", "graphs" ], "primaryTopicId": "trigonometry", "summary": "Read the coordinates of the marked minimum Q on y = 4 cos x.", "answerGuide": "Use the cosine minimum and its degree position on the negative side of the y-axis.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q04c", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Count the solutions of three cosine equations over their stated degree intervals.", "answerGuide": "Rearrange each to a cosine value, count the periodic solutions in the interval, and handle included or excluded endpoints.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q05a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the line through the origin perpendicular to the given line.", "answerGuide": "Read the given gradient, take its negative reciprocal, and use the origin to fix the intercept.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q05b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "coordinate-geometry", "algebra" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the intersection coordinates of the two lines by algebra.", "answerGuide": "Solve the two line equations simultaneously, then substitute to obtain the second coordinate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q05c", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the shortest distance from the origin to the given line as a simplified surd.", "answerGuide": "Use the perpendicular intersection from the previous part and calculate its distance from the origin.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=11", "qpPages": [ 10, 10 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q06a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Show that the central angle COA is 1.847 radians to three decimal places.", "answerGuide": "Use CO = OB + BC, apply the sine rule to find angle CAO, then subtract it and 0.6 from π.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q06b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Find the landing pad area by combining a triangle with the major sector.", "answerGuide": "Calculate triangle AOC using 15, 17 and the included angle 0.6; add the major-sector area using its reflex angle and radius 15.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q06c", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "geometry", "trigonometry" ], "primaryTopicId": "geometry", "summary": "Find the perimeter of the landing pad to three significant figures.", "answerGuide": "Add AC, BC and the major arc length; use the cosine rule for AC and radius times reflex angle for the arc.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=13", "qpPages": [ 14, 14 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q07a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Find the cubic f(x) that touches the x-axis at zero, crosses at x = 4, and passes through (10, 120).", "answerGuide": "Represent the repeated root by a factor x² and the other root by (x − 4); substitute the point to find the scale factor.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes this part as the main stumbling block: fewer than half formed the repeated-root cubic correctly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q07b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "graphs", "quadratics" ], "primaryTopicId": "graphs", "summary": "Sketch y = 1.2x(8 − x) on the supplied axes.", "answerGuide": "Recognise a downward-opening parabola with roots at 0 and 8 and a maximum midway between them.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q07c", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 5, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Solve algebraically for every intersection of the cubic and the given parabola.", "answerGuide": "Set the expressions equal, factor out x, solve the remaining quadratic, and substitute each x value into either curve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=15", "qpPages": [ 18, 19 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q08a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "differentiation", "functions" ], "primaryTopicId": "differentiation", "summary": "Use the tangent at P to show k = 1/4 and find its intercept c.", "answerGuide": "At x = 4 equate f′(4) to the tangent gradient 10 to determine k; substitute P(4,12) into y = 10x + c.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q08b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Find the second derivative of f at P.", "answerGuide": "Differentiate the supplied f′(x), then evaluate at x = 4 using k = 1/4.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q08c", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 4, "topicIds": [ "integration", "functions" ], "primaryTopicId": "integration", "summary": "Find f(x) from its derivative and the point P.", "answerGuide": "Integrate f′(x), include a constant, then use f(4) = 12 to determine it.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=17", "qpPages": [ 22, 22 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q09a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "transformations", "functions" ], "primaryTopicId": "transformations", "summary": "Describe the transformations from the two parent square-root graphs to C1 and C2.", "answerGuide": "For C1, use a vertical stretch by factor √2 (equivalently a horizontal compression by factor 1/2); for C2, translate the reflected square-root graph up by 12.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=19", "qpPages": [ 26, 26 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports substantial difficulty describing the transformations in this part; correct transformation language and scale were often missing.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q09b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 6, "topicIds": [ "functions", "indices", "algebra" ], "primaryTopicId": "functions", "summary": "Show the intersection x-coordinate satisfies the supplied surd equation, then find the exact coordinates of P.", "answerGuide": "Equate the curves, factor out √x, and rationalise 12/(√2 + 1); then x = 144(3 − 2√2) and y = 12(2 − √2).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=19", "qpPages": [ 26, 26 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies Question 9 as the paper’s most challenging question and reports that fully correct exact intersection coordinates were rare.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q10a", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "indices", "quadratics" ], "primaryTopicId": "indices", "summary": "Solve the equation exactly when k = 4.5.", "answerGuide": "Set u = x³, factor 7u² + 8u + 1 = 0, then return to x; the exact roots are −1 and −1/∛7.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA11-01-q10b", "paperId": "ial-2025-6-WMA11-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA11", "unit": "Pure Mathematics 1 (P1)", "year": 2025, "session": "May/June", "paperCode": "WMA11/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 4, "topicIds": [ "quadratics", "algebra" ], "primaryTopicId": "quadratics", "summary": "Find the values of k for which the equation has no real roots.", "answerGuide": "Set u = x³, require the quadratic discriminant 4k(5 − k) to be negative, giving k < 0 or k > 5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P1_MS.pdf#page=20", "qpPages": [ 30, 30 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes this as a very challenging part for lower grades and a strong discriminator; sign errors in the discriminant region were common.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma11-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q01a", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 4, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Find the first four terms of (1 − 4x)^7 in ascending powers of x.", "answerGuide": "Use the binomial theorem through the x³ term and simplify all coefficients: 1 − 28x + 336x² − 2240x³.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q01b", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the expansion to find k when the x² coefficient in (5 + kx)(1 − 4x)^7 is 1316.", "answerGuide": "Combine the x² contributions 5(336) and k(−28), then solve 1680 − 28k = 1316 to get k = 13.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q02a", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 5, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the equation of the circle whose diameter endpoints are (−2, 5) and (4, 15).", "answerGuide": "Find centre (1, 10) and radius squared 34; the circle is (x − 1)² + (y − 10)² = 34.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q02b", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "coordinate-geometry", "geometry" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the exact point on the circle nearest the x-axis.", "answerGuide": "The circle lies above the x-axis; move one radius down from its centre to get (1, 10 − √34).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=8", "qpPages": [ 4, 4 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q03", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Find the area between y = 3x + 16/x² − 8 and the chord joining P(1, 11) to Q(4, 5).", "answerGuide": "The chord is y = −2x + 13. Integrate line minus curve from 1 to 4 and take the positive area, 13.5 square units.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q04a", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "algebra", "proof" ], "primaryTopicId": "algebra", "summary": "Use the factor theorem to show b = 3a + 2 for the given cubic with factor x + 3.", "answerGuide": "Set f(−3) = 0 and rearrange the resulting relation between a and b.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q04b", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Use the given remainder on division by 2x − 1 to find a and b.", "answerGuide": "Set f(1/2) = 7/4 and solve with b = 3a + 2; this gives a = 2 and b = 8.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=11", "qpPages": [ 8, 8 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q04c", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Divide the resulting cubic by x − 2 and state the quotient and remainder.", "answerGuide": "Use polynomial long division (or synthetic division); the quotient is 2x² + 12x + 29 and the remainder is 55.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q05a", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "integration", "data-analysis" ], "primaryTopicId": "integration", "summary": "Estimate the integral of 3(1/2)^x from −2 to 7 using all tabulated ordinates.", "answerGuide": "Use the trapezium rule with width 1.5: half the width times the two endpoint ordinates plus twice the interior ordinates, giving 18.8.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q05b", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 3, "topicIds": [ "integration", "data-analysis" ], "primaryTopicId": "integration", "summary": "Use part (a) to estimate both the shifted exponential integral and the integral of 2^(−x) + 2x over the same limits.", "answerGuide": "For (i), the integrand is one quarter of the original, giving 18.8/4 = 4.7. For (ii), use 2^(−x) = y/3 and add the exact integral of 2x, giving 51.3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=13", "qpPages": [ 12, 12 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q06i", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(i)", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Solve the base-2 logarithmic equation with its domain restrictions.", "answerGuide": "Combine logarithms to obtain (4 − x)² = 4(x + 11); the roots are 14 and −2, but only x = −2 is in the domain.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=14", "qpPages": [ 14, 14 ], "msPages": [ 14, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q06ii", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(ii)", "marks": 5, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find the exact intersection of y = 3^(2x+1) and y = 6·3^x.", "answerGuide": "Set the curves equal and divide by 3^x to get 3^x = 2; hence P = (log₃2, 12).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=15", "qpPages": [ 14, 14 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q07ia", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)(a)", "marks": 2, "topicIds": [ "sequences" ], "primaryTopicId": "sequences", "summary": "Find the sum to infinity of the geometric series 10 + 8 + 6.4 + ….", "answerGuide": "The common ratio is 0.8, so S∞ = 10/(1 − 0.8) = 50.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q07ib", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(i)(b)", "marks": 3, "topicIds": [ "sequences", "algebra" ], "primaryTopicId": "sequences", "summary": "Find the smallest k for which the kth term of the geometric series is below 0.0005.", "answerGuide": "Solve 10(0.8)^(k−1) < 0.0005, retain the strict inequality when taking logarithms, and choose the smallest integer: k = 46.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies this exact subpart as particularly challenging; candidates often reversed the inequality after dividing by log(0.8) or failed to choose the next integer.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q07ii", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(ii)", "marks": 4, "topicIds": [ "sequences", "quadratics" ], "primaryTopicId": "sequences", "summary": "Find the greatest possible sum of the first n terms of 850 + 843 + 836 + ….", "answerGuide": "Use a = 850 and d = −7 in S_n = n/2(1707 − 7n); the integer n that maximises the sum is 122, giving S_122 = 52033.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=16", "qpPages": [ 18, 18 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q08i", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(i)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Solve 3tan(2x + π/5) = √3 for 0 ≤ x < π in exact rational multiples of π.", "answerGuide": "Set the tangent equal to 1/√3, use its π-period, then apply the interval: x = 29π/60 and 59π/60.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=18", "qpPages": [ 22, 22 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q08ii", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(ii)", "marks": 5, "topicIds": [ "trigonometry", "quadratics" ], "primaryTopicId": "trigonometry", "summary": "Solve 5sinθ tanθ = cosθ + 4 for 0 ≤ θ < 360°.", "answerGuide": "Replace tanθ by sinθ/cosθ, multiply through by cosθ, and use sin²θ = 1 − cos²θ. Keep the valid cosine root and give both angles, about 50.3° and 309.7°.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=19", "qpPages": [ 22, 22 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q09a", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "geometry", "algebra" ], "primaryTopicId": "geometry", "summary": "Derive the open container’s metal area as S = Px² + Q/x and find P and Q.", "answerGuide": "Use triangular cross-section area 3x² and volume 3Lx² = 12, so L = 4/x². The two end triangles and two rectangular sides give S = 6x² + 20/x.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=20", "qpPages": [ 26, 26 ], "msPages": [ 20, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that candidates struggled to model the open prism, form its volume equation, and derive the required surface-area formula in this exact part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q09b", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 5, "topicIds": [ "geometry", "differentiation" ], "primaryTopicId": "geometry", "summary": "Use calculus to find the minimum surface area as x varies.", "answerGuide": "Differentiate S = 6x² + 20/x, set 12x − 20/x² = 0 to get x = (5/3)^(1/3), and substitute to obtain a minimum of about 25.3 m².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=20", "qpPages": [ 26, 26 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q09c", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 2, "topicIds": [ "geometry", "differentiation" ], "primaryTopicId": "geometry", "summary": "Justify that the value of S found in part (b) is a minimum.", "answerGuide": "Show S″ = 12 + 40/x³ is positive for x > 0 at the stationary point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=21", "qpPages": [ 26, 26 ], "msPages": [ 21, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA12-01-q10", "paperId": "ial-2025-6-WMA12-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA12", "unit": "Pure Mathematics 2 (P2)", "year": 2025, "session": "May/June", "paperCode": "WMA12/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": null, "marks": 4, "topicIds": [ "proof", "algebra" ], "primaryTopicId": "proof", "summary": "Prove by exhaustion that m² + 3m + 2 is divisible by 3 whenever positive integer m is not a multiple of 3.", "answerGuide": "Write m as 3p + 1 or 3p + 2. Substitution gives 3(3p² + 5p + 2) or 3(3p² + 7p + 4), so both cases are divisible by 3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P2_MS.pdf#page=22", "qpPages": [ 30, 30 ], "msPages": [ 22, 23 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson names Question 10 as a particularly challenging question; the scored group is the whole four-mark proof.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma12-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q01a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Evaluate the composite function gf(1) for the given f and g.", "answerGuide": "Find f(1) first, then substitute that value into g; gf(1) = 15/4.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q01b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "functions" ], "primaryTopicId": "functions", "summary": "Find the complete range of f(x) for x ≥ 0.", "answerGuide": "Use the increasing rational function f(x) = 2x/(3x + 1): its minimum is 0 and its upper limit 2/3 is not attained, so 0 ≤ f(x) < 2/3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson says identifying both bounds of this exact range was demanding; candidates often missed or included the upper bound.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q01c", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Find the inverse function f⁻¹(x).", "answerGuide": "Set y = 2x/(3x + 1), rearrange for x, and swap variables to obtain f⁻¹(x) = x/(2 − 3x).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q01d", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 2, "topicIds": [ "functions", "algebra" ], "primaryTopicId": "functions", "summary": "Solve f⁻¹(x) = f(x) over their valid domains.", "answerGuide": "Equate x/(2 − 3x) and 2x/(3x + 1), retain the x = 0 root, and solve the remaining factor; x = 0 or 1/3.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q02a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "trigonometry" ], "primaryTopicId": "trigonometry", "summary": "Write 7 cos x − 24 sin x as R cos(x + α), with exact R and α to 3 d.p.", "answerGuide": "Match coefficients: R = √(7² + 24²) = 25 and tan α = 24/7, so α ≈ 1.287 radians.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q02b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Use the rewritten function to find the minimum of g(x) and the least positive x where it occurs.", "answerGuide": "With f(2x) = 25 cos(2x + α), the denominator is largest at cosine −1, so min g = 1/33; solve 2x + α = π for x ≈ 0.927 radians.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=9", "qpPages": [ 6, 6 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q03a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "exponentials", "coordinate-geometry" ], "primaryTopicId": "exponentials", "summary": "Find the linear equation relating log₁₀ y and log₁₀ x from two plotted points.", "answerGuide": "Use the points (−3.5, 0) and (0, −2) to get gradient −4/7 and intercept −2: log₁₀ y = −(4/7)log₁₀x − 2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q03b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Express y as p x^q using the logarithmic linear relation.", "answerGuide": "Convert powers of ten: y = 10⁻²x^(−4/7), so p = 1/100 and q = −4/7.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=10", "qpPages": [ 8, 8 ], "msPages": [ 10, 12 ], "hardestPart": null, "studyPriority": 4, "difficultyEvidence": "Pearson describes this exact conversion as more challenging, with varied success using logarithm laws and comparing coefficients.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q04a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra", "quadratics" ], "primaryTopicId": "algebra", "summary": "Combine and factor the rational expression to show the stated simplified form.", "answerGuide": "Factor x² + x − 12 = (x + 4)(x − 3), combine over a common denominator, then cancel x + 4 to get 7x/(x − 3).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q04b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Differentiate the simplified rational function.", "answerGuide": "Apply the quotient rule to 7x/(x − 3); the derivative simplifies to −21/(x − 3)².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=13", "qpPages": [ 10, 10 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q05i", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(i)", "marks": 2, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Integrate sin²(3x).", "answerGuide": "Use sin²(3x) = (1 − cos 6x)/2, then integrate to x/2 − sin(6x)/12 + C.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson says this exact trigonometric integral was the most challenging part of Question 5; many did not use the double-angle identity.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q05ii", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(ii)", "marks": 2, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Integrate x(x² + 4)^(3/2).", "answerGuide": "Set u = x² + 4 so du = 2x dx; the antiderivative is (x² + 4)^(5/2)/5 + C.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=15", "qpPages": [ 12, 12 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q06a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Find A in the cooling model when the initial temperature is 75°C.", "answerGuide": "Set t = 0 in θ = 21 + Ae^(−kt), giving A = 75 − 21 = 54.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q06b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "exponentials", "algebra" ], "primaryTopicId": "exponentials", "summary": "Use the five-minute temperature drop to find k to 3 significant figures.", "answerGuide": "Set θ = 25 at t = 5: e^(−5k) = 2/27, so k = ln(27/2)/5 ≈ 0.521.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=16", "qpPages": [ 14, 14 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q06c", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "exponentials", "differentiation" ], "primaryTopicId": "exponentials", "summary": "Find when the model predicts the temperature is falling at 9°C per minute.", "answerGuide": "Differentiate to θ′ = −54ke^(−kT), set θ′ = −9, and solve T = ln(6k)/k ≈ 2.19 minutes.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=17", "qpPages": [ 14, 14 ], "msPages": [ 17, 17 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q07a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "numerical-methods", "differentiation" ], "primaryTopicId": "numerical-methods", "summary": "Use calculus to show the stationary point x-coordinate satisfies the given iteration equation.", "answerGuide": "Differentiate e^(−x²)sin 3x, set the derivative to zero, and rearrange tan 3x = 3/(2x) to the stated arctan form.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=18", "qpPages": [ 18, 18 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q07b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "numerical-methods" ], "primaryTopicId": "numerical-methods", "summary": "Apply the iteration from x₁ = 0.4 to find x₂ and x₄ to four decimal places.", "answerGuide": "Iterate xₙ₊₁ = (1/3) arctan(3/(2xₙ)); x₂ = 0.4367 and x₄ = 0.4304.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=18", "qpPages": [ 18, 18 ], "msPages": [ 18, 19 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q07c", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "numerical-methods", "functions" ], "primaryTopicId": "numerical-methods", "summary": "Verify that the stationary x-coordinate is 0.430 to three decimal places using a suitable interval and function.", "answerGuide": "Define F(x) = x − (1/3)arctan(3/(2x)); show a sign change across the rounding interval 0.4295 < x < 0.4305.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=19", "qpPages": [ 18, 18 ], "msPages": [ 19, 19 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson says very few candidates gained marks in this exact verification part; most did not identify a suitable function.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q08a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 3, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Prove the triple-angle identity for tan 3x and state its excluded angles.", "answerGuide": "Apply the tangent addition formula to tan(2x + x), use tan 2x = 2tan x/(1 − tan²x), and simplify; exclude poles at x = (2n + 1)π/6.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=20", "qpPages": [ 22, 22 ], "msPages": [ 20, 21 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q08b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "trigonometry", "algebra" ], "primaryTopicId": "trigonometry", "summary": "Use the identity to solve the given equation for 0 < θ < π/2.", "answerGuide": "Set t = tan 3θ and use sec²3θ = 1 + t² to form 2t² − t − 6 = 0; enumerate the valid angles in range: θ ≈ 0.37, 0.72, 1.42 radians.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=21", "qpPages": [ 22, 22 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q09a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 3, "topicIds": [ "trigonometry", "functions" ], "primaryTopicId": "trigonometry", "summary": "Find the exact y-coordinates where the parametric curve meets the y-axis.", "answerGuide": "Set x = 0, so sin(3y + π/4) = 0; within the stated interval the coordinates are y = π/4 and 7π/12.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=23", "qpPages": [ 26, 26 ], "msPages": [ 23, 23 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q09b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "trigonometry", "differentiation" ], "primaryTopicId": "trigonometry", "summary": "Show that (dy/dx)² has the stated reciprocal quadratic form and find p and q.", "answerGuide": "Differentiate x with respect to y and invert: dy/dx = 1/[2cos(3y + π/4)]. Use sin(3y + π/4) = 3x/2 to obtain 1/(4 − 9x²), so p = 4 and q = 9.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=24", "qpPages": [ 26, 26 ], "msPages": [ 24, 25 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q09c", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "trigonometry", "coordinate-geometry" ], "primaryTopicId": "trigonometry", "summary": "Find the exact x-coordinate where the normal at A meets the tangent at B.", "answerGuide": "At A the tangent gradient is −1/2, so the normal gradient is 2; at B the tangent gradient is 1/2. Intersect the lines through (0, π/4) and (0, 7π/12) to get x = 2π/9.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=25", "qpPages": [ 26, 26 ], "msPages": [ 25, 27 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports frequent errors in this exact part from confusing normal and tangent gradients and ignoring the sketch; it was not well attempted.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q10a", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(a)", "marks": 3, "topicIds": [ "transformations", "functions" ], "primaryTopicId": "transformations", "summary": "Find the y-intercept and vertex coordinates of y = |kx − 10| + k.", "answerGuide": "At x = 0 the y-intercept is 10 + k; the vertex occurs when kx − 10 = 0, giving (10/k, k).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=28", "qpPages": [ 30, 30 ], "msPages": [ 28, 29 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q10b", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(b)", "marks": 3, "topicIds": [ "transformations", "algebra" ], "primaryTopicId": "transformations", "summary": "Find the x-values satisfying |kx − 10| + k ≥ 2k.", "answerGuide": "Since k > 0, divide |kx − 10| ≥ k into two branches: x ≤ 10/k − 1 or x ≥ 10/k + 1.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=29", "qpPages": [ 30, 30 ], "msPages": [ 29, 30 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA13-01-q10c", "paperId": "ial-2025-6-WMA13-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA13", "unit": "Pure Mathematics 3 (P3)", "year": 2025, "session": "May/June", "paperCode": "WMA13/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": "10(c)", "marks": 4, "topicIds": [ "transformations", "functions" ], "primaryTopicId": "transformations", "summary": "Find the k-range for which y = 3x + 1 intersects the modulus graph at two distinct points.", "answerGuide": "Require the right-arm gradient k to exceed 3 and the line to lie below the vertex, k² − k − 30 < 0; hence 3 < k < 6.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P3_MS.pdf#page=30", "qpPages": [ 30, 30 ], "msPages": [ 30, 31 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson calls this exact boundary analysis a discriminator at A/A*; the required two-intersection condition and vertex boundary were often missed.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma13-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q01a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 2, "topicIds": [ "differentiation", "algebra" ], "primaryTopicId": "differentiation", "summary": "Find the two possible y-coordinates on the implicit curve when x = 1/2.", "answerGuide": "Substitute x = 1/2 into the curve equation, form the resulting quadratic in y, and solve for both values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q01b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 6, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Find the tangent at the point above the x-axis, in the required integer-coefficient form.", "answerGuide": "Differentiate the curve implicitly, substitute the positive y-coordinate and x = 1/2 to find the tangent gradient, then use the point-gradient equation and simplify.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=7", "qpPages": [ 2, 2 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q02", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 4, "topicIds": [ "differentiation" ], "primaryTopicId": "differentiation", "summary": "Show that the radius of a melting spherical ice ball decreases at a rate inversely proportional to r².", "answerGuide": "Differentiate V = 4πr³/3 with respect to time, use the constant volume-decrease rate and the chain rule, then state why the remaining coefficient is constant.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=9", "qpPages": [ 4, 4 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q03", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": null, "marks": 6, "topicIds": [ "differentiation", "integration" ], "primaryTopicId": "differentiation", "summary": "Solve the given differential equation with y = 4 at x = π/6, expressing the result as y² = g(x).", "answerGuide": "Separate variables, integrate the trigonometric expression, apply the initial condition to find the constant, and present the result for y².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=10", "qpPages": [ 6, 6 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q04a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "algebra" ], "primaryTopicId": "algebra", "summary": "Decompose the rational function into the requested partial-fraction form and find A, B and C.", "answerGuide": "Set the numerator equal to the proposed partial-fraction numerator, compare coefficients or substitute convenient values, and solve for the constants.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q04b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "integration", "algebra" ], "primaryTopicId": "integration", "summary": "Use algebraic integration to evaluate the definite integral exactly in the requested logarithmic form.", "answerGuide": "Integrate each partial-fraction term, substitute the limits, and combine the logarithms into the required expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=12", "qpPages": [ 8, 8 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson describes part (b) as more discriminating than the usual partial-fractions work and lists several common integration and logarithm errors.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q05a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "differentiation", "coordinate-geometry" ], "primaryTopicId": "differentiation", "summary": "Use parametric differentiation to find the normal at t = 2 in the required integer-coefficient form.", "answerGuide": "Find dx/dt and dy/dt, form dy/dx at t = 2, take the negative reciprocal for the normal gradient, and use the coordinates at t = 2.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson reports many issues in part (a), especially differentiating x(t), simplifying dy/dx and taking the negative reciprocal for the normal.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q05b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "algebra", "functions" ], "primaryTopicId": "algebra", "summary": "Show that the parametric curve has the requested Cartesian equation and state its valid domain.", "answerGuide": "Rearrange x(t) to make t the subject, substitute into y(t), simplify to the stated rational form, and identify the excluded x-value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=14", "qpPages": [ 12, 12 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says full marks in part (b) were extremely rare; many candidates found the equation but missed the required domain restriction.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q06a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 2, "topicIds": [ "vectors" ], "primaryTopicId": "vectors", "summary": "Write a vector equation for the line through A and B.", "answerGuide": "Use A as a position vector and B − A as a direction vector, then express the line with a scalar parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q06b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 6, "topicIds": [ "vectors", "geometry" ], "primaryTopicId": "vectors", "summary": "Find the exact possible values of α given that angle BAC is 45°.", "answerGuide": "Form vectors AB and AC, use the scalar-product angle formula and their magnitudes to obtain a quadratic equation in α, then solve it.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=16", "qpPages": [ 16, 16 ], "msPages": [ 16, 17 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says this part required problem-solving, offered few marks without vector AC, and had more numerical and algebraic slips than any other part.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q07a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "trigonometry", "integration" ], "primaryTopicId": "trigonometry", "summary": "Show that u = tan x transforms the given trigonometric integral into the stated rational expression in u.", "answerGuide": "Use du/dx = sec²x and sec²x = 1 + tan²x, substitute throughout, and simplify to the required integrand in u.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 3, "difficultyEvidence": "Pearson notes that the proof in part (a) required both the derivative of tan x and the identity tan²x = sec²x − 1; candidates omitting these steps often lost marks.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q07b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "integration", "trigonometry" ], "primaryTopicId": "integration", "summary": "Hence evaluate the trigonometric integral.", "answerGuide": "Integrate the transformed expression in u, substitute u = tan x, and include the constant of integration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=18", "qpPages": [ 20, 20 ], "msPages": [ 18, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes part (b) as equally discriminating; common errors included leaving the result in u or integrating the transformed polynomial incorrectly.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q08a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Find the first three non-zero terms of the binomial expansion in ascending powers of x.", "answerGuide": "Factor out the constant term, apply the generalized binomial series, and simplify the first three coefficients.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=19", "qpPages": [ 22, 22 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q08b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 1, "topicIds": [ "binomial" ], "primaryTopicId": "binomial", "summary": "Use the expansion in part (a) to state the corresponding expansion with the sign of x reversed.", "answerGuide": "Replace x by −x in the first three terms from part (a), preserving the resulting alternating signs.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=19", "qpPages": [ 22, 22 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q08c", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 2, "topicIds": [ "binomial", "algebra" ], "primaryTopicId": "binomial", "summary": "Use the two expansions to show the requested quadratic approximation with simplified fractional coefficients.", "answerGuide": "Multiply the two truncated series, collect the constant and x² terms, and simplify the fractions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=19", "qpPages": [ 22, 22 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q08d", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 3, "topicIds": [ "binomial", "numerical-methods" ], "primaryTopicId": "binomial", "summary": "Use the approximation at x = 1 to obtain the requested fully simplified rational estimate for 135.", "answerGuide": "Substitute x = 1 into the product approximation, then rearrange the resulting relation to express the target square root as a simplified fraction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=19", "qpPages": [ 22, 22 ], "msPages": [ 19, 20 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls part (d) very discriminating and says many candidates did not connect the target number with the required approximation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q09a", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(a)", "marks": 4, "topicIds": [ "integration", "geometry" ], "primaryTopicId": "integration", "summary": "Show that the volume of the solid of revolution has the stated integral form and determine its constants.", "answerGuide": "Write the disk-method volume integral, expand the squared radius, use the double-angle identity, and match coefficients.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=21", "qpPages": [ 26, 26 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q09b", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(b)", "marks": 4, "topicIds": [ "integration", "exponentials" ], "primaryTopicId": "integration", "summary": "Evaluate the integral of eˣ cos x.", "answerGuide": "Use integration by parts twice, then rearrange to isolate the original integral.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=21", "qpPages": [ 26, 26 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q09c", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 9, "questionPart": "9(c)", "marks": 4, "topicIds": [ "integration", "geometry" ], "primaryTopicId": "integration", "summary": "Find the paperweight volume formed by joining two solids, to two significant figures.", "answerGuide": "Combine the results from (a) and (b), interpret the two joined solids as twice the volume from one half, evaluate the definite integral, and round as requested.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=26", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=21", "qpPages": [ 26, 26 ], "msPages": [ 21, 22 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies part (c) as perhaps the paper’s most demanding aspect: interpreting that the paperweight is twice the volume of S caused difficulty.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WMA14-01-q10", "paperId": "ial-2025-6-WMA14-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WMA14", "unit": "Pure Mathematics 4 (P4)", "year": 2025, "session": "May/June", "paperCode": "WMA14/01", "variant": "Standard paper", "questionNumber": 10, "questionPart": null, "marks": 4, "topicIds": [ "proof", "trigonometry" ], "primaryTopicId": "proof", "summary": "Use proof by contradiction to show |cos 2x / (cos x − sin x)| < 1 for every angle with 90° < x < 180°.", "answerGuide": "Assume the reverse inequality for an angle in the interval, use the double-angle identity and handle the modulus correctly to derive an impossibility, then conclude the stated inequality.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_QP.pdf#page=30", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_P4_MS.pdf#page=23", "qpPages": [ 30, 30 ], "msPages": [ 23, 24 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson reports that setting up the contradiction and handling the modulus were very demanding in this question.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wma14-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q01a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find Q’s speed after the collision from the given impulse.", "answerGuide": "Use impulse equals change in momentum for Q, with a consistent positive direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q01b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the mass constant k for P.", "answerGuide": "Apply impulse–momentum to P, or use conservation of momentum with the result from part (a).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=8", "qpPages": [ 2, 3 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q02a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Find A’s constant velocity from its two position vectors.", "answerGuide": "Subtract the position vectors and divide by the elapsed time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q02b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Write A’s position vector at any time t.", "answerGuide": "Use the constant velocity with one of the given position vectors.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q02c", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "vectors", "kinematics" ], "primaryTopicId": "vectors", "summary": "Find the bearing of B from A at t=5.", "answerGuide": "Evaluate both position vectors at t=5, form the displacement from A to B, then use trigonometry for the bearing.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=9", "qpPages": [ 4, 7 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q03a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the launch speed for a particle projected vertically upward and landing 12 m below its start.", "answerGuide": "Choose upward as positive and use displacement, time and acceleration in a constant-acceleration equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q03b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the particle’s impact speed at the ground.", "answerGuide": "Use a constant-acceleration equation with the launch speed and the 12 m downward displacement; report speed as positive.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Questions 3 and 6 as the greatest challenge. In Q3(b), sign and speed-versus-velocity errors persisted when finding the impact result.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q03c", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 2, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Sketch the speed–time graph from launch until impact.", "answerGuide": "Mark the instant of zero velocity at the highest point and draw the two linear speed segments.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=10", "qpPages": [ 8, 11 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q04a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the angle of the second rope attached to the barge.", "answerGuide": "Resolve the two rope tensions perpendicular to the barge’s motion and equate the components.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q04b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics", "vectors" ], "primaryTopicId": "mechanics", "summary": "Find the barge’s acceleration along the canal.", "answerGuide": "Resolve both tensions in the direction of motion, subtract resistance, and apply Newton’s second law.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=11", "qpPages": [ 12, 15 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q05", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": null, "marks": 6, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass position of a non-uniform rod from two tipping conditions.", "answerGuide": "Take moments about A in the first limiting case and about C in the second, then eliminate P and W to find d/a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=12", "qpPages": [ 16, 19 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q06a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show the deceleration of a package moving up a rough inclined plane.", "answerGuide": "Resolve the weight and limiting friction along the plane, use the normal reaction, and apply Newton’s second law.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q06b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the projection speed needed to travel 1.5 m before stopping.", "answerGuide": "Use the deceleration from part (a) in a constant-acceleration equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q06c", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the smallest force that holds the package at rest at B.", "answerGuide": "At limiting equilibrium resolve along and perpendicular to the slope and use the correct direction of friction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=13", "qpPages": [ 20, 23 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Questions 3 and 6 as the greatest challenge and specifically reports that the friction direction in Q6(c) was a key error.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme01-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q07a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 2, "topicIds": [ "kinematics", "vectors" ], "primaryTopicId": "kinematics", "summary": "Find the magnitude of boat A’s constant acceleration.", "answerGuide": "Read the acceleration vector from the coefficient of t and take its magnitude.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q07b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "kinematics", "vectors" ], "primaryTopicId": "kinematics", "summary": "Find boat B’s acceleration vector from its velocities at t=2 and t=5.", "answerGuide": "Divide the change in velocity by the elapsed time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q07c", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "kinematics", "vectors" ], "primaryTopicId": "kinematics", "summary": "Find boat B’s velocity at noon.", "answerGuide": "Use constant acceleration to work back from a given velocity to t=0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q07d", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 4, "topicIds": [ "kinematics", "vectors" ], "primaryTopicId": "kinematics", "summary": "Find when the two boats have the same speed.", "answerGuide": "Form each velocity vector as a function of time, equate squared magnitudes, and solve for the positive time.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q07e", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 3, "topicIds": [ "kinematics", "vectors" ], "primaryTopicId": "kinematics", "summary": "Show the quadratic condition for the boats to move in the same direction.", "answerGuide": "Set the velocity vectors parallel, equate the component ratios, and simplify to the required quadratic.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=15", "qpPages": [ 24, 27 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q08a", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 2, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Write an equation of motion for P in the two-mass pulley system.", "answerGuide": "Resolve the forces on P in the direction of its acceleration.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q08b", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find k from the acceleration of the two-mass system.", "answerGuide": "Write an equation of motion for the other mass as well and solve the pair.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q08c", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the force exerted on the pulley by the string.", "answerGuide": "Use the equation of motion to find the tension, then combine the two vertical string forces.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q08d", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(d)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the impulse on Q when it rebounds from the ground.", "answerGuide": "Find its speed just before impact from its acceleration and fall distance, then use change in momentum with the rebound direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME01-01-q08e", "paperId": "ial-2025-6-WME01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME01", "unit": "Mechanics 1 (M1)", "year": 2025, "session": "May/June", "paperCode": "WME01/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(e)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the total time until P first comes to rest.", "answerGuide": "Find the time before Q hits the ground, then the post-rebound motion time until P’s velocity becomes zero and add the intervals.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_QP.pdf#page=28", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M1_MS.pdf#page=17", "qpPages": [ 28, 32 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q01a", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Show that the acceleration constant is k=4.", "answerGuide": "Integrate acceleration to obtain velocity, use the initial rest condition, then substitute t=1.5 and speed 13.5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q01b", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "kinematics", "integration" ], "primaryTopicId": "kinematics", "summary": "Show the stated displacement function.", "answerGuide": "Integrate the velocity from part (a) and use s(0)=0.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q01c", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the total distance travelled before P returns to O.", "answerGuide": "Solve s(t)=0 for the positive return time, find when v changes sign, then add the outward and return distances.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q02", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 7, "topicIds": [ "coordinate-geometry", "energy-momentum" ], "primaryTopicId": "coordinate-geometry", "summary": "Find the speed immediately after the impulse.", "answerGuide": "Differentiate the position vector for the pre-impulse velocity at t=4, apply J=m(v−u) componentwise, and take the magnitude of the new velocity.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=9", "qpPages": [ 4, 5 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q03a", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the centre-of-mass distance of the wire framework from AB.", "answerGuide": "Take moments about AB using each component’s mass and centroid distance, including both semicircular arcs and straight connectors.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q03b", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the exact tan θ when the framework is suspended from A.", "answerGuide": "Take moments about A in the hanging equilibrium, using the framework centre of mass and the right-angle geometry at E.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=10", "qpPages": [ 6, 9 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Questions 3 and 6 as the greatest challenge; in Q3(b), many candidates confused tan(45°−OAG) with tan θ or used an incorrect compound-angle relation.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q04a", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the constant speed of the car climbing the slope with its engine running.", "answerGuide": "Set engine power equal to the resisting force times speed, including weight’s component down the slope.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q04b", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the car’s acceleration at 10 m s⁻¹.", "answerGuide": "Use P=Fv to determine the engine force at that speed, then apply Newton’s second law along the slope.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q04c", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 4, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find how far the car travels after the engine is switched off.", "answerGuide": "Apply work-energy: initial kinetic energy is spent against gravity and the constant 200 N resistance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=12", "qpPages": [ 10, 13 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q05a", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Show the speed of Q after the collision.", "answerGuide": "Apply conservation of linear momentum and Newton’s law of restitution, taking the initial directions into account.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q05b", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 5, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find k in the kinetic-energy loss expression.", "answerGuide": "Write the initial and final kinetic energies using the post-collision speeds from part (a), then factor out (1−e²)mu².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q05c", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Use the impulse on P to find e.", "answerGuide": "Use impulse=change in momentum for either particle and equate its magnitude to the given expression.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q05d", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 3, "topicIds": [ "energy-momentum", "mechanics" ], "primaryTopicId": "energy-momentum", "summary": "Find the wall’s coefficient of restitution f.", "answerGuide": "Relate Q’s speed after rebounding from the wall to its speed before impact, then equate it to P’s speed.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=13", "qpPages": [ 14, 17 ], "msPages": [ 13, 14 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q06a", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the rail’s normal reaction S.", "answerGuide": "Take moments about A for the uniform rod, using the contact point distance 1.5a and the centre-of-mass distance a.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q06b", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Show the vertical force component V at A.", "answerGuide": "Use vertical force balance with the rod’s weight and the vertical component of the reaction at C.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q06c", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 6, "topicIds": [ "mechanics", "trigonometry" ], "primaryTopicId": "mechanics", "summary": "Derive the required quadratic in tan θ for limiting equilibrium.", "answerGuide": "Use the limiting friction relation F=μR, horizontal force balance, the expressions from (a) and (b), then divide by cos²θ and collect powers of tan θ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=15", "qpPages": [ 18, 21 ], "msPages": [ 15, 16 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson identifies Questions 3 and 6 as the greatest challenge. In Q6(c), many solutions omitted forces or could not complete the required trigonometric identity.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme02-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q07a", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 4, "topicIds": [ "kinematics", "energy-momentum" ], "primaryTopicId": "kinematics", "summary": "Show that the vertical velocity magnitude is V=6 when the stone is 11 m above ground.", "answerGuide": "Apply conservation of mechanical energy between launch and the point at height 11 m, and use the component ratio in the question.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=17", "qpPages": [ 22, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME02-01-q07b", "paperId": "ial-2025-6-WME02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME02", "unit": "Mechanics 2 (M2)", "year": 2025, "session": "May/June", "paperCode": "WME02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 6, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find the height where the projectile hits the plane.", "answerGuide": "Use the velocity components and the perpendicular-impact condition to determine the launch angle and time, then calculate vertical displacement.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M2_MS.pdf#page=17", "qpPages": [ 22, 24 ], "msPages": [ 17, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q01a", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 3, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the tension in the vertical elastic string.", "answerGuide": "Take moments about the hinge for the rod in equilibrium.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q01b", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 4, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the elastic string’s natural length L.", "answerGuide": "Use Hooke’s law with the tension from part (a) and the stated string length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=7", "qpPages": [ 2, 3 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q02", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": null, "marks": 7, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Find the table’s force on a particle moving in a horizontal circle.", "answerGuide": "Resolve vertically to relate tension and normal reaction, then resolve horizontally for centripetal force.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=8", "qpPages": [ 4, 5 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q03a", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 4, "topicIds": [ "kinematics" ], "primaryTopicId": "kinematics", "summary": "Find v in terms of x for the particle with acceleration 4x−2.", "answerGuide": "Use a=v dv/dx, integrate with respect to x, and apply the initial condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q03b", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 5, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Use a differential equation to connect velocity with displacement and time.", "answerGuide": "Use v = ds/dt to change the independent variable, integrate the resulting relation, and apply the given condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=9", "qpPages": [ 6, 7 ], "msPages": [ 9, 10 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls Q3 the first major challenge; many candidates could not progress in part (b) without simplifying the result from part (a).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q04a", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Show the change in elastic energy as the particle moves from C to D.", "answerGuide": "Find the string lengths and extensions at C and D, then compare their elastic potential energies.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q04b", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 5, "topicIds": [ "mechanics", "energy-momentum" ], "primaryTopicId": "mechanics", "summary": "Find the particle’s speed at D with air resistance included.", "answerGuide": "Apply work–energy with the changes in kinetic energy, gravitational potential energy, elastic energy, and work against drag.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=8", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=11", "qpPages": [ 8, 11 ], "msPages": [ 11, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q05a", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Show the speed of a particle when it leaves the sphere.", "answerGuide": "Use conservation of energy between A and B.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q05b", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Show the stated value of tan α at the leaving point.", "answerGuide": "Use the radial equation of motion with zero reaction at separation and substitute the speed from part (a).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q05c", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 7, "topicIds": [ "kinematics", "mechanics" ], "primaryTopicId": "kinematics", "summary": "Analyze projectile motion after a particle leaves a sphere.", "answerGuide": "Resolve the launch velocity into components, use horizontal and vertical motion equations, and apply the impact condition.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=13", "qpPages": [ 12, 15 ], "msPages": [ 13, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson says the circular-to-projectile transition in Q5(c) challenged many candidates and required a multi-step velocity-component argument.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q06a", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 5, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Find the centre of mass of a uniform solid hemisphere.", "answerGuide": "Integrate the circular cross-sectional area moments and divide by the hemisphere volume.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q06b", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 4, "topicIds": [ "mechanics", "geometry" ], "primaryTopicId": "mechanics", "summary": "Show the centre-of-mass distance for the solid after a smaller hemisphere is removed.", "answerGuide": "Use the two hemisphere masses and their centre positions in a composite-body moment equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q06c", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 5, "topicIds": [ "mechanics" ], "primaryTopicId": "mechanics", "summary": "Use centre-of-mass geometry to analyze equilibrium of a solid.", "answerGuide": "Locate the centre of mass from the component masses, then take moments about the relevant support or pivot.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=16", "qpPages": [ 16, 19 ], "msPages": [ 16, 18 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson reports substantially fewer successful solutions and many blank or zero-mark responses for the exact equilibrium part Q6(c).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q07a", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 7, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Show that the particle executes SHM with the stated centre and period.", "answerGuide": "Write the equation of motion at a general displacement using both elastic tensions and show acceleration is proportional to displacement toward equilibrium.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=19", "qpPages": [ 20, 24 ], "msPages": [ 19, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q07b", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 3, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find the maximum speed in the SHM.", "answerGuide": "Use the amplitude and angular frequency in v_max=ωA.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=19", "qpPages": [ 20, 24 ], "msPages": [ 19, 22 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WME03-01-q07c", "paperId": "ial-2025-6-WME03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WME03", "unit": "Mechanics 3 (M3)", "year": 2025, "session": "May/June", "paperCode": "WME03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 6, "topicIds": [ "mechanics", "kinematics" ], "primaryTopicId": "mechanics", "summary": "Find an SHM time interval subject to a speed bound.", "answerGuide": "Use the SHM displacement and velocity relations, turn the speed restriction into a displacement condition, then find the allowed interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_M3_MS.pdf#page=19", "qpPages": [ 20, 24 ], "msPages": [ 19, 22 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson describes Q7 as a significant SHM challenge and notes that many candidates struggled to complete the speed-restriction time interval in part (c).", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wme03-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q01a", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the cumulative distribution value F(3.5).", "answerGuide": "Add the probabilities for outcomes at or below 3.5.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q01b", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that 3X−3 is less than X+2.", "answerGuide": "Solve the inequality for X, then add the probabilities of the qualifying values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q01c", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find E(X).", "answerGuide": "Multiply each outcome by its probability and sum.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q01d", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Show the variance of X.", "answerGuide": "Find E(X²) and use Var(X)=E(X²)−[E(X)]², showing the calculation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q01e", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find Var(5−2X).", "answerGuide": "Use Var(a+bX)=b²Var(X).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q02a", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the mean b for Class B.", "answerGuide": "Use the combined mean equation for the 54 students and solve for b.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q02b", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the standard deviation for all 54 students.", "answerGuide": "Combine the supplied sums and square sums to calculate the overall variance, then take its square root.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q02c", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain the effects of adding 2 to every Class B mark on the variance, combined mean and combined standard deviation.", "answerGuide": "Treat each of the three requested effects separately, giving the change or no change and a reason.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03a", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Give another reason for using statistical models.", "answerGuide": "State a valid benefit such as simplifying the real world or showing relationships.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03b", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the coded gas-consumption sum of squares Sww.", "answerGuide": "Use the coded totals to calculate Sww and show the numerical substitution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03c", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find Std and Sdd from the coded summaries.", "answerGuide": "Undo the linear coding for w, then use the scale factors for the covariance and variance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03d", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(d)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the product-moment correlation between temperature and gas consumption.", "answerGuide": "Calculate r from the corrected covariance and sums of squares.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03e", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(e)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Interpret the correlation in context.", "answerGuide": "State the direction and strength of the association in terms of temperature and winter gas use.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03f", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(f)", "marks": 3, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show the regression line of w on t.", "answerGuide": "Use the regression gradient and intercept formulas with the given summary values.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03g", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(g)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Write the regression line of d on t.", "answerGuide": "Convert the coded regression equation using w=11.5d.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q03h", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(h)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Estimate consumption at 2°C and interpret the effect of a 1°C rise.", "answerGuide": "Substitute t=2 for the estimate, then interpret the gradient with units.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q04a", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(C∩D) from the conditional probability.", "answerGuide": "Use P(C|D)=P(C∩D)/P(D).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q04b", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 2, "topicIds": [ "probability" ], "primaryTopicId": "probability", "summary": "Find P(C) using the union probability.", "answerGuide": "Rearrange P(C∪D)=P(C)+P(D)−P(C∩D).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=9", "qpPages": [ 10, 11 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q05a", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 3, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Find the height of the second tallest histogram bar.", "answerGuide": "Compare frequency densities, use the given tallest-bar height to set the scale, and calculate the second tallest height.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q05b", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Estimate the number of cars travelling between 35 and 57 km/h.", "answerGuide": "Interpolate within the relevant grouped intervals and combine the estimated frequencies.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q05c", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Estimate the median speed by linear interpolation.", "answerGuide": "Locate the median position and interpolate within its class interval.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q05d", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(d)", "marks": 4, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Estimate the upper quartile and interquartile range.", "answerGuide": "Interpolate Q3 in its class and subtract the given lower quartile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q05e", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(e)", "marks": 2, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Describe the data’s skewness with a reason.", "answerGuide": "Compare the median and quartiles or use the mean–median relationship to support the direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q05f", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(f)", "marks": 1, "topicIds": [ "data-representation", "data-analysis" ], "primaryTopicId": "data-representation", "summary": "Comment on whether a normal model is suitable.", "answerGuide": "Refer to the skewness result and whether it is consistent with a symmetric normal distribution.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=12", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=10", "qpPages": [ 12, 15 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q06ai", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(i)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(X>508) for a small flour bag.", "answerGuide": "Standardize 508 using the mean and standard deviation, then find the upper tail.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_QP.pdf#page=16", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S1_MS.pdf#page=11", "qpPages": [ 16, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST01-01-q06aii", "paperId": "ial-2025-6-WST01-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST01", "unit": "Statistics 1 (S1)", "year": 2025, "session": "May/June", "paperCode": "WST01/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)(ii)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find P(4960.6).", "answerGuide": "Use one minus the CDF value at 0.6.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q01c", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Form the median equation and find the median.", "answerGuide": "Set F(m)=0.5, form the required polynomial, then select the root in the support.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q01d", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(d)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the mode using calculus.", "answerGuide": "Differentiate the density on its support, set the derivative to zero, and check the valid point.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q01e", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(e)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Determine the distribution’s skewness.", "answerGuide": "Compare the mean with the median and mode, then state the skew direction.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=6", "qpPages": [ 2, 5 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q02a", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State a Poisson model for customers in a half-hour.", "answerGuide": "Halve the hourly mean rate to set the Poisson parameter.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q02b", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State an assumption for the Poisson model.", "answerGuide": "State that arrivals occur independently or singly at a constant rate, in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q02c", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(c)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State hypotheses for a two-sided change in arrival rate.", "answerGuide": "Express H0 and H1 using the half-hour mean rate.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q02d", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(d)", "marks": 3, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the critical region at the 3% level.", "answerGuide": "Use the null Poisson distribution to find suitable lower and upper tail cutoffs.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q02e", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(e)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the test’s actual significance level.", "answerGuide": "Add the two tail probabilities for the critical region.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q02f", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(f)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Interpret the observed 16 customers using the critical region.", "answerGuide": "Decide whether 16 is in the rejection region and state the conclusion in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=7", "qpPages": [ 6, 9 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q03a", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find n and p from the binomial mean and variance.", "answerGuide": "Use np=3 and np(1−p)=2.55, then solve.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q03b", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that a medium bag has more green sweets than its expected count.", "answerGuide": "Use the binomial model and sum the probabilities above the mean.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q03c", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(c)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Use and justify a Poisson approximation for the yellow sweets.", "answerGuide": "Set λ=np, calculate the upper tail, and explain why n is large and p is small.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=8", "qpPages": [ 10, 13 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q04a", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the sampling distribution of points from three beads.", "answerGuide": "List possible totals and calculate each probability using the binomial count of black beads.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q04b", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the smallest n for which at least one set scores 9 with probability above 0.95.", "answerGuide": "Find the success probability from part (a), use the complement probability for n sets, and solve the inequality.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=9", "qpPages": [ 14, 17 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q05a", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 5, "topicIds": [ "distributions", "data-analysis" ], "primaryTopicId": "distributions", "summary": "Show a parameter for a two-piece probability density function.", "answerGuide": "Use the total area under both pieces to form the normalization equation, then solve and show the parameter working.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson notes that Q5(a) caused candidates difficulty: the full two-piece density had to be integrated and the algebra shown; the report warns against an invalid shortcut.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst02-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q05b", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find E(X) by algebraic integration.", "answerGuide": "Integrate x f(x) over both parts of the support and add the results.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q05c", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find Var(X) using the given mean.", "answerGuide": "Find E(X²) by integrating x²f(x), then subtract [E(X)]².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=10", "qpPages": [ 18, 21 ], "msPages": [ 10, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q06a", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the endpoints a and b of the uniform interval.", "answerGuide": "Use the given interval probability and the uniform mean to form two equations.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=12", "qpPages": [ 22, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q06b", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find c from E(cX+1)=3.", "answerGuide": "Use linearity of expectation and E(X) from the question.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=12", "qpPages": [ 22, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q06c", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find Var(X) exactly.", "answerGuide": "Use the variance formula for a continuous uniform distribution on [a,b].", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=12", "qpPages": [ 22, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q06d", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find E(X²) exactly.", "answerGuide": "Use Var(X)=E(X²)−[E(X)]².", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=12", "qpPages": [ 22, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q06e", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(e)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Evaluate the requested probability for the uniform variable.", "answerGuide": "Write the event interval and divide its length inside [a,b] by the full interval length.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=22", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=12", "qpPages": [ 22, 23 ], "msPages": [ 12, 12 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q07a", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State a binomial model for insurance purchases in a sample of 40.", "answerGuide": "Identify the number of trials and the historical purchase probability.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q07b", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 1, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "State an assumption for the binomial model.", "answerGuide": "State constant purchase probability or independent customer decisions.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q07c", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the largest r for which fewer than r customers has probability below 0.05.", "answerGuide": "Use the binomial CDF and compare consecutive cumulative probabilities.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q07d", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(d)", "marks": 5, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find t using the stated normal approximation.", "answerGuide": "Approximate the binomial distribution by a normal model with continuity correction and solve for t.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST02-01-q07e", "paperId": "ial-2025-6-WST02-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST02", "unit": "Statistics 2 (S2)", "year": 2025, "session": "May/June", "paperCode": "WST02/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(e)", "marks": 5, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Complete a binomial hypothesis test and interpret its result.", "answerGuide": "Set the null distribution and critical region, calculate the required tail probability, then state the conclusion in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S2_MS.pdf#page=13", "qpPages": [ 24, 28 ], "msPages": [ 13, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q01a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(a)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Give a reason for systematic sampling in this club.", "answerGuide": "Explain how systematic sampling can reduce bias or create a representative selection.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q01b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(b)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Find the interval between selected members.", "answerGuide": "Divide the population size by the required sample size.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q01c", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 1, "questionPart": "1(c)", "marks": 1, "topicIds": [ "data-analysis", "probability" ], "primaryTopicId": "data-analysis", "summary": "Find the probability of selecting two adjacent listed members.", "answerGuide": "Use the fixed sampling interval and the specified starting member.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=2", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=6", "qpPages": [ 2, 3 ], "msPages": [ 6, 6 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q02a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(a)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Calculate Spearman’s rank correlation coefficient for the judges.", "answerGuide": "Find the squared rank differences and substitute into the Spearman formula.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q02b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 2, "questionPart": "2(b)", "marks": 4, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Test for positive rank correlation at the 1% level.", "answerGuide": "State hypotheses, compare the coefficient with the critical value, and conclude in context.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=4", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=7", "qpPages": [ 4, 5 ], "msPages": [ 7, 7 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q03a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(a)", "marks": 2, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Find the expected frequency for diesel cars with an early engine problem.", "answerGuide": "Use row total times column total divided by the grand total.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q03b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 3, "questionPart": "3(b)", "marks": 7, "topicIds": [ "statistical-testing", "probability" ], "primaryTopicId": "statistical-testing", "summary": "Test for an association between engine type and problem timing.", "answerGuide": "Use the supplied cell contributions with the observed contribution, degrees of freedom and critical value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=6", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=8", "qpPages": [ 6, 9 ], "msPages": [ 8, 8 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q04a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(a)", "marks": 3, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Find unbiased estimates of the red-squirrel mean and variance.", "answerGuide": "Use the sample total and corrected sum of squares to calculate the sample mean and unbiased variance.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q04b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(b)", "marks": 7, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "Test whether the grey-squirrel mean exceeds the red mean by more than 200 g.", "answerGuide": "Set hypotheses for the difference, calculate the test statistic and compare with the one-tailed critical value.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q04c", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 4, "questionPart": "4(c)", "marks": 1, "topicIds": [ "data-analysis", "statistical-testing" ], "primaryTopicId": "data-analysis", "summary": "State an assumption for the test in part (b).", "answerGuide": "Give the population or sampling assumption required for the large-sample test.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=10", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=9", "qpPages": [ 10, 13 ], "msPages": [ 9, 9 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q05a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(a)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Estimate the proportion of male puppies in the litters.", "answerGuide": "Calculate the sample proportion from the total number of male puppies divided by the number of puppies.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q05b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(b)", "marks": 2, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Find the missing expected frequency r.", "answerGuide": "Use the total expected frequency of 250 and subtract the other listed expected frequencies.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q05c", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 5, "questionPart": "5(c)", "marks": 7, "topicIds": [ "distributions", "statistical-testing" ], "primaryTopicId": "distributions", "summary": "Test whether a binomial distribution is a suitable model.", "answerGuide": "Complete the chi-squared goodness-of-fit test, combining classes where required and using the adjusted degrees of freedom.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=14", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=10", "qpPages": [ 14, 17 ], "msPages": [ 10, 10 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q06a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(a)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Explain why the stated expression is a statistic.", "answerGuide": "Point out that it depends only on observed sample values and has no unknown population parameters.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q06b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(b)", "marks": 2, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Show that R is a biased estimator of μ.", "answerGuide": "Take expectations of the weighted sample observations and compare E(R) with μ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q06c", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(c)", "marks": 1, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the bias of R.", "answerGuide": "Subtract μ from E(R).", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q06d", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 6, "questionPart": "6(d)", "marks": 4, "topicIds": [ "data-analysis" ], "primaryTopicId": "data-analysis", "summary": "Find the coefficients that make both estimators unbiased.", "answerGuide": "Set the expectation equations equal to μ, form two simultaneous equations, and solve for the coefficients.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=18", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=11", "qpPages": [ 18, 19 ], "msPages": [ 11, 11 ], "hardestPart": null, "studyPriority": 1, "difficultyEvidence": "Pearson calls Q6 the most challenging question overall and specifically says part (d) caused difficulty forming the two unbiasedness equations.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20250814.pdf", "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q07a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(a)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Show the standard error relation for the sample mean.", "answerGuide": "Convert each given probability to a z-score using σ/√n, form two equations, and eliminate μ.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q07b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(b)", "marks": 2, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find μ from the result in part (a).", "answerGuide": "Substitute the standard error relation into either probability equation.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q07c", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 7, "questionPart": "7(c)", "marks": 3, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that the sample mean exceeds 59.", "answerGuide": "Standardize 59 using the mean and standard error found earlier, then find the upper tail.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=20", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=12", "qpPages": [ 20, 23 ], "msPages": [ 12, 13 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q08a", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(a)", "marks": 4, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that three sand bags total more than 54.1 kg.", "answerGuide": "Find the mean and variance of the sum, standardize, and calculate the upper tail.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q08b", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(b)", "marks": 5, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Find the probability that two cement bags differ in weight by more than 0.02 kg.", "answerGuide": "Model the difference of independent normal variables, account for both tails, and standardize.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": null, "difficultyEvidence": null, "difficultySourceUrl": null, "tagConfidence": "reviewed" }, { "id": "ial-2025-6-WST03-01-q08c", "paperId": "ial-2025-6-WST03-01", "board": "Edexcel IAL", "qualification": "Pearson Edexcel International A Level Mathematics", "syllabus": "WST03", "unit": "Statistics 3 (S3)", "year": 2025, "session": "May/June", "paperCode": "WST03/01", "variant": "Standard paper", "questionNumber": 8, "questionPart": "8(c)", "marks": 6, "topicIds": [ "distributions", "probability" ], "primaryTopicId": "distributions", "summary": "Work with a linear combination of normally distributed variables.", "answerGuide": "Find the combination’s mean and variance, then standardize the required probability or quantile.", "qpUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_QP.pdf#page=24", "msUrl": "https://archive.paperlords.org/library/IAL/Maths/MayJune%202025/IAL_MATHS_2025_MayJune_S3_MS.pdf#page=14", "qpPages": [ 24, 28 ], "msPages": [ 14, 15 ], "hardestPart": null, "studyPriority": 2, "difficultyEvidence": "Pearson identifies Q8(c) as challenging and a strong discriminator; most full-mark solutions came from the strongest candidates.", "difficultySourceUrl": "https://qualifications.pearson.com/content/dam/pdf/International-Advanced-Level/Mathematics/2018/Exam-materials/wst03-01-pef-20250814.pdf", "tagConfidence": "reviewed" } ] }